163 Workbook 5.0 Revised June 2019

Content Pg Ch 21: Electric forces 3 Ch 22: Electric fields 11 Ch 23: Gauss’s Law 29 Ch 24: Electric Potential & Electric Potential Energy ( V & U) 55 Ch 25: Capacitors 77 Ch 26: Resistivity and Current 93 Ch 27: Resistor Circuits 105 Still Ch 27: KVL 113 Still Ch 27: RC Transients 116 Ch 28: Magnetism affecting moving charges 121 Ch 29: Moving charges as the source of magnetism 133 Ch 30: Induced EMF 151 Still Ch 30: RL Transients 163 Still Ch 30: LC Oscillators 165 Ch 31: AC Circuits 167 Still Ch 31: Filter Circuits 173 Ch 32: Review into Types of Magnetism & Maxwell’s Eqt’ns 181 Ch 33: EM Waves 189 Equation Sheet Back Cover 2

Chapter 21: Electric Forces Learn how to correctly use the equation for Coulomb force . • Distinguish between charge and magnitude of charge . • Distinguish between force and magnitude of force . • Note how symmetry can simplify computation. • Use appropriate prefixes and conversions factors in numerical data. • Plot force (or force magnitude ) for a range of different cases.

Coulomb Force VECTOR 1 Coulomb Force VECTOR 2 Coulomb Force MAGNITUDE

= The hat changes to a vector & the = = distance squared changes to distance cubed . means the force of charge 1 on charge 2 means the magnitude of the force between charges 1 and 2 means the vector from the center of 1 to the center of 2 means the distance (no direction) from 1 to 2 means the direction only from 1 to 2…this is a unit vector means charge 1…you should include the sign of the charge means magnitude of charge 1…you should NOT include the sign of the charge

Coulomb constant Permittivity of free space Relationship

N ∙ m C = = 8.99 × 10 = 8.85 × 10 = = C N ∙ m Proton mass Electron mass MAGNITUDE of electron charge

= 1.673 × 10 kg = 9.109 × 10 kg = +1.602 × 10 C

While these are just constants, I do have a few comments. • I think of as 9E9 (like “ninety nine”…while cheesy, maybe it helps that number stick) • In some online homework services the epsilon can be a little tricky. Epsilon looks like his ( ) or this ( ). This ( ) is “capital script E” while this ( ) is the “Euler constant”. Notice the very subtle differences. ℰ ℇ When doing homework online, you may want to mouse over the epsilon before clicking on it to ensure you are actually using epsilon. Don’t blame me…I voted for ZOG! • Obviously has another meaning relating to natural logarithms. The context of a problem statement usually makes clear which is to be used in a problem. • It is worth restating that in E & M problems is the magnitude of the charge on an electron. o The charge on a proton is . + o The charge on an electron is . – o The magnitude of electron charge is . + o Remember how is the magnitude of the acceleration due to gravity? The acceleration = +9.8 due to gravity in freefall was . = − 4

21.1 Determine the units of the Coulomb constant by looking at . =

21.2 Three particles are shown at right. Particles 1 & 2 are 5.00 m apart while 5.00 m particles 1 & 3 are 12.00 m apart. 1 2 a) Determine the displacement vector that points from 2 to 3.

b) Determine the unit vector that points from 2 to 3. c) Determine the unit vector that points from 1 to 2. 12 .00 m d) Determine the unit vector that points from 1 to 3. e) Determine the unit vector that points from 3 to 2. 3

21.3 Assume , , and . (cm) = +1.00C = −2.00C = −3.00C For now, assume only electrical forces act on the charges. a) Without using any math, sketch the estimated directions of

and in the figure. Try to estimate which force arrow should be larger (or are they about the same size)? 4 b) Sketch the estimated direction of the NET force ON ? Hint: tail-to-tip, tail-to-tip.

c) Determine the net force (magnitude and direction) on q3. 3 Note: unless otherwise specified, when asked for a direction you are expected to determine a numerical value for an angle and include a small sketch labeling your angle for clarity. 2

(cm)

1 2 3 5

PhET Activity: Use the Electric Field Sensor to show direction and relative size of Force 21.4a Start with a single positive charge in the middle of the screen. Place an electric field sensor 1 major unit to the right of the first positive charge. a) How much should the force change (on the sensor) if you move the sensor so it sits 2 major units away? Does the force increase or decrease? By what factor (2× bigger, 2× smaller, or something else)? b) With the sensor sitting 2 major units from the initial positive charge, how far must you move the sensor to cut the force in half?

21.4b Place two positive charges separated vertically by about 4 units (2 m). a) Predict the direction of the net force on a positive charge placed halfway between the two masses. b) Predict the direction of the net force on a positive charge if it is placed lightly left or right of the midpoint. c) As you slide it left or right, where do you put the positive charge to maximize the net force on it? d) How do your previous results change if the top charge is negative instead of positive?

21.4c Create a square with positive charges at 3 of the corners. I used sides with length of 4 major units (2 m). Predict the direction of the net force if positive (or negative) charge is placed: a) At the remaining corner. b) At the center of the square. c) Which of the above two arrangements produces a larger force on the 4 th charge? d) How should your results for the previous parts change if the top left corner is negative instead of positive?

21.4d Create a square with positive charges at each corner. I used sides with a length of 4 major units (2 m). Predict the direction of the net force on a fifth charge located: a) At the center of the square. b) At the middle of the left side of the square. c) At the middle of the bottom of the square. d) One major unit (0.5 m) from one of the corners. 21.4e Create a square with positive charges on the top two corners and negative charges on the bottom two corners. I used sides with a length of 4 major units (2 m). Predict the direction of the net force on a fifth charge located: a) At the center of the square. b) At the middle of the left side of the square. - c) At the middle of the bottom of the square. d) How would things change if the left two charges were negative and the right two were positive?

21.4f An equilateral triangle may be made by placing two positive charges 4 units (2 m) apart with a 2 third positive charge halfway between and up units (1.5 m). Create such a triangle. Predict 3.6 3.5 the direction of the net force on a positive charge placed a) At the center of the triangle (halfway between and 1.15 units 0.6 m up). b) At the midpoint of a side of the triangle. 1 3 c) How would your above answers change if the top charge was negative?

21.4g Place two charges separated horizontally by about 9 units (4.5 m according to the simulation’s scale). Place an extra charge on the left to double the charge there. Now imagine you are going to place an

additional positive charge (the sensor charge) somewhere between the two. 2 a) Predict the direction of the force if the sensor charge is placed midway between and . 2 b) At what location is the sensor charge in equilibrium? Should it be (3 units from the right end, which is also 6 units from the right end) or somewhere else? Is the equilibrium stable or unstable? c) How should your answers change if the bottom charge is negative? 6

21.5 Welcome to the land of pain Two point charges are located at fixed positions on the -axis (see upper figure at right). Assume . One charge is distance above the origin while the other lies = = + distance below the origin. A third point charge, , lies at position on the -axis. This charge is also positive with magnitude . To be clear, we will want to consider both = positive and negative values of for this problem. 2 a) Think before computing. Which way should the net Coulomb force on (due to the other two charges) point? Be sure to consider the direction of the net force for , , and . 0 0 = 0 b) Determine an algebraic expression for the net Coulomb force exerted on by the other two charges. Check the units of your final answer. Also, compare your result to what you expected from part a. c) What value of x gives a maximum force? What is the maximum value of the force? d) Assuming , use the binomial expansion to determine an approximate equation for the force. The answer should be of the form . = ∙ e) For , use the binomial expansion to determine an approximation for the force. f) Plot the Coulomb force vs assuming and . Use values of ranging = 1.0m = +1.6 × 10 C from up to in increments. Assume positive values of force indicate a force to −5.0m +5.0m 0.1m the right while negative values of force indicate a force to the left. Think after computing. i. Do the signs on your plot match your intuition from part a for the directions of the force? ii. Does the force maximum at the appropriate position? iii. Can you get a numerical value for the slope close to the origin and compare it to part d. iv. You should also be able to plot the function from part e on the same graph. For large values of we expect the function from part e and our function from part b to be nearly identical. g) Think about what would happen to charge if it was free to move along the -axis and was released from rest. Be sure to consider all three cases: , , and . 0 0 = 0 h) WHAT IF??? Suppose while . Think through the entire problem again. Some = = + = − parts might not change that much or at all. What would change? How would the plot change? How would your answers to part h change? Don’t redo the work, just think then check my work in the solutions. i) WHAT IF??? Suppose while . Think through the entire problem again. Some = = + = − parts might not change that much or at all. What would change? This one is a little trickier to discuss…part h is wildly different, no? Don’t re-compute; just think then check my work in the solutions.

Calculus based electricity and magnetism is no joke. You must understand concepts, figures, geometry, trig, calc, various notations, prefixes, and numerical computation. PLEASE, PLEASE, PLEASE take this class seriously and start practicing at least 2 hours a day for at least 5 days a week.

Remember this: you must get a C or better to transfer. Why not play it safe and over -study for the first month? This is a wiser plan of attack than waiting to see how you do on test one and then having your career goals put on hold for a year.

And, let’s not forget, you have other classes, too! By grinding it out 2 hours a day, 5 days a week you should still have plenty of time for other classes when all the exams end up on the same week (or day).

21.6 Pain can be one dimensional: Three point charges are arranged as shown in the figure. Two charges are fixed in place. These charges are and = − = . The charge can be located at any arbitrary position along the +2 = − horizontal axis (left or right of the origin). To be clear, is the horizontal position of and can be positive or negative while is a distance (always positive). = 2 = − a) Think before you compute. i. Which way will the force point for ? − ii. Which way will the force point for ? − iii. Which way will the force point for ? iv. Will there be 0, 1, 2, or 3 equilibrium points? Recall, an equilibrium point implies . = 0 v. Does your answer to part iv change your mind about part iii? What happens if ? b) Determine the net Coulomb force acting on for . c) Determine the net Coulomb force acting on for . − d) Determine the net Coulomb force acting on for . − e) At what location is in equilibrium? f) Assume . Plot vs for using increments of 0.1 nm. Note: the = 1.0m −5 +8 formula will not make sense if is exactly on top of one of the other two charges (when )! You = may have trouble making the plot. If you do have trouble plotting, delete the value of for . = g) Think after you compute. Does your plot agree with your reasoning on part a? h) WHAT IF??? Suppose instead of being a negative charge. Rethink the entire problem and = + consider what, if anything, would change. i) WHAT IF??? Suppose you kept but charge was changed to . Rethink the entire = − = −2 problem and consider what, if anything, would change. 2 + 21.7 Pain can be 3D: Four identical positive charges (with magnitude ) are arranged on a square of side that lies in the -plane as shown in the 1 figure. A fifth identical point charge is located at position from 5 0 0 the origin on the -axis. + a) Determine the Coulomb force of 1 on 5. It will look ugly . + 4 b) What direction is the net Coulomb force on 5? Hint: symmetry. 3 c) Determine the net force on 5. Isn’t symmetry awesome?!?!? d) Think: How would the problem change if the fifth charge was negative instead of positive? e) Think: How would things change if charges 1 and 2 were negative while charges 3, 4, and 5 were positive? Still assume all charges have equal magnitude . f) Think: How would things change if charges 1 and 3 were negative while charges 2, 4, and 5 were positive? Still assume all charges have equal magnitude . g) Now make some plots. Start by assuming all charges are positive with the same magnitude as an electron. Assume one side of the square is . = 1.0m

21.8 Consider a square of sides charges on three of the corners. The top left corner has charge while – the other two corners each have charge . + a) Determine the net force (magnitude and direction) on a fourth charge ( ) placed at the fourth + corner. b) Suppose, instead of placing the fourth charge at the fourth corner, the fourth charge is placed at the center of the square. Determine the net force, magnitude and direction. c) Is there an equilibrium position for the fourth charge? Is there more than one? Is there no equilibrium position? If yes, where is it (or them)? If no, explain heuristically why no equilibrium position exists. Note: if you are a clever person, on the last part you might think to re-align the figure such that the main diagonal of the square becomes the -axis. If you don’t see this clever trick, don’t worry; things are messier but should still be doable.

21.9 Three point charges are arranged as shown in the figure. The distance are noted as multiples of the arbitrary distance . You may assume has units of meters. a) Determine the magnitude of the net electric force exerted on the mass at the 3 12 origin. Answer with a 3 digit decimal times . b) Determine the direction of the net electric force exerted on the mass at the origin. Express your final answer in degrees with three sig figs. Include a sketch showing the angle to help clarify your answer.

Note: when someone tells you to determine a direction, you are usually expected to get a −5 number for an angle and draw a sketch unless otherwise specified.

−6

−2

21.10 Now consider an electron in a circular orbit around a proton (a crude model of the hydrogen atom). The proton and electron are typically separated by about . The 5.29 × 10 m mass of a proton is while the mass of an electron is . The 1.67 × 10 kg 9.11 × 10 kg magnitude of the charge (on both the proton and electron) is given as . = 1.6 × 10 C + a) Determine the magnitude of the force on the electron due to the proton using Coulomb’s law. b) Determine the gravitational force acting on the electron. Assuming it is near the earth’s surface you can just use . c) Take the ratio of the weight to the electrical force. Do you think it is reasonable to ignore gravitational forces for this type of problem? d) Determine the velocity of the electron in its orbit. You may assume constant speed for this crude model.

21.10½ Pretend for a moment that gravity does not exist. Suppose the moon is held in circular orbit about the earth using a Coulomb force instead. Assume the earth and moon each carry the same magnitude of charge uniformly distributed. Assume the earth has mass and radius . The moon has mass 5.97 × 10 kg 6.37 × 10 m 7.35 × and radius . The center-to-center distance from earth to moon is . Assume the 10 kg 1.7 × 10 m 3.8 × 10 m moon orbit requires 27.3 days. a) What magnitude of charge would be required to keep the moon in orbit using a Coulomb force? b) If we assume the moon is negative, how many electrons are added to the moon?

21.11 Two balls, each mass , hang from the ceiling using light strings. Each ball has static charge with unknown magnitude . As a result, the two balls repel from one another and hang at the angle shown in the picture. The distance of separation is and the length of each string is . Assume the size of each ball is negligible compared to the length of the string. Assume the origin of the coordinate system is the point where the two strings attach to the ceiling. To be clear, the figure shows the equilibrium state. a) Determine the angle between each string and the vertical in terms of and . This angle is called the half-angle as it is half of the angle between the two strings. b) Determine the magnitude of static charge on each ball. Hint: FBD. c) Are the balls both positive, both negative, or one of each? d) Suppose one ball had more charge than the other. Would the balls hang at different angles from the vertical or would one ball swing out more than the other?

21.12 Three equal negative charges are arranged at the corners of an equilateral triangle of − − side . A fourth charge is placed distance from the top charge. + a) Is there some value for such that the 4 th charge is in equilibrium? b) What value of is required for the fourth charge to be halfway between the bottom + two corners of the triangle? c) What is the net force on the 4 th charge when the 4 th charge is halfway between the − − bottom two corners? d) Write an equation for the net force on the fourth charge for arbitrary position inside the triangle . Do not worry about above the triangle ( or below the triangle ( ). 0

21.13 A positive charge is placed at the top corner of the equilateral triangle with negative + charges (each ) placed at the bottom corners of the triangle. A fourth charge, positive with + – unknown magnitude , is placed distance from the from the top. Assume . a) Without doing any math, do you think there should be an equilibrium point? If so, + would it be inside or outside of the triangle? Is there more than one possibility for an equilibrium point? b) GNARLY…AVOID: As a particularly brutal challenge, determine the equilibrium − − point (or points). The next part is much better practice for the test for typical undergrad courses…do that instead. c) Really good practice! You are told the net force magnitude on the 4 th charge is when it is located at the centroid. Determine an expression for in terms of , , , and .

21.14 This time three balls (each mass ) are hanging from strings. Each ball again has static charge with unknown magnitude . The balls separate from each other and form an equilateral triangle in the horizontal plane. The side of the equilateral triangle is and the length of each string is . Assume the size of each ball is negligible compared to the length of the string. Assume the origin of the coordinate system is the point where the three strings

attach to the ceiling. To be clear, the figure shows the equilibrium state. a) Determine the distance between the center of triangle and any one of the balls in terms of . b) Determine the distance between the center of the triangle and the origin in terms of and . c) At what angle, from the vertical axis, does each ball hang in terms of and ? d) Determine the magnitude of charge on each ball. e) If the balls each had slightly different charges, would the problem still be symmetric? Would the spheres still lie in an equilateral triangle?

21.15 Consider three charges placed as shown in the figure. a) Where should you place a fourth charge ( ) to put charge at the origin into = + 3 equilibrium? Give your answer as an coordinate where & are each written in terms of . th 5 b) Where should you place the 4 charge if it had the same magnitude but was negative? c) What if? Suppose we wanted the 4 th charge to be exactly distance from . What −2 new value of would need to be used? Would the angle also need to change?

21.16 Assume four charges lie on a circular arc of radius . A fifth charge is at the origin. All charges have the same magnitude but different signs. Use symmetry to determine the direction

of the net force on the charge at the origin for each case below. a) First assume all are positive. b) What if & are negative and the rest are positive? c) What if , & are negative and the rest are positive? d) What if & are negative and the rest are positive? e) What if , & are negative and the rest are positive? The point of this problem is to emphasize how symmetry considerations can dramatically

simplify problems.

21.17 A simple pendulum is made from a string of length and a small ball of mass and charge magnitude . At the point where the string attaches to the ceiling a Unknown second point charge (unknown magnitude and sign) is located. The system is free to charge swing normally; the upper point charge doesn’t add friction at the pivot or anything like that. In this problem the ball is always released from rest at an angle from the vertical. a) Will the presence of the charges change the speed of the ball at the bottom of the swing? Explain why (or why not). b) After doing several experiments, student learn the maximum tension without the unknown charge present is . When the unknown charge is present is reduced by 10%. Are the charges the same sign or opposite signs? c) Determine the unknown charge magnitude.

Chapter 22: Electric Fields What is a field (in mathematics)? Some maps show how fast the wind is blowing and in what direction (try a web search “wind map”). The “wind field map” is the name for a map showing wind speed & direction at many points. Here is the useful thing about a wind field map. 1) Different planes experience different amounts of wind force. 2) The wind force depends on two things: the plane (speed, orientation, geometry, etc) and the wind field (the wind speed/direction at every point on the map) 3) Using the wind field allows one to find the wind force on many different planes at many different spots. The wind field map is difficult to obtain (requires many measurements of wind speed and direction). Once done, however, the field map is easily used by pilots to find the wind force on their plane. Maybe someone could look at the map and decide where it is economically feasible to put up a wind generator. Someone else could use it to determine where trees are desired to abate seasonal winds. They might also use it to determine types of trees able to withstand the typical peak wind force.

Electric field versus electric force. The Coulomb force refers to the electric force on objects (this is like the wind force). The are many different positive and negative charges which can experience forces (this is like the type of plane). The electric field is only part of the force calculation to get the electric force (it is like the wind field map with direction and speed). Mathematically it is simple:

= ℎ ×

= One tricky aspect of electric fields is difficult to describe using an analogy to wind fields: Charges cause the electric field AND charges feel the electric field. We could rework our analogy like this… On a perfectly calm day with no wind, arrange a bunch of fans on a grid. Some of the fans are on high speed, some are on low speed, and the fans all point in different directions. Each fan causes wind that exerts a wind force on the other fans. Each fan experiences a wind force caused by all the other fans.

FOR POINT CHARGES ONLY: Electric Field VECTOR 1 Electric field VECTOR 2 Electric field MAGNITUDE

= = The hat changes to a vector & the distance = squared changes to distance cubed . means the electric field CAUSED BY charge 1 at some random point P means the magnitude of the electric field caused by charge means the vector from the center of to the point of interest P means the distance (no direction) from to P means the direction only from to P …this is a unit vector means magnitude of charge 1…you should NOT include the sign of the charge • Note: there need not be a charge at the point where the -field is determined!!! • The electric field points away from + and towards – . • It points in the direction a positive charge (placed at the point of interest) would move. • The electric field gets weaker with increasing distance from the charge. 12

22.1 Sketch the field of Sketch the field of the the proton all by itself electron all by itself

Sketch the field of the negative & positive charge near each other (called a dipole)

22.2 Assume and . Notice there is no = +1.00C = −2.00C (cm) charge at the origin. There is still an electric field at the origin caused

by q1 and q2. 4 a) Sketch the direction of and in the figure. b) Which direction will the net field point at the origin? Hint: tail-to-tip, tail-to-tip. 3 c) Determine the net field (magnitude and direction) at the origin. d) Now imagine a charge is placed at the origin. = −3.00C Determine the force on . Hint: use the definition of the 2 electric field . Compare your result to the 21.3 . =

(cm)

1 2 3

22.3 A positive charge and a negative charge are separated by distance Z 2a on the -axis as shown in the figure. Again, we call two opposite charges separated by a small distance an electric dipole. To be clear, let y us assume that both charges have equal magnitude q. The point P is distance x to the right of the origin. The point Z is distance y above the + origin. To be clear, there are no charges at the origin, point P, or point Z. Note . 2 a) Determine the net electric field (mag & direction) at the origin. b) Determine the net electric field (mag & direction) at the P. P c) Determine the net electric field (mag & direction) at the Q. x

22.4 Two point charges are each distance from the origin. The white charge is negative with unknown magnitude while the black charge is positive with magnitude . To be clear, is 2 2 also unknown but we do know the charges are opposite in sign with the black charge being twice as large. A sensor is used to determine the net electric field magnitude at the origin is .

a) Determine the direction of the field at the origin. Remember, when a teacher asks for the direction of a vector angle they typically expect you to give a number for an angle. In my classes, you should also include a sketch showing the approximate direction with the numerical value of the angle labeled in the figure. b) Determine the charge magnitude . 14

22.5 Look at the following charge configurations. Sketch the direction of the electric field at the center of each rectangle. If there is no field, put a dot. To be clear, consider each rectangle as an isolated system. That is, in reality the rectangles are not right next to each and as such the charges form one rectangle have no effect on the other rectangles.

q q q -q -2q q

-q -q -q q -q 2q

22.6 Four charges of equal magnitude are arranged on the corners of a square. Determine the direction of the electric field halfway between the two charges on the left side of the square (at the black dot). To be clear, there is no charge at the black dot.

22.7 Suppose you have many, many charges arranged on two plates as shown in the figure. It is known that for this situation the charges combine to make a net electric field given by where is constant. Said another way, between the plates there is a uniform electric field to the right with magnitude . A ball with charge magnitude and mass hanging on a string is placed in this known electric field. It is assumed is miniscule compared to all the charges on the plates. This allows one to ignore the effects of the ball polarizing the plates. The ball quickly reaches an equilibrium position as shown in the right figure below.

The ceiling The ceiling

a) Is the ball positively or negatively charged? b) Determine the angle in terms of , , , and . c) It is possible to adjust the direction of the electric field between the plates (by reorienting the plates). Is it possible to increase the deflection angle (assuming field strength & charge on the hanging ball remain unchanged)? d) Setting it up is good practice for all, seeing it all the way through is a challenge. Requires Calculus: What orientation gives maximum angle of deflection for a given charge? e) Suppose you put an electrically neutral ball between the plates (instead of a charged ball). Will the angle be the same, significantly less dramatic, or will there be no angle (ball hangs vertically)? How, if at all, is your answer to this question affected if the neutral ball is a conductor versus an insulator ? Explain.

22.8 You are given a set of charged parallel plates. Remember that for parallel plates the electric field (inside the plates) is constant. Assume that the magnitude of the electric field inside the plates is . For this problem, assume the gravitational force on the ball is at least a little larger than the electrical force. A ball with charge and mass is hanging on a string of length between the plates. The distance between the plates is . The ball is released from rest at the angle shown in the figure. You are told the ball then swings back and forth like a simple pendulum. At the lowest point in the swing the ball is observed to have speed . Tip: when not specified, assume . If you later find the charge is negative, usually the equations you derived still work when you put in a negative . a) Which direction should the field point between the plates? b) Determine the tension in the string at the lowest point in the swing? c) Now assume someone very gradually increases the voltage applied to the plates. What should happen to the period of oscillation of the pendulum? d) What changes if the plate polarity is flipped?

22.9 You are given a set of charged parallel plates. Assume the magnitude of the electric field inside the plates is . An ion with charge and mass is moving with speed to the right as it enters the plates. Assume gravitational forces are negligible for this problem. Assume fringing fields are negligible. If we assume fringing fields are negligible we assume outside the plates but the magnitude is constant everywhere inside the plates. This is a good approximation is plate spacing is small compared to plate length/width. a) How long will it take to reach the other side of the plates? Answer in terms of and . Assume the charge does make it through the plates without smashing into either plate. b) What is the direction of the acceleration while is between the plates? c) What is the magnitude of the acceleration while is between the plates? d) How far vertically will the charge fall as it passes through the plates? Answer in terms of the constants given. e) Assume the distance fallen is about 1/5 of the plate spacing. Sketch the path the charge will travel between the plates. In particular, is the path a straight line, an arc of a circle, parabolic, or something else?

How would you figure out the electric field due to a whole bunch of point charges in a line? This is called “ determining the electric field due to a continuous distribution of charge ”. Note: I have a more succinct summary on page 25 or so that will only make sense after we do a lot of problems… 1) Choose a coordinate system and stick with it for the entire problem. Note: if charge density is uniform, the best choice is often to put origin at the point of interest . 2) Select an arbitrary slice of the object to use as (a tiny bit of charge). 3) Define the distance from the origin of your coordinates to the arbitrary slice . 4) Now select a point on interest. This is the point at which you want to know the electric field. 5) Now assume that your is a tiny amount of charge. Write the tiny electric field ( ) caused by that little bit of charge . Note: will vary depending on how the object is sliced up. Use the formula for the electric field caused by your slice’s shape. Also, don’t forget the vector nature of (e.g. )…ignoring this can give incorrect answers upon integration! 6) Determine using the far right column in the table below. Your choice of slice and the geometry of the object will combine to determine . Substitute this result into . Type of correct symbol continuous for charge if uniform if non-uniform ? distribution density line charge = some function of , , or ℎ (rod or arc) (usually given by instructor) ℎ 2D surface (plates = some function of , , or ℎ & shells) (usually given by instructor) 3D object (solid = some function of , , or ℎ cylinder or sphere) (usually given by instructor)

7) Now integrate. Use the coordinate system to determine the limits. If two people chose different coordinate systems, their answers will look very different at this stage (but should be the same in the end). 8) Carefully check the following things on your result: a. Units (compare to ). Should have on top, Coulombs on top, and meters 2 on bottom. b. Direction (does field point away from plus/towards minus). c. Also, look carefully at any unusual subtraction terms to see if they contribute an extra + or -. For example, while . < Using a common denominator cleans up these types of subtractions to make signs more obvious. d. Think about the size of the field as you get far away from the object. Usually one wants to verify the result for the field becomes small far from the charge.

Shortcuts and tricks: 1) Use symmetry to figure out if the or part will be unimportant. You still have to do all the work of the set-up with both the and parts. You would simply not need to do the integral for the or part (if dictated by the symmetry of the problem).

2) You could simply determine the MAGNITUDE of the electric field and then think about the direction afterwards!

3) Build up objects out of slices for which you already have an -field equation. For example, if you know the -field of a disk, one could build up a solid cylinder out of many disks. Make your slice a disk. Use the known equation for the disk and make that your . 18

General technique for doing calculus on rod shape 1. Pick a an arbitrary chunk of the rod and label it . 2. Write down from source to the point of interest (the point at which you wish to compute ). 3. Determine . 4. If uniform, determine . If non-uniform, is typically some function written down in the problem statement. Use lambda to convert . 5. Build the rod out of point charges. Write down

COMMON MISTAKE

6. You have to use the full field when doing the integral. Not just the magnitude. 7. Plug in the pieces for the integrand. 8. Pick correct limits based on the coordinate system you chose. 9. Turn the crank and spit out the answer. 10. Check the answer (units, direction, and large/small distance limits if possible)

22.10 Determine the electric field caused by a uniformly charged rod at the point A, distance to the left of the rod. Assume total charge on the rod is . So everyone does this the same way, choose the point A as the origin of your coordinate system. A a) Determine (displacement vector from source to point of interest). b) Determine (distance from source to point of interest). c) Determine (unit vector from source to point of interest). d) Determine linear charge density (charge per unit length). e) Set up the integral and determine the electric field at point A. Remember to write your final answer in terms of only variables given in the problem statement! If possible, think of ways to check your answer before reading the solution.

22.11 Suppose we redid the previous problem with a slightly different coordinate system. This time choose the left end of the rod as the origin of your coordinate system. a) What changes (or doesn’t change) in the above derivation? A b) Should you get the same final result or not? c) In your opinion, which coordinate system is nicer to work with a L for this case? 19

22.12 Let’s mix it up a bit and try the other side of the rod and be off axis. Show the set-up for determining the electric field caused by a uniformly charged rod at the point B as shown in the figure. Again, assume total charge . Choose the right end of the rod as the origin of your coordinate system. Notice the scalar is a distance (positive) while the vector is negative. ℎ General technique for doing calculus on ring or arc shape 1. Pick a an arbitrary chunk of the rod and label it . BBB 2. Write down from source to the point of interest (the point at which you wish to compute ). 3. Determine . 4. If uniform, determine . If non-uniform, is typically some function written down in the problem statement. Use lambda to convert . 5. Build the arc (or ring) out of point charges. Write down

COMMON MISTAKE

You have to use the full field when doing the integral. Not just the magnitude. 6. Plug in the pieces for the integrand. 7. Pick correct limits based on the coordinate system you chose. 8. Turn the crank and spit out the answer. 9. Check the answer (units, direction, and large/small distance limits if possible)

22.13 Determine the electric field on axis a distance from a uniformly charged ring Slanted Side View Pure Side View (total charge Q and radius R). Check units, what happens as goes to zero (does it make sense?), limit as becomes much, much larger than R.

General technique for doing calculus on disk or washer shape 1. If uniform, determine . If non-uniform, is typically some function written down in the problem statement. Use lambda to convert . 2. Build the electric field of the washer/disk out of ring electric fields. Write down

WATCH THE CHANGES IN THE STEPS BELOW CAREFULLY

COMMON MISTAKE

You have to use the full field when doing the integ ral. Not just the magnitude. 3. Plug in the pieces for the integrand. 4. Pick correct limits based on the coordinate system you chose. 5. Turn the crank and spit out the answer. 6. Check the answer (units, direction, and large/small distance limits if possible)

22.14 Determine the electric field distance (on-axis) from the center of disk. Assume the disk has radius , uniform charge density and total charge .

22.15 Suppose you instead had a washer with inner radius and outer radius carrying the same total charge . What steps of the previous derivation would change?

While not usually a focus of my class, I will include this example. I think it worthwhile to see how this case sets up. I strongly feel it lends insight to the changes between, rods, arcs, and washers which tend to be the focus on exams.

General technique for doing calculus on solid cylinder, pipe or cylindrical shell shape 1. If uniform, determine . If non-uniform s, is typically some function written down in the problem statement. Use lambda to convert 2. Build the electric field of the washer/disk out of ring electric fields. Write down

COMMON MISTAKE

USUALLY you have to use the full field when doing the integral…not just the magnitude. 3. Plug in the pieces for the integrand. 4. Pick correct limits based on the coordinate system you chose. 5. Turn the crank and spit out the answer. 6. Check the answer (units, direction, and large/small distance limits if possible)

22.16 A solid cylinder has length , radius , and carries total charge . The cylinder is NOT a thin rod. Assume the charge is uniformly distributed along the cylinder. Point P

is distance from the top face of the cylinder along the central axis of the cylinder. Determine the electric field at point P.

22.17 A long thin rod with total length carries a non-uniform charge distribution. When aligned with the coordinate system shown in the figure, the charge distribution is given by . To be clear, is some positive constant. Point P is distance to the right of the origin. Determine the electric field at P.

22.18 A thin rod is bent into an quarter circle of radius as shown in the figure. The arc carries total charge uniformly distributed. For ease of communication, I have labeled the angle which is measured clockwise from the positive vertical axis. Determine the electric field at the origin. Note: tips regarding versus for arcs are on the pages that follow.

22.19 A thin rod is bent into an quarter circle of radius as shown in the figure. The arc carries a non-uniform charge distribution given by where is a positive constant. The angle is measured clockwise from the positive vertical axis.

a) Think: before you start integrating, get a feel for how the charge is distributed. Is it positive, negative, some mix of the two? Where is the charge most concentrated? b) Determine the electric field at the origin. Note: There are two things one almost always needs to do in a non-uniform problem. c) Determine the units of any constants in the density. Why? So you may check the units on your final answer. d) Determine the equation relating total charge to density. Why? So you may rewrite the final answer in terms of the total charge on the rod. Why? So you can compare the field strength to a point charge, ring or some other known object carrying the same total amount of charge. This gives you a better feeling for the result and helps one determine if the result seems reasonable. Going Further 22.19d: What if the density was instead ? How would the distribution of charges change? Where is the charge most concentrated? Is it still symmetric? Do both & terms survive the integration for this case? If only one component survives integration, which component is it? Going Further 22.19e: For this particular problem, which direction does the electric field at the origin point if the charge density is an even (or odd function). Going Further 22.19f: A way I could mix this problem up on a test is to put the arc on the -axis and have the angle run positive as it goes counter-clockwise from the -axis. You’d pretty much have to start all over and do similar steps. It forces you to know the whole derivation and the principles behind setting up the integral rather than memorizing this case. Essentially, I expect you to know the process rather than how to copy one solution to one special case. See example below…

24.20 Imagine you have an arc of charge with radius and positive charge uniformly distributed. A point charge carries charge distance to the right of the origin. Note: the magnitude of the point charge is of the arc’s charge magnitude. The half-angle of the arc is some unknown angle . The system is designed to have zero electric field at the origin. a) Is the point charge positive or negative? b) Determine the numerical value of the half-angle required to make the electric field zero at the origin. To be clear, if you discover the tips of the arc would lie in the first and fourth quadrants. 22.21 A straight rod has length and carries total charge uniformly distributed. Point P is distance below the left end of the rod as shown in the figure. Determine the electric field at P. Note: tips for determining limits of integration for rods are the following pages.

Tips for Understanding (from source to POI) and Integration Limits for Arcs CHOICE 1: We start with the standard coordinate system shown at right. A typical problem asks you to determine electric field at the origin. An positive angles ( ) arbitrary slice of the arc is chosen for a positive angle in the first quadrant. The radius of the arc is . on positive x-axis i. The angle is defined as positive going counter-clockwise from the positive -axis. negative angles ( ) ii. The vector is drawn going from the source (the slice of arc) to the point of interest (the origin). WATCH OUT! This is opposite of the slice’s position vector which goes from origin to arc. Subtle difference… The combination of choices made in i. and ii. gives

Before doing anything else, see if this makes sense for both positive and negative angles. • If I put in an angle for between , . We see correctly points from source to POI ( down and to the left ). • If I put in an angle for between , . < We see correctly points from source to POI ( up and to the left ). • If I put in an angle for between , . < We see correctly points from source to POI ( down and to the right ).

CHOICE 2: A typical problem asks you to determine electric field at the origin. An arbitrary slice of the arc is chosen. The radius of the arc is . i. The angle is defined as positive going clockwise from the positive -axis. ii. The vector is drawn going from the source (the slice of arc) to the point of interest (the origin). The combination of choices made in i. and ii. gives

Before doing anything else, see if this makes sense for both positive and negative angles. • If I put in an angle for between , . We see correctly points from source to POI ( down and to the left ). • If I put in an angle for between , . We see correctly points from < source to POI ( down and to the right ). • If I put in an angle for between , . We see correctly points from < source to POI ( up and to the left ).

CHOICE 3: A typical problem asks you to determine electric field at the origin. positive angles ( ) An arbitrary slice of the arc is chosen. The radius of the arc is . i. Angle is defined as positive going clockwise from the negative -axis. ii. The vector is drawn going from the source (the slice of arc) to the point of interest (the origin). The combination of choices made in i. and ii. gives

negative angles ( ) Verify this makes sense for both positive and negative angles. • If I put in an angle for between , correctly points from source to POI ( down and to the right ). • If I put in an angle for between , correctly points from source to POI ( up and to the right ). • If I put in an angle for between , correctly points from source to POI ( down and to the left ).

Tips for limits of integration for rods. There are several common variations regarding problem 22.21 requiring you to think carefully about the angles.

Integration Limits for an Infinite rod

Integration Limits for a Semi-infinite rod

Integration Limits for a Rod with one end at the origin

SUMMARY for continuous charge distributions

Electric charge creates electric fields Coulomb’s law was used for point charges. The point charge formula was used to determine electric fields created by continuous charge distributions. • Build up rods and arcs out of point charges o Use o points from the source (the charge) to the point of interest (where you want to determine ) o for a rod along -axis or for an arc or ring o If uniform …if non-uniform expect some formula for o For an arc, where must be expressed in radians o We assume rods are 1D objects and your slice is also 1D. We assume arcs/rings are simply 1D rods bent into a circular shape. Therefore, assume arcs/thin rings have 1D charge density. • Build up thin disks out of rings o Use o is the radius of the ring (radius of your ring shaped slice of the disk) o is the distance from the center of the ring to the point of interest o …notice the area of a ring shaped slice of radius is o Notice a thin plate is 2D and our ring shaped slice is now considered as 2D!?!?!? o The dimensionality of a slice always matches the dimensionality of the object sliced…

General challenge for all continuous distribution of charge problems: In general we know but . In certain special cases, however, the electric field magnitude can be correctly obtained simply by integrating . For which of the previous problems did would you have gotten the correct answer by simply integrating ?

Why does a balloon stick to a wall? Electrical polarization. Initially all three objects – the wall, the balloon, and the sweater – are electrically neutral.

When an object is rubbed, friction tends it to either Think: should there be a force on the balloon now? Which way? shed electrons to or attract electrons from the object rubbing it. Rubber tends to attract electrons, glass tends to shed electrons. Rub a rubber balloon on the sweater and it attracts electrons from the sweater. Since rubber is an insulator the charges don’t move around freely and redistribute all over the balloon, they stay concentrated where you rubbed it. In this case notice that the left portion of the balloon must be the part that rubbed on the left side of the sweater.

Lastly, bring the balloon close to the wall (which is also an insulator). The electrons in the wall are bound to the Sketch what you think should happen to the protons in the wall (just like the balloon) because it is electrons in the wall! an insulator. HOWEVER, the electrons in the atoms in the wall can re-orient themselves to all be as far as possible from the negatively charged balloon. When the balloon gets close enough to the wall it feels an attractive force from the protons in the wall and a repulsive force from the electrons in the wall. Since the electrons are slightly further away they produce slightly less force. The protons win out and pull the balloon against the wall so that friction can keep it from sliding down. Electrical Polarization is the name for the organization of charge in the wall. Even though the wall has no net charge it is organized in such a way that it can cause a net force.

Note: electrical polarization occurs much more dramatically in conductors since electrons are more free to move in conductors! That said, it does also occur in insulators; the balloon wouldn’t stick to the wall without it!

Some loose ends: Good conductors: charges move freely (metals)

Weak conductors: charges move but only with great difficulty (graphite)

Insulators: charges move but only with the greatest of difficulty (rubber, glass, air)

Semi-conductors: can vary from insulator to conductor depending on situation/environment (Si, GaAs, solar cells, computer chips)

Super-conductors: expel all magnetic fields (and also have the properties of a perfect conductor)

How do things acquire charge? Generally speaking, electrons are removed from one object and placed on another. One object becomes negatively charged while the other becomes positively charged.

Electric Field is zero inside conductors (wrap your phone in hardware fabric ¼” grid and radio waves (which employ electric fields) usually get blocked out…no reception…should also work on the classroom! Might need several layers to be effective. Similarly, might need several layers of aluminum foil to be effective. To learn more try a web search for “penetration depth” or “skin depth”. It gets complicated in a hurry.

Excess charges tend to go to outside of conductors (like charges repel, since free to move in a conductor they move to the outside).

Electric fields tend to point perpendicular to the surface of a conductor (if not true the charges at the surface would start moving to the side until all sideways electric fields are canceled).

Charge tends accumulates at sharp points on a conductor (lightning rods have sharp points where charge is concentrated to attract lightning so it doesn’t hit the house).

Gauss’s Law Yes, it really has all those s’s. Open question: is there any word with a single letter repeated more times in the English language…or any language? Electric flux: Φ = ∙ flux means amount of something permeating a surface (water through cross-sectional area of a pipe) here the electric flux can be thought of as a measure of the number of electric field lines thru a given surface

23.1 Solve in class. Suppose a constant electric field is . A flat plane of dimension 12.00 m by 5.00 m = 10.0 is oriented in the -plane 1) does the electric field penetrate the plane or run parallel to it? 2) what is the area vector? 3) what is the electric flux? is it a vector? 4) what is the flux through a similar sized plane oriented along the -plane remember: flux will come up again with magnetic fields so stay frosty and learn this well!

23.2 Write down the equation for the point charge electric field. Think: is this the field caused by or felt by the charge? Suppose a ball of diameter 20.0 cm concentrically surrounds a proton. What is the area vector of the ball? What is the magnitude of the electric field at the surface of the ball? Ans = 1.44e-7 N/C What is the direction of the electric field at the surface of the ball? Answer: radialy outwards = + What is the electric flux through the surface of the ball caused by the point charge? 1.81E-8 N .m2/C What is the numerical value of ? 1.81E-8 N .m2/C

For any CLOSED surface, regardless of the shape of the surface! This is called Gauss’s law. Φ = However, in most problems either a computer computes the flux OR there is a convenient symmetry.

For spherically symmetric charge distributions, spherical surfaces can be used to compute the electric flux.

For cylindrically symmetric or long wire situations, cylindrical surfaces can be used to determine electric flux

For plates or large planes, a pillbox shape (a small box with flat sides) can be used to determine flux.

23.2½ Before using Gauss’s Law, understand how charge density works in 3D (in solutions in color as well). Consider volume charge density (in units of ) for a tiny chunk of an object ℎ ℎℎ = =

= This last line can be expressed in sentence form as follows: The total charge on an object ( ) is the sum total (integral) of all the little chunks. The charge on a single chunk is the density of that particular chunk ( ) time the volume of that chunk ( ).

In this class we will determine the volume of the chunks based on the geometry of the slices we use.

For a sphere :

= ∙ Cross-sectional ∙ view of a sphere ∙ The charge on a sphere is thus

If the density is uniform Notice the density of the uniform object is

ℎ

How to handle non-uniform density begins on the next page… 31

For this class we may also consider non-uniform cases. The density could depend on the radius. As an example, perhaps the charge density increases with radius according to the equation . If this density equation applies: WATCH OUT! When you plug in the density it is criti cal you use the same radius in the density and in the shell volume . Think: we are building up the charge on the entire sphere using charge of each shell. The charge on each shell is the density at the radius of the shell times the volume of the shell with that radius .

Finishing this problem out we find

Now check the units. We know the density was defined as

The units of the constant can be found by doing

We see the units of in the equation are indeed Coulombs.

Relating charge to charge density in spherical shells Cross-section of spherical shell The charge integral now becomes Inner radius , outer radius This integral is perfectly valid for uniform or non-uniform densities. If non-uniform, plug in whatever density is given and do the integral.

If uniform: WATCH OUT! The total volume of a shell is

Thin versus thick cylindrical shells Side view of cylindrical shell Consider the cylindrical shell shown at right. The thickness of the shell is . The volume of the shell is given by . + = . = . . . End view of cylindrical shell . A thin shell is any shell where the second term is negligible.

Ignoring this smaller second term gives

. This answer makes sense if you consider the lower half of the figure at right (showing the area of the end). Make a cut Roll it flat

Going Further: When is it reasonable to call something a thin shell? Said another way, what does it mean to say ? Thin-walled method only valid if If , we know . By ignoring the second term we do introduce a small error. Let us The trapezoidal area above essentially becomes rectangular with area ! assume we want this error to be less than 1%. We simply assume the second must be less than 1% of the first term! The volume of the pipe is < 0.01 = < 0.0 This means must be smaller than 2% of for us to assume the cylindrical shell is thin (and keep errors under 1%).

Cross-sectional view of spherical shell Thin vs thick spherical shells At right is a cross-sectional view of a spherical shell. You could think of a spherical shell as a basketball with some inner radius and some outer radius . + = Show the volume of the shell is given by =

= + + For a thin shell, we expect causes the last two terms to be much smaller than the first term. = Think: the surface area of a sphere is . If you were somehow able to slice a basketball ball and lay it flat it would be a pancake of area and thickness . This makes sense.

If the second term is less than 1% of the first term we find 0.01

0.01 This means must be smaller than 1% of for us to assume the spherical shell is thin (and keep errors under 1%). FYI – that third term has cubed in it. When the third term will be approximately a 0.01% correction. 0.01

NOTE: Practical summary on page 35… Case 1 ( ) 23.3 Suppose we have a solid, insulating sphere of radius with positive The Gaussian charge distributed uniformly throughout it’s volume. To match soln’s surface is outside later use red for Gaussian & blue for limits. the object a) In lecture: For Case 1, draw several electric field vectors on the Gaussian surface. b) In lecture: Draw several vectors normal to the surface and label them as . c) In lecture: What do you notice about the all cases? Why is ∙ it reasonable to say in this case? ∙ = d) In solution: Use Gauss’s law to determine the magnitude of the electric field for all points . Tip: recall . = e) In lecture: Check units by comparing your result to the point charge electric field.

23.3½ Case 2 requires more effort to determine the amount of charge that is Case 2 ( ) enclosed by the Gaussian surface. Think about an onion made up of layers of spherical shells. We are still using a uniformly charged, solid sphere. The Gaussian a) Use Gauss’s law to determine the electric field magnitude for all . surface is inside the object • Use the fact that, in general, . = • Determine in terms of and for a uniform charge. For a uniform charge we can use . = R • If you had a non-uniform case is given to you by some equation. • For a uniform case, density is constant and pulls out of the integral. In the non-uniform case density does not pull out of the integral. r • For case 2 you can see the charge enclosed by the Gaussian surface (dotted line) can be built up by spherical shells (thick solid line). The volume of one such spherical shell is just × . = = • We can see that the smallest spherical shell that could exist inside the Gaussian surface would have a radius of while the = 0 largest such radius a shell could have (while still being inside the Gaussian surface) is . Use this = to determine the lower and upper limit of the integral. • Finally, plug in all these results and determine an equation for when . b) What is the electric field at r = R ? Do the two formulas agree at ? = c) Plot the magnitude of the electric field as a function of radial distance from the center of the sphere. Don’t forget to use the result from the previous problem d) What would be different if the charge Q was negative? e) How would you modify your computation if the sphere was hollow at the center?

If you tried to use for Case 1 : It makes no sense to allow the upper limit to be ? Once you go = = beyond the radius there is no more charge. The density ρ drops to zero for . As such you would get = two different integrals that looked like = = + 0 = 34

For a cylinder to be considered as “long” we require

23.4 Now let’s consider the Gaussian surface (dotted lines) for a solid cylinder. Notice that Gaussian surfaces are only appropriate for long cylinders ( ). a) In lecture: For Case 1, draw several electric field vectors on the Gaussian surface (above, below, & at ends). b) In lecture: Draw several vectors normal to the surface and label them as . c) In lecture: What do you notice about in MOST cases? In what region of space is it reasonable to say ∙ in this case? ∙ = In order for our Gauss’s law computations to be approximately valid, the contributions to the electric flux from the end caps must be negligible compared to the rest of the surface. This is only true if L>> R. d) If we ignore the areas of each end cap, what is the area of the Gaussian surface? e) Determine the electric field magnitude both outside and inside the cylinder. f) Plot the electric field magnitude versus position for all (both . g) What computation technique should be used to determine the electric field near the cylinder if is not much, much greater than R?

23.4½ A insulating cylindrical shell of has inner radius and outer radius carries unknown total charge distributed uniformly throughout its volume. Assume is the radial position vector from the central axis. The electric field distance from the origin is measured to be . You may assume the length of the cylinder is . a) What is the linear charge density on the cylindrical shell? Hint: you should know a memorized formula for . Use this to relate to . Your final answer should be in terms of . b) Determine volume charge density in terms of . c) Determine the electric field for . Answer in terms of …not . d) Determine the electric field for . Answer in terms of …not e) Determine the electric field for all . Answer in terms of …not

23.5 Use Gauss’s law to find the electric field inside and outside a non-uniformly Er>b charged, insulating cylindrical shell. Suppose the insulating cylindrical shell has End View of Cylinder positive charge density ( is a positive constant). The inner radius of the = shell is and the outer radius of the shell is . a) Determine the units of in order to check the units at the end of the problem. b) Use to determine the total charge in terms of , , and . = c) Now use Gauss’s law to determine .

d) Now assume the Gaussian surface has radius . Correctly label the Ea

g) Lastly consider a Gaussian surface for . How much charge is enclosed? Determine . h) Tabulate results. Check the units of each magnitude. Verify when or = the appropriate cases match. =

i) Plot the magnitude of the field versus

position ( vs ). Clearly indicate the two notable radii and on your graph. Be sure to include the values of the E for these special radii on the vertical axis. To determine if a section of the graph is concave up or down you can use the second derivative test.

Practical Gauss’s Law Summary for insulating spheres, cylinders, spherical shells, & cylindrical shells: Gauss’s law is given by

Φ = ∙ = where is electric flux, is an integral over a closed surface, is charge enclosed in that surface . Φ ∙ The “surface” can be thought of as an imaginary net you place around some or all of a charged object. Usually some electric field lines penetrate through this imaginary net. The electric flux is a measure of how many field lines penetrate the surface AND how big the surface is. The vector is a vector pointing perpendicular to a tiny portion of the surface. The dot product is used because it only captures portions of the electric field parallel to . Said another way, the dot product only captures portions of the field perpendicular to the surface.

For exams, students typically use Gauss’s law only for charged objects with nearly perfect symmetry (large slabs, spheres or spherical shells, and long cylinders or long cylindrical shells). To effectively use Gauss’s law for these cases, one must use a surface (imaginary net) which matches the symmetry of the charged object. If such a surface is chosen, the following reasoning applies: 1) The vectors are parallel to each other at every point on the imaginary surface. When this is true …we can get rid of the dot product. ∙ = 2) The magnitude of the electric field, , is constant for every point on the surface. = When true, we can pull that constant value out of the integration: . ∙ = = = 3) Gauss’s law becomes the much friendlier equation = Several results are worth memorizing (see table immediately below). These results are useful as it gives you a way to check more complicated problems. Notice in these results , and are constants while little is the only parameter free to vary. In the last column that symbol “ ” means “is proportional to”…it’s not alpha ( ). ∝ Proper way to describe it Nickname Equation Proportional to: Outside a solid sphere of radius Outside uniform sphere 1 Total charge uniformly distributed = ∝ Inside a solid sphere of radius Inside uniform sphere Total charge uniformly distributed = ∝ Outside a solid cylinder of radius Outside uniform cylinder 1 Total charge uniformly distributed = ∝ Inside a solid cylinder of radius Inside uniform cylinder Total charge uniformly distributed = ∝

Some tricks for Gauss’s Law: 1) Whenever your Gaussian surface is outside the object (true for both uniform and non-uniform cases). = 2) For uniform distributions of charge only: when your Gaussian surface is inside the object one can use

= 3) For a system with nested shells, can get tricky. Be sure to include from all shells completely inside the Gaussian surface as well as the appropriate from a shell that is only partially enclosed by the Gaussian surface. 4) Inside a conductor, . Use this fact to determine (rather than using a known to find ). Any static = 0 charge on a conductor resides on the surfaces and not in the body of the conductor. Four worked examples follow on the next few pages. Please work through these examples. 37

In hollow region No charge enclosed, no electric field Inside shell = = = = 1 = Cross-section of 1 Insulating spherical shell = Spherical Shell with Uniform = ∙ Charge Units check Distribution Matches boundary (plug in , get …matches hollow region) = = 0 Outside shell = = = = =

= 1 = = Units check, result for matches previous upon plugging in =

WATCH OUT: Outside the spherical shell the electric field has magnitude where little could be any radius greater than = Only at the surface of the sphere (when ) can we say = =

In hollow region No charge enclosed, no electric field Inside shell End view of = = = cylindrical shell Insulating = Cylindrical Shell 1 with Uniform = Charge 1 Distribution = Assumptions: = ∙ = ∙ Cylinder is long Units check: on top, Coulombs on top, meters squared on bottom ( ) Matches boundary (plug in , get …matches hollow region) = = 0 Results valid Outside shell when = = = =

= =

1 = = = Units check, result for matches previous upon plugging in =

WATCH OUT: Outside the cylindrical shell the electric field has magnitude where little could be any radius greater than = Only at the surface of the sphere (when ) can we say = =

1st step: determine units for We know the equation = = = = = 2nd step: determine relationship between Why? Useful for boundary checks & comparing to memorized results. = = = = Now we know = Insulating In hollow region Spherical Shell No charge enclosed, no electric field with Cross-section of Inside shell NON -Uniform spherical shell Charge = = = Distribution = = 1 = = 1 = Assumptions: = = ∙ = = ∙ Units check, matches boundary (plug in , get ) = = = 0 Outside shell Hopefully you already memorized the formula for outside of spheres Write down

= Units check, result for matches previous upon plugging in =

If you want to see how I derived it, read below.

= = = = 1 = =

1st step: determine units for We know the equation = = = = = 2nd step: determine relationships between Why? Useful for checking units and doing boundary checks. = = = = Insulating = = Cylindrical Shell with Recall, from our listed assumptions . = = NON -Uniform In hollow region Charge Distribution No charge enclosed, no electric field Inside shell End view of Assumptions: cylindrical shell Cylinder is long = = = ( ) = Results valid when 1 = 1 = = = = ∙ = = ∙ = Units check, matches boundary (plug in , get ) = = 0 Frisky note: I believe the change of base theorem gives = = Outside shell Hopefully you already memorized the formula for outside of cylinders = Write down

= Units check, result for matches previous upon plugging in =

If you want to see how I derived it, read below.

= = = = 1 = =

Electric fields in conductors (IN EQUILIBRIUM)

New version with diagrams on next page…

If there is an electric field inside a conductor you have current flow (discussed later in the class). Conversely, if there is no current flow inside a conductor (in equilibrium) there must be no electric field present. This principle is used in Faraday cages (why your cell phone often works poorly in metal boxes…for instance an elevator).

I think of it this way, suppose for an instant that there was an electric field in the conductor. The charges would move around due to the force exerted by the electric field in the conductor. Eventually they would all build up on the edges of the conductor and create their own electric field that cancels out the field that caused them to move in the first place. We can see then, for conductors in equilibrium all charge will reside on the surfaces. In this manner the charge will spread out as much as possible.

Note: it turns out that charge will tend to accrue at sharp points (regions with smallest curvature). This is the principle behind lightning rods or why you can scuff your feet on the carpet and get the shock at your finger.

Up to now we’ve discussed insulators with charge somehow distributed throughout the entire body of the object. These charges have been either uniformly distributed or non-uniformly distributed. For spheres and cylinders we note the non-uniform distributions are functions of (radial charge distributions).

Solid conducting sphere with total charge At the instant a negative charge For simplicity, assume some excess negative charge with total is transferred onto the conductor magnitude is transferred onto a conducting sphere. That negative charge is comprised of a whole bunch of electrons. In a conductor, the electrons are free to move about. Since like charges repel, the electrons will repel each other. Eventually the electrons will reach equilibrium. This occurs when the electrons are spread out as much as possible on the surface of the sphere with none in the body of the sphere.

Note: if a conductor was positively charged a similar process would occur. Instead of electrons added to the sphere, they would be Immediately after charge repel each other. removed to make it positive. The electrons remaining in the sphere In a conductor the electrons are free to move! would rearrange themselves in such a manner as to leave the excess positive charge on the outside surface of the sphere.

If you had a single spherical shell the same thing would happen. This is interesting because no charge is left on the inner surface of a single conducting. If you have multiple, concentric conducting shells things get trickier (discussed later).

Now that we know the charge distribution we can apply Gauss’s law to a conducting sphere! Electrons reach the surface of sphere and spread out uniformly with no electrons left inside! If Gaussian surface is outside sphere then . For , how much charge is enclosed? ALL OF IT! We know . = For a sphere the Gaussian surface has area . = = = Same thing happens with a single spherical = shell. If (outside sphere) the field is the same as a point charge!

If Gaussian surface is inside sphere then . For , how much charge is enclosed? NONE OF IT! We know . = 0 0 =

= 0 In short no electric field inside the sphere (when ). 43

23.6 Using Gauss’s Law for conducting shells. Often Gauss’s law is used to find charge on each surface of each conducting shell. In the example at right assume the inner spherical shell has charge and the outer spherical shell has charge . = + = + a) Use Gauss’s law to determine the electric field for all regions of interest. b c d List your results at the bottom of the page. Almost no calculation is a required! You can use that information to plot your results. Assume = , , , , and . As far as the plot is concerned, let us 1 = 1 = = = ignore units and worry about the shape. b) Now think about the inner shell. The electrons in that shell should reorganize themselves such that the excess charge on the shell will be as far apart as possible. On which surface will the excess charge be? c) Check your intuition with Gauss’s law. You know that . Does = it match up for the two innermost regions of interest in the problem? d) Now think about Gauss’s law for some radius between and . You know that the electric field is zero. Therefore the enclosed charge must be______. This is where things get interesting… e) You know you are enclosing some charge from the first shell already. The total charge enclosed must be zero. All charge on conductors resides on the surfaces (not necessarily the outer surfaces). Use this information to determine the charge on the inner surface of the outer shell (at radius ). The result will seem bizarre since you know the total charge on the outer shell is positive! f) Now use the fact that the total charge on the outer shell is . Use this to determine the amount of charge = + on the outer surface of the outer shell with radius . Notice how this amount of charge relates to the total charge in the problem. This is valid for any combination of positive and negative charges as well! g) Determine the surface charge density at each important radius ( , , , and ). = = = = 44

USEFUL GAUSS’S LAW TRICK FOR UNIFORM CASE ONLY WHEN ONLY PART OF TOTAL CHARGE IS ENCLOSED

= +

23.7 Example for a Uniform Cylindrical Shell End View of Cylinder Assume you have an insulating cylindrical shell with uniform charge density throughout the volume of the shell. The inner radius of the shell is and the outer radius is . Assume and the charge per unit length on the entire shell is . Note: very often for cylinders the total charge is expressed per unit length . It is easy to change between total charge and using or . = = Determine the electric field magnitude for .

When can you use Gauss’s law versus having to do the old method? Pretty much any spherical cases you can use Gauss’s law. The one exception would be any spherical case where charge density is non-uniform AND the non-uniform density depends on angle (not radius). This is typically discussed for the first time in a junior or senior level undergraduate physics class.

For cylinders it is a bit more complicated. Gauss’s law applies only far from the ends of the cylinders AND when the Gaussian radius is much smaller than the total length of the cylinder ( ). Furthermore, Gauss’s law will not work if the non-uniform density is a function of (or ) instead of a function of .

I think pictures will help here .

In the figure at right I tried to sketch in an approximate region of validity for any Gauss’s law work on a cylinder.

In the figure below we see two different non-uniform charge distributions on cylindrical insulators. The one on the left varies over the length of the rod while the one on the right varies radially. Gauss’s law would work only for the radial distribution on the right. Density varies over length of rod Density varies over radius of rod

USEFUL GAUSS’S LAW TRICK FOR UNIFORM CASE ONLY WHEN ONLY PART OF TOTAL CHARGE IS ENCLOSED

= +

23.8 A point charge is surrounded by a neutral concentric conducting spherical shell with inner radius and outer radius . Outside of these is a Cross -section of spherical shells concentric insulating spherical shell with inner radius and outer radius . Assume the insulating shell has charge distributed uniformly throughout its + volume. a) Determine the charge density on each surface of the conductor. b) Determine an equation for the charge density of the insulator. c) Ignoring the region inside the point charge, determine the electric field at all points of interest. d) Plot assuming radially outwards fields are positive and radially inwards fields are negative.

23.9 Extra problem forcing you to deal with , and on cylinders An insulating cylindrical shell has outer radius and inner radius with End view of cylindrical shells . This shell carries non-uniform charge distribution throughout the volume of the shell governed by the equation . The insulating shell is = surrounded by a concentric conducting shell with inner radius and outer radius . The outer shell has linear charge density . Assume and are positive. a) Determine the units on . b) Determine the total charge on the inner shell. c) Determine the charge per unit length on the inner shell. d) Determine the charge on the inner surface of the outer conductor. e) Determine the charge per unit length on the inner surface of the outer conductor. f) Determine the charge and charge per unit length on the outer surface of the outer conductor. g) Determine the magnitude of the electric field in all regions of interest and plot it. Assume all constants are 1 to simplify the plotting.

23.10 If you are a physics major (or perhaps electrical engineer) you might like this ones, everyone else should skip this. I solved it and wrote up a pretty decent solution.

Imagine a solid insulating sphere with an off-axis spherical hole cut out of it. The insulating object has uniform charge density throughout its volume. For this problem I am only interested in learning about the electric field inside the hole. Assume the hole has radius and the insulator has radius . The center of the hole is distance from the center of the insulator.

Trick for solving: this object is the superposition of a solid insulating sphere of radius with positive charge density ( ) and a second solid insulating sphere + with radius and negative charge density ( ). Get the field for each one separately and add them up. To be painfully clear, you need to electric field vector (not just the magnitude) before you add them up. Don’t forget, since the hole is off axis, you must correctly account for that using a coordinate transformation. The final result is really wild!

P.S. – This problem is not really that bizarre from a real world application.

23.10 ½ CHALLENGE: If you are a physics major (or perhaps electrical engineer) you might like this one, everyone else should skip this. I solved it and wrote up a pretty decent solution. Extra practice on the coordinate transformation part: I did solve this once but must’ve deleted my nice color coded solution…ouch. Imagine you have two nearly infinite lines of charge. One has charge density and the other has + charge density . The two lines are separated by distance . The field outside a single line charge is . Determine the electric field from the combination of two line charges = anywhere on the -axis. Note: I haven’t done this so it might look really ugly…but who cares what it looks like.

Think: this isn’t that crazy of a problem from a real world application standpoint…determine the electric field between two wires each carrying charge… +

23.11 Yes, it’s long. But, do you want to know what you doing or not? Some killer nuggets in here…A line charge End view of line charge and coaxial is surrounded by three concentric conducting cylindrical conducting cylindrical shells shells. The line charge, for the purposes of this problem, is an infinitely thin conducting metal wire. The outermost shell has linear charge density distributed uniformly + along its length. The next largest shell is electrically neutral. The smallest shell carries linear charge density . The wire at the center has linear charge density . Assume + the total length of each cylindrical shell is much larger than any of the radii. Assume all charges are distributed uniformly along the length of each shell. a) Determine the electric field for the seven regions of interest. The regions are numbered for ease of

communication. Ignore the region inside the central wire as we assumed it was infinitely thin. b) Plot the electric field vector versus distance from

the central wire. I will assume positive values are radially outwards. c) Determine the surface charge density at each radius. d) Sketch the charges and how they distribute on each surface. In my solution I will assume + corresponds to 4 excess plus signs.

Gauss’s Law in Slabs & Plates Consider a square thin insulating plate area and thickness . Total charge on only one side. A Gaussian surface is drawn with a dotted line in both the slanted view and the pure side view. In the pure side view, I have drawn several electric field vectors at various positions on the Gaussian surface. Note: I am assuming the plate is actually infinitely tall and that is why the field vectors always point right or left. I set the coordinate axes such that the charged portion of the insulator is on the -axis. Lastly, notice that both sides of the shaded faces of the Gaussian surface both lie outside of the insulator.

Slanted view Pure side view

= = 0

Notice that the electric field vectors are always perpendicular to the shaded faces of the Gaussian surface. Also, the electric field vectors are always parallel to the non-shaded faces of the Gaussian surface. Assume the area of the shaded face is (not necessarily ). We find ∙ = = Why the factor of 2? We get a contribution from both the left and the right face. There are no contributions from any other face since the dot product in the integrand is zero when runs parallel to the face.

What is ? Since all charge is on plate surface we determine with surface charge density. = =

= Plugging this into Gauss’s law gives

= =

Consider a square thin conducting plate area and thickness . Total charge spreads out over both sides. A Gaussian surface is draw with a dotted line in both the slanted view and the pure side view. In the pure side view, I have drawn several electric field vectors at various positions on the Gaussian surface. Note: I am assuming the plate is actually infinitely tall and that is why the field vectors always point right or left. There is no electric field inside the conductor. Lastly, notice that both sides of the shaded faces of the Gaussian surface both lie outside of the conductor.

Slanted view Pure side view

= = +

What is ? Since all charge is on plate surface we determine with surface charge density. Here is where things differ from the insulator! The conductor has total charge which spreads out over both faces. The surface charge density on either face is thus

= = = HOWEVER, there is also contribution from each face to the charge enclosed

= Plugging this into Gauss’s law gives

= = Notice the final answer has no factor of 2 this time. RESULTS SUMMARIZED ON THE NEXT PAGE.

Type of plate on single surface in terms of in terms of Insulator with on a single surface = = = Conductor with spread out over both surfaces = = =

The two results are identical when written in terms of & . That said, is almost always written in terms of for plates. When someone refers to on a plate, first consider the type of plate (insulator vs conductor).

Inside a slab? For a conductor assume the electrostatic field inside is zero. Electrostatic implies none of the charges are moving so we are considering after the charges in the conductor have moved around and reached equilibrium. If the electric field was non-zero, we would not yet have the electrostatic field. The charges would move around until such point they reach equilibrium. In matter of nanoseconds the charge distribution inside the conductor would stabilize and, at that point , we have achieved the electrostatic condition and the electric field inside the conductor would be zero.

Consider an insulating slab with charge distributed uniformly throughout it’s volume (not just on the surface). The Gaussian surface shown at right. This Pure side view time the Gaussian surface is entirely inside the slab which is centered on the

coordinate system. We again assume the only contributions to the electric field of the slab come from the left and right faces giving

∙ = = 1 = 1 = The left side of the equation always uses the Gaussian area. On the right side of the equation is usually the Gaussian area times a differential thickness. Since the width of our Gaussian box relates to the variable we don’t want to use that same variable name for our slices. The area is constant for the + case of the slabs and factors out. We then find 1 = If the charge density is symmetric (if is an even function) we may simplify this to 1 = The factor of 2 from changing the limits cancel the factor of 2 from the left and right faces.

If you have a uniform slab

= = 52

While it might be tempting to rewrite using , don’t do it. Strictly speaking, only use if the charge is on a = surface (not distributed throughout the volume). This might seem strange since, with cylinders for example, we are always rewriting . Convention dictates type of substitution is ok for cylinders but not for = the interior of slabs. y 23.12 A slab of insulating material is somehow given a non-uniform charge density described by . Here is measured from the center of the slab as shown in the figure and is a = positive constant. The slab is essentially infinite in the and directions.

Before cranking this out, think about an arbitrary point inside the slab to the right of the origin. a) Which way will the field point? Consider the contributions of nearby charges. b) Which way will the field point at the origin? x c) Where is the magnitude of the field largest & smallest? d) Are there any locations inside the slab where no electric field is present?

e) Derive an expression for the electric field in the interior ( ) region of the slab. f) Derive an expression for the field in the exterior ( ) region of the slab. g) Plot the field versus (see below). Include both positive and negative values of . Clearly d indicate the values of the field at on the vertical axis of the plot. Assume positive = values are fields pointing to the right and negative values are fields pointing to the left.

Going further …how would things be different if the density function was to some odd power…

23.13 Two very large conducting discs are arranged as shown. Each disc has total area . Note: the figure is a side view; you are viewing each disc on edge. I will assume the distance between the discs is much smaller than the radius of each disc (figure not scale).

Because , you may assume the discs extend infinitely far in and out of the page as well as up and down. The disc has charge distributed uniformly on the inner surface. The right disc has charge distributed uniformly on the inner surface.

For this problem assume .) =

Determine the electric field in the three regions of interest. Make a plot of . Assume positive values are fields pointing to the right and negative values are fields pointing to the left.

Summary comments for Slabs/Plates • For isolated conductor, excess charge resides on outer surface. • For conductor, external electric field points perpendicular to surface ( ). Within conductor . = = 0 • For plates see the table below. Type of plate on single surface in terms of in terms of Insulator with on a single surface = = = Conductor with spread out over both surfaces = = = • For slabs/plates we assume ; the 2 comes from using both sides of Gaussian surface. ∙ = • Inside insulating slabs/plates we use and = = • For parallel plates we use either where is the surface charge on a single plate or where = = is the spacing between the plates and is the potential difference between the plates.

23.14 Three very large conducting discs are arranged as shown. Each disc has total area . Note: the figure is a side view; you are viewing each disc on edge. You may assume the discs extend infinitely far in and out of the page as well as up and down. The leftmost and rightmost slabs have charge distributed uniformly on the inner surface. The center slab has charge . Notice charge is distributed uniformly over each surface of the inner disc!

For this problem assume . =

Sketch a plot of vs along the horizontal axis shown in the figure. Assume electric field vectors pointing left are negative while electric fields pointing right are positive.

Variation 2: What if you redid this problem but the left most plate had charge , the middle plate had charge and the right plate had charge . All positive charges. Consider how the charges would spread out over the faces. Hint: you know the net electric field inside each conductor must be zero. The electric field from a single face (or surface) is given by

. How do the pictures and the plot change?

Variation 3: What if you redid this problem but the left most plate had charge , the middle plate had charge and the right plate had charge . How would the charges align themselves? How does the plot change?

Electric Potential

In PHYS 161 we started with forces and kinematics (vector equations) and worked up to energy (a scalar equation). Now, in E & M, we started with forces and fields (both vector equations). Some problems can be solved much more easily using energy methods (just like 161). It will seem like we are getting more and more abstract but in the end it will come together (fingers crossed).

Analogy with GPE:

= For no good reason I will call term “the potential”. It is not the potential energy, just “the potential”. Depending on coordinate system could be negative. However, all coordinate systems showed the same CHANGE in . In general, if was positive the object converted kinetic energy into potential and slowed down. Δ

Δ = Δ Here I would say that “(the change in potential energy) = (mass)×(the change in potential)”.

Read problems carefully for these words: Electric potential = voltage Electric potential difference = difference in voltage = Δ = Electric potential energy (in Joules) CHANGE in EPE = difference in Joules = Δ = center-to-center distance between ’s OR distance from charge to some point P out in space =

Magnitude of the E-field caused by (or surrounding) an isolated point charge at some distance = from the center of the point charge Magnitude of Coulomb force that charge 1 exerts on 2 (which happens to equal the magnitude of = the force that 2 exerts on 1)…assumes charges have center-to-center separation Electric potential (Voltage) at some point in space a distance from the center of an isolated point = charge …can be + or -! Electric potential energy shared by a pair of charges (1 & 2) that have center-to-center separation = …can be + or -! Magnitude of force that acts on a point charge in the presence of an electric field with magnitude = Potential energy of a point charge that is located at a position in space where the voltage (or = electric potential) is known to be Change in potential energy of a point charge that has moved. Here . This is the Δ = − change in voltage when comparing the final position to the initial position. Surprisingly, this does Δ = Δ not depend on the path traveled by the charge ! The magnitude of the electric field between two plates relates to the electric potential difference Δ (voltage difference ) between the plates and the spacing of the plates . Note: this is only valid = Δ if the size of the plates is much greater than the spacing between the plates. Another way to write the previous equation. Here means the charge per unit area on one of the = plates. It is assumed one plate has charge density while the other has charge density . + − Note: sometimes you will see re-written in terms of the constant using = Work done on a charge in uniform electric field moving in -direction = Δ = Δ ∆∆∆U= ∆∆∆EPE=change in elec pot erg IS OPPOSITE OF WORK BONE BY ELECTRIC FORCE (by def’n) remember the delta Use this stuff to derive U and V for point charges up above. 56

24.1 Three point charges are arranged as shown in the figure. The distance are noted as multiples of the arbitrary distance . You may assume has units of meters. A point P is shown in the figure. No charge is located at P. a) Determine the electric potential at P. 1 b) Determine the electrical potential energy associated with this group of point charges. Answer with a 3 digit decimal times .

24.2 Two point charges are separated by a distance . Nipomo is distance above the midpoint of the line between the two charges (see figure). Lompoc Nipomo is some distance to the right of the origin. a) Find electric potential in Nipomo .

b) Find the electric potential at Lompoc . c) Think: Without doing any computation, what should your result be when ? Does your intuition match the result? What gives? = 0 d) Find an approximation for valid when ? Do you need to use the binomial expansion in this case? Lompoc

e) Find the electric potential energy stored in the fields created by the +

two point charges.

24.3 Seting this up is really good practice…Two point charges are each distance from the origin. The white charge is positive with magnitude while the black charge is negative with magnitude . At some point (or points) on the -axis the electric potential is zero. How far is this point (or points) from the origin? Note: the potential is zero infininitely far from the origin. I am looking for the any points not infinitely far from the origin.

24.4 Two vertical parallel plates set up a constant electric field of 1000 N/C. The plates are spaced 5.00 cm apart. A proton starts from rest right next to the positive plate. Gravitational forces are negligible. a) Determine the force on the particle. b) Determine the work done on the proton by the electric field just q before it impacts the negative plate. c) Determine the final kinetic energy. d) Determine the final velocity (magnitude and direction). e) Determine the potential difference between the two plates in units of Volts. f) Oftentimes we say the negative plate is at ground. This means we say the electric potential at the negative plate is 0 Volts. This is just like saying the bottom part of a gravitational energy problem is zero height. Use this fact to determine the voltage (with the sign) of the other plate. g) In this problem is said to be the electrical potential energy of the proton-plate system while is the electric potential at the position of the proton. Determine the initial electrical potential energy of the proton. Hint: if the voltage is known this is simplest to calculate using . = h) Suppose involved an electron starting from rest at the negative plate. What changes? What doesn’t?

f) Your previous answer is negative. What does that sign mean? Specifically, if the two charges are brought together from infinity, would a person have to be pushing them together (keeping them from flying apart) or holding them back (keeping them from smashing together)? g) Worthless Challenge: if figure is drawn to scale, determine and . h) More Worthless challenge: determine magnetic north in above figure…this is an E & M course ya know.

24.5 Three electrons are at rest on the -axis. The spacing between adjacent electrons is . Mass of electron is . a) Find the initial energy stored in the fields between the three charges. Answer in terms of , , and . b) Now assume the one on the right is released from rest while the other two are held fixed in place. Determine the speed of the moving electron when it gets very far away from the other two stationary electron.

Initial state, all Final Picture? electrons at rest.

24.6 Four charges are arranged at the corners of a square as shown in the figure. - q a) What is V at the center of the arrangement? a b) What is the direction of at the center? c) What happens to your answers if the bottom two charges are switched? a d) What happens to your answers if the left two charges are switched (from the initial configuration)?

q - q

van de Graaff demos: charged hair, pie pans, cup of Styrofoam peanuts, little piece of cotton, Faraday cage?

24.7 What if you shoot a proton directly at much heavier positive charge from far Befor away? It would come close then be repelled away. Compute the distance of closest = + = + approach. Assume perfect alignment so the proton center is fired directly at the = center of the stationary positive charge. Assume and . = = You may assume gravitational forces are negligible.

At this point, do some web research for “Rutherford scattering”. When Rutherford and some homeys fired positively charged alpha particles at a thin piece of gold foil After

they were shocked to discover some alpha particles essentially bounced backwards. = This did away with the plum pudding model of the atom and made people realize atoms have a tiny dense, positively charged core called a nucleus. I believe the

quote about this experiment is: = It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. Another classic Rutherford quote: All science is either physics or stamp collecting. Why do people say physicists are arrogant? I just can’t understand…

24.8 Four identical point charges with situated on the corners of a square of side . All four charges are released from rest . Assume these are ions and gravity is negligible. a) Determine the speed of each charge after configuration has doubled in size. b) Why is the previous answer unaffected by the sign of the charges? c) Why is it inappropriate to determine the force, find acceleration, use a kinematics equation such as to find the final speed? = + d) Challenge/At least consider and read solution: What would change if charges 1, 2, & 3 are fixed in place & only charge 4 moves?

24.9 Four point charges with situated on the corners of a square of side . This time all four charges are initially moving outwards with low speed away from the center. This time, + however, the sign of the charges alternate as you go around the square as shown in the figure. The charges still have the same magnitude but not the same signs! All charges again have the same mass . Assume these are ions and gravity is negligible.

a) Before looking at the solutions, take a guess what happens. Once all four charges are released, what will happen to the charge configuration? In particular: • Will it still be a square?

• Will the charges fly apart forever, smash into each other or something else? + b) Determine the largest charge spacing the charges experience. If they fly apart forever, call this spacing . c) Challenge/Think: Eventually, the charges reverse direction and head back towards the center of the square where they collide. Why is it impossible, without relativity, for us to determine the max speed each charge achieves after the instant of release? d) Suppose, instead of all equal masses, protons and electrons were used as the positive and negative charges. Would your answer to part a change? Why is this a much tougher problem? How might one in theory learn about the behavior of such a system?

24.10 Equilateral triangle with identical negative charges on the corners. Top charge is released from rest. Speed of top charge once it is far from the other charges. Assume all charges have same mass . Assume these are ions and gravity is negligible.

24.11 Top charge released from rest. Bottom two charges are fixed. Assume all charges have same mass . Assume these are ions and gravity is negligible. a) Max speed of top charge. b) How far below the two neg charges will the plus charge go? c) Describe the motion of the plus charge assuming perfect alignment? d) Challenge: is the motion simple harmonic motion or no? Does or = something else?

e) Challenge: Describe the motion of the top charge is the alignment is ever so slightly − − imperfect to the right or left of the centerline. Hint: you could try to run this on a simulation and observe the behavior.

24.12 A glimpse of brutality…Four identical point charges with situated on the corners of a square of side . Charges 1, 2 and 3 are locked in place and only charge 4 is released from rest . What is speed of charge 4 after it has traveled distance ? Assume these are ions and gravity is negligible.

Note: you might find it interesting to read about “denesting radicals”…deliciously brutal!

24.13 Physics/chemistry majors. Determine the electric potential at P due to the electric dipole shown. a) Write down the vectors and in terms of the vector and the distance a. P b) Determine the distances and in terms of the r and a. c) Write down the electric potential in terms of k, q, r, and a.

What happens in the limit ? You might think you could simply set all the a terms to zero but you would be incorrect! Because you have a subtraction between two terms a very interesting thing happens. Use the binomial expansion in this type of case. The binomial expansion

says that ONLY WHEN . To use the binomial expansion you must first get a term that is much le ss than one.

In your problem we know that . Do you see that this implies - + ? Going a step further we can see that we also have That means a we can use the binomial expansion if we can make the electric potential

have terms in it that look like that or . d) Use the binomial expansion to simplify the resul t for the electric potential in the limit .

24.14 Equipotential Stuff (Next page is variation of that matches simulation) This stuff is fairly straight forward…students can do it at home on their own. Show a simulation to explain what an equipotential is then have students do 21.14-17 on their own? In the figure below we see a single point charge with . Point P is shown for convenience. No = + charge exists at P. In the figure the point charge is the small black dot; for now you can ignore the circles around the dot. Recall that k= 8.99 ×10 9 N m2/C 2. Plugging in numbers one finds the equation for electric potential (as a function of radius) is · = This expression is a mix of numbers and variables we require units. Purely symbolic answers require no units.

Note: the units of are the same as the units of . When plugging in r in m, the units will be Volts=V. Part A: All points on any given circle have equal potential. What is a clever name for any line (or surface in 3D space) which contains points of equal potential? Part B: The first circle has a radius of 1.0 cm from charge q, the next has radius 2.5 cm, and the last has radius 5.0 cm. Determine the electric potential on each circle. Label the circle with the voltages. Part C: One can use equipotentials to APPROXIMATELY determine the magnitude and direction of the electric field. • The magnitude or strength of the field is approximated by • The direction of the field points from high electric potential to low electric potential. Point P is halfway between the 2.00 V and 4.00 V equipotentials. Estimate the size of the field at P using . Part D: Point P is about 3.75cm from the point charge. Compute directly using . = Part E: Determine the % difference between the estimate of part C and the direct computation of part D.

Going further: One can improve the precision of the estimate in part C by using equipotentials which are closer together.

Note: To make computations from the equipotentials easier, typically one spaces the equipotentials by equal potential differences. For instance, it is easier for most people to use equipotentials drawn at 5V, 10V, 15V, & 20V instead of 2V, 5V, 10V, and 20V.

24.15 These numbers match a simulation online. Now assume that . This time we find = − 8 = − 99 a) Use this to determine the radius of the , , and , equipotentials. Note: in the figure below assume the gird size is− approximat −5ely − in each direction.−5 Use a pencil in case you mess up your circles. Label the voltage on each equipotential.5 If time, add in the equipotential. b) What do you notice about the spacing of the equipotentials and the strength of the −5electric field? In particular, when the equipotentials are closely spaced the electric field is strong or weak? c) What do you notice about the direction of the electric field compared to the equipotentials? d) Use to estimate the electric field at a point halfway between the & − 5 − equipotentials. Compare this result to the value obtained using with a % difference.

Equipotentials (Summary) When drawing equipotentials, use a constant between equipotentials. Δ The electric field points perpendicular to the equi potentials The electric field points towards lower voltages The magnitude of the electric field is approximately given by E≈| |/| ∆r| Δ The electric field is strong when the equipotentials are most closely spaced

Now consider a topographic map (see figure below). In a topographic map, lines of equal elevation (equal gravitational potential energy) are shown. The lines are usually spaced out by a convenient change in height such as 40 ft or 100 ft (or something comparable). When the lines are close together, is that a steep hill or a flat plain? If a marble was placed on the hill at a point where the lines are close together, will the marble accelerate rapidly or slowly?

24.16 Compare topo maps to electric field maps. Assume the electric field map uses equipotentials spaced with a constant . Instead of a marble, one places a tiny charge (test charge) near the source charges (the ones used to Δ create the field map). a) Suppose a test charge is placed in a location where equipotential lines are close together. Will the test charge be in the presence of a large or small electric field? b) Would the test charge tend to accelerate rapidly or slowly when placed where equipotentials are closely spaced? c) If we assume the test charge is positive, will it tend to move towards higher or lower electric potential. d) If a negative charge was used, instead of a positive charge, would it tend to go to higher or lower potential?

One last note: The electric field map of a point charge is very simple. In real life, charge distributions get very complex. As such, the corresponding field maps are rarely a series of perfectly shaped circles. That said, if you look at a field map and see a small section with circular shapes it must correspond to either to a deep valley (of negative charge) or a tall mountain (of positive charge). In this sense the analogy with a topo map is quite good.

24.17 Three charges are arranged as shown in the figure below. Compare your answers to a simulation online. a) Without overthinking it, sketch the electric field by using field lines connecting the three charges. Just guess. The field lines should be most dense where the field is strongest. Indicate with arrows on the field lines the direction of the field. Hint: if you have twice the charge, you should have twice the amount of field lines… b) Is there a line or curve between the three charges where the potential is equal to zero? If no such line exists, state NO SUCH LINE EXISTS. If such a line exists, draw it. Do a simulation to find the answer!!!

-q

+2 q

-q

Continuous charge distributions 24.18 Determine the electric potential caused by a uniformly charged rod at the point A distance to the left of the rod. Simplify the work and consider what checks can be done on your final result.

A Choose the point A as the origin of your coordinate system.

24.19 Determine the electric potential distance (on-axis) from the center of a ring of charge. Ring has total charge and radius . a) Check units. b) What happens as goes to zero? Does it make sense? c) Limit as becomes much, much larger than ? d) If done early, sketch a plot of versus .

24.20 Determine the electric potential distance (on-axis) from the center of a

disk of charge. Disk has total charge and radius . Going Further: what would change if this was a washer?

24.21 A thin rod is bent into an quarter circle of radius as shown in the figure. The arc carries a non-uniform charge distribution given by where is a positive constant. The angle is = measured in radians clockwise from the positive vertical axis.

a) Think: before you start integrating, get a feel for how the charge is distributed. Is it positive, negative, some mix of the two? Where is the charge most concentrated? b) Determine the units of any constants in the density. c) Determine the equation relating total charge to density. d) Determine the electric potential at the origin. e) What is the electric potential if a different density ( ) is used? = 24.22 A straight rod has length and carries total charge uniformly distributed. Point P is distance below the left end of the rod as shown in the figure. Determine the electric potential at P. You should at least get as far as setting up the integral and plugging in the limits. And know which integral from a table to grab… Think: what would be different if the problem had non-uniform density ? =

24.23 A straight rod has length and carries total charge uniformly distributed. Point P is distance below the left end of the rod as shown in the figure. Determine the electric potential at P. You should at least get as far as setting up the integral and plugging in the limits. The point is to see what changes compared to the last problem. Think: what would be different if the problem had non-uniform density ?

There are several common variations on the previous problem requiring you to think carefully about the angles.

Infinite rod

− −

Semi-infinite rod

24.24 Imagine you have an arc of charge. The arc has radius and carries charge with unknown uniform density. A point charge carries positive charge and is located distance to the right of the origin. Assume the half-angle of the arc is . The system is designed such that the electric potential is zero at the origin. a) Is the arc positively or negatively charged? b) Determine the charge density of the arc.

24.25 Calculating electric field from electric potential. Same stuff in solutions using color. Point charge at origin. Point of interest P on -axis distance from charge. =

= − That weird symbol is called the “del operator” (also called “nabla”). When used on a scalar, such as , we say you are taking the gradient. In 2D Cartesian form = + Those slightly italicized derivatives are called partial derivatives. They are actually easier to use than total derivatives. I will explain how they work in this and the next example.

= − + = − + = − − Here is how that partial derivative works. A partial with respect to means treat everything except in the equation as a constant. A partial with respect to means treat everything except in the equation as a constant. = − −

= − − + = + Go back and look at the picture. Verify that is the correct electric field at point P.

24.26 Calculating electric field from electric potential. In color in solutions… Point charge at origin. Point of interest P at the coordinate . = = = + + +

= − = − + = − + − + = − + − +

= − − Notice each term has a 2 to cancel the ½’s from the derivatives. Also notice each term has which can be factored out. Lastly, notice each ½ from the derivatives has a minus sign to cancel the minus sign out front!

Now use the fact that in polar coordinates and move that ugly crap to the basement to get Go back and look at the picture. Verify that is the correct electric field at point P. 68

What is the point of using ? It is pretty easy to write down electric potential compared to electric field. Computers can pretty much take any derivative you can imagine and do it symbolically. If you want to find the electric field of some crazy assembly of charge you could first find the electric potential at some arbitrary point with the coordinate . Then take the 3D gradient of the electric potential and you know the electric field. From that point, one uses a computer model to predict how something behaves in such an assembly of charges. Practice for this type of thing on this page.

24.27 Challenge: Set-up until point when you need integral table is good practice for all.

a) Find the electric potential the point P due to a rod of length carrying charge . Show all work. Hint: set-up integral then shift to . Then use

Don’t forget to shift your limits (if you are hip to that style). b) Find the electric field at the same position. Write the field in Cartesian. Hint: change and in your previous answer to and . Then you can use Note: when performing assume everything is constant except . When performing assume everything is constant except . Why care? In general it is easier to do integrals with (as opposed to ) because it is a scalar. Computers are great at taking derivatives for you these days. This now gives template for how to determine for any crazy group of charges: first determine potential using calculus (or summing over point charges) then have a computer turn the crank to compute .

24.28 Previously we determined the electric potential at point P distance (on- axis) above a ring of charge with radius and uniformly distributed charge is a) By symmetry, which way does the field point at P? b) Use the gradient operator in that direction to determine the electric

field.

24.29 Challenge Apply the same process shown above to a disk. First write down the electric potential at point P distance (on-axis) above a disk of charge with radius and uniformly distributed charge . Use the gradient to determine the electric field at P. 69

Another practical application using … Another way is useful is in understanding how charges behave in structures built by electrical engineers. First the engineer determines a plot of potential versus position ( vs. plot) is known for a particular device (say a solar cell or a transistor). If you recall, taking derivatives of is equivalent to getting the slope of the plot! ofvs . One can use such a plot to predict the behavior of chargesslope inside the device.plot Ok…I am dumbing this down a bit. If you are interested, read about “energy band diagrams” or take a semiconductor device course. This topic does foreshadow it quite well. I like a quote attributed to Nobel prize winner Dr. Herbert Kroemer: “If, in discussing a semi-conductor problem, you cannot draw an energy band diagram, this shows you don’t know what you are talking about.” 24.29¼ Consider the plot of voltage versus position shown at right. V (mV) Assume there is no electric field in the - or -directions. 80 a) Over what spatial interval(s) is the electric field zero? b) Over what interval(s) does the electric field point ? 60 c) Suppose a proton of mass is initially 1.673 × 10 kg located at the origin travelling to the right. What is the initial 40 potential energy of the proton?

d) What minimum initial speed is required for the proton to 20 reach the position ? = 1000nm 24.29½ Consider the same plot of voltage versus position shown at 0 x (nm) right. Assume there is no electric field in the - or -directions. 0 200 400 600 800 1000 a) Suppose an electron with a mass is .11 × 10 kg released from rest at . Will9 it move to the right or left? = 300nm b) Determine the acceleration of the electron from the previous part just after its release. c) How far does the electron travel before it reverses direction? d) What is the electron’s max speed?

24.29¾ Consider the plot of versus shown at right. a) Suppose you want to release a charge from rest at . V (µV) 10.0cm 16 You want this charge to travel to the right. Should you use a positive charge or a negative charge? 12 b) Describe the motion of such a charge after it is released. In particular, when will the charge be moving right (or left) and 8 speeding up/slowing down, moving right (or left) with constant speed, and/or at rest? How far left or right will the charge move? 4 c) Assume the charge has mass . Now assume 2.00 × 10 kg 0 the initial speed is at . The charge reaches x (cm) 285 = −10.0 cm -20 -10 0 10 20 30 40 50 position with final speed . Determine the -4 50.0cm 80.0 charge. Answer as a decimal with 3 sig figs times . -8 24.29 Use the same plot as the previous question. An electron is initially at moving to the left . = +50.0cm a) What initial speed is required for the electron to reach ? .00cm b) Assume the electron’s initial speed is slightly greater than the5 minimum required to reach . .00cm Describe the motion the electron. Specifically, where does the electron reverse direction? Where5 is travelling the fastest? Where is it travelling the slowest?

Determining from The following two mathematical statements are equivalent:

Getting from Getting from

= − = − Some useful info on the right side equation: • is NOT electric potential…it is electric potential DIFFERENCE. • is not . People all over, myself included, get lazy and use the two interchangeably. For right now let’s be careful with the distinction. • Electric potential DIFFERENCE is useful in practical applications such as circuits, capacitors, etc. • If the line of integration is parallel to the -axis use (similar for -axis). • If the line of integration is radially outwards use . • If performing this type of integral in relation to a spherical or cylindrical object, I recommend choosing to integrate radially outwards to reduce the chances of making sign errors. • Typically we use a known electric field (e.g. from Gauss’s law or Ch 22 methods) and use it to determine electric potential difference.

Example 1: Near a large plate To the right of the plate on the -axis are two points A & B. Assume the distance between these two points are . Assume the plate carries positive charge density .

B A − 2 Grouping the ’s we see . Also, is constant and pulls out of integral. 1 = − 2 Always integrate in the “normal” direction (left to right or radially outwards). Also remember that is not …its . = − = − 2 2 If I want the potential difference going the opposite way I simply flip the sign!

= − = + 2

Example: use the electric field above a ring to determine the electric potential above a ring: First I look up the electric field from a ring of charge.

Then I use The limits of integration should be whatever random points you want. For now let us call them and . Note: my recommendation is to always go upwards (form lower point to higher point) to avoid confusing minus signs issues. − Now comes the clever part. Assume we are going from distance to infinity. At infinity, we are so far from the ring there is zero potential ( ). 0 − = − + − − Do the integral first. THEN change and . From here you could do the integral to determine the electric potential above the ring. Since we have already solved this another way…don’t worry about finishing this integral right now.

24.30 Consider two parallel plates. Assume the left plate carries positive charge density while the right plate carries opposite charge density Assume the distance between the two plates is . Assume each plate is a conductor with thickness . Watch out: all charges will reside on the inner surfaces of each plate due to polarization. a) Determine the electric field in all three regions of interest (to the left, between, and to the right of the plates). b) Determine the magnitude of the electric potential difference between the plates. c) Write a relationship between the magnitude of the electric field between the plates and the magnitude of the potential difference between the plates. d) Plot both vs and vs .

24.31 Using electric field to determine both inside and outside of an insulating sphere

R A B C D

Far from a sphere we expect 0

From Gauss’s law we determined Inside Sphere Outside Sphere

a) Use to determine the electric potential difference between point C at radius and infinity. Integrate radially outwards using . Use this result to obtain . b) Notice point B is a special case where either electric field equation is valid. The equation for should be valid at point B. What is the special value of electric potential at point B?

c) Use to determine . d) Now you know the electric potential outside and inside a sphere. Plot it! Assume and 1 ignore units. If feeling gung ho, also plot the magnitude of electric field vs for comparison. e) How would the plot change if a conducting sphere was used instead of an insulating sphere? Explain why.

24.32 If you can do this one you probably know pretty well. Consider a uniformly charged insulating spherical shell with inner radius and outer radius . The total Cross-section of sphere charge on the sphere is .

a) Determine the electric field in all three regions of interest. b) Determine the electric potential as a function of in all three regions of interest. c) Plot vs and vs .

24.33 Why would it be significantly harder for you to do calculation for charge in an arc geometry? Under what circumstances would such a calculation be fairly doable?

24.34 An unusual arrangement of charges has created an electric field given by B A − where is the horizontal position based on the coordinate system shown. Assume and are integers and and are positive constants. The electric field equation is valid over the entire shaded region shown in the figure. a) Determine the unis of & . b) Determine the electric potential difference going from A to B. c) What about going from B to A? d) Are there any integers that might be particularly troublesome for either or ? If so, which integers cause noteworthy issues for which exponents and how would you handle the issues. e) Suppose an electron made the entire transit from A to B. Under what circumstances would the electron’s speed increase?

24.35 Worth Knowing: Why can one say the electric potential at infinity is zero for a sphere but not for an infinite line charge (cylinder)?

24.36 Going Further Regarding above question: In reality, the electric potential should be zero far from a line of charge. How could you set up a problem to more accurately determine electric potential differences far from a line of charge? Assume the line of charge is on the -axis and you are at points far away on the -axis. To be clear, we are no longer in the limit . I can think of two ways to set this up…

24.37 Common in many physics texts. Two conducting spheres have radius and respectively. The sphere of radius carries charge while the sphere of radius carries charges . The two spheres are far apart (the distance between them is much greater than either radius). a) Determine the electric potential at each sphere’s surface. Assume they are very far apart. b) Determine the electric potential (magnitude is fine) between the surfaces of the two spheres. c) Would there be any difference if the spheres were solid insulators? d) Challenge: how would you have to handle this problem if the spheres were two insulators close together? e) Brutal Challenge: If you had two conducting spheres close together, polarization would be non-negligible. Instead of trying to calculate how this changes things, describe why your calculations would get messed up. Do you think the potential difference would be slightly higher, about the same, or slightly lower than if no polarization occurred?

24.38 Challenge: Two very long, solid, conducting cylinders are parallel to each other and are separated by distance (distance between surfaces of conductors). Each cylinder has radius . The left cylinder carries linear charge density while the right cylinder carries linear charge density . Is it possible to determine the electric potential difference between the surfaces of the two cylinders? If so, do it. If not, show me why not? Side note: this problem relates to the capacitance per unit length between adjacent electrical cables. In many experiments this is critical information.

24.39 Imagine you have eight identical point charges symmetrically arranged on a circle. Assume the radius of the circle is and each point charge has positive charge . I numbered each of the charges for ease of communication. I was interested in learning about two interesting points in the circle. The first is the easy one, point A, at the center of the circle. The second point, B, is located halfway between charges 2 and 8 and directly below charge 1. To be clear, no charge is located at A or B.

NOTE: I would use coding to actually solve this problem. That said, I think it can be useful to consider how you might set it up. In particular, think about how you would get , , and . Think: if coding, could you extend your code to analyze a spherical shell ? The things I was curious about are as follows: a) What is the electric field and the electric potential at point A? b) What is the electric field and the electric potential at point B? In particular, I expect at least some cancellation of the fields from all the charges at B but will there be zero field? c) My guess is the electric field contributions from each charge at B will not perfectly cancel. What if you replaced the point charges a thin ring of uniform linear charge density? Do you think such a ring would have no field at B? If the field is non-zero at B for a ring, does this contradict the prediction from Gauss’s law which states “no electrostatic field exists inside a charged conducting cylinder or sphere”? d) What is the electric potential energy associated with this assembly of charges? This seems very time consuming so I won’t bother with it. Think: you need the charge to charge spacing for every possible pair which gives 4 distinct possible distances. Also, if I did the math correctly, there are 28 distinct pairs of charges to keep track of. OUCH. Skip this part but you might enjoy verifying I counted pairs correctly… e) What if you redid this problem with a uniform ring of charge, found the potential at B, then took the gradient to find the electric field at B? Can you set this up? See how this technique, despite being challenging, could be pretty sweet compared to direct calculations?

Capacitance Capacitance ( ) between two conductors plates is defined by = where is magnitude of charge on one of the conductors and is the potential difference between the conductors. Usually this potential difference could be applied by a battery or other power source.

Note: when we say a capacitor has charge we mean the positive conductor carries charge and the negative + conductor carries charge . Capacitance is positive.

Note: For a particular arrangement of conductors, capacitance measures how easy it is to store charge (or electrical energy). Rearranging the capacitance equation gives . As gets bigger you can store more charge (or = more electrical energy) for the same size battery.

25.1 Consider two parallel plates carrying charges densities respectively. The + area of each plate is . Assume the plate spacing is small enough for the plates to be considered infinite in size.

a) Determine the electric field (vector) between the plates. b) Use to determine the potential difference between the two = plates.

c) Use the definition of capacitance, . Note: capacitance is a positive quantity. d) Think: as decreases, what happens to capacitance (increase, decrease, stay the same)?

e) Think: should you increase or decrease the area of the plates to increase ? f) A common classroom demo uses plates with diameter with a spacing of . Determine a numerical value of capacitance in engineering notation with appropriate prefix. g) The air between the plates will break down and begin to conduct electricity when the electric field between the plates is about . If we connect a battery across the two plates (potential difference of ), how close can we bring the two plates together before a spark jumps the gap? h) At the spacing determined in the previous part, what charge magnitude is on each plate? i) How many excess electrons reside on the negative plate?

25.2 Assume you have two concentric conducting spherical shells. The inner shell has radius and carries charge . The outer shell has radius and carries charge . a) Determine the electric field in the region between the two shells. b) Integrate using to determine the potential difference between the two shells.

c) Use the definition of capacitance, . Note: capacitance is a positive quantity. As such, feel free to use when doing these types of problems to get rid of any minus signs. d) What if the outer shell is really large? Consider the limit as …

25.3 Consider two conductors shown I the figure at right. The inner conductor is a solid metal wire of length and radius . The outer conductor is a thin cylindrical shell of radius . Notice the two conductors share the same central axis. This arrangement of conductors is called a coaxial cable. For this entire problem let us assume the cable is very long ( ). Assume the inner conductor carries linear charge density while the outer conductor carries linear charge density .

a) Determine the total charge on the inner cylinder, the volume charge density and the surface charge density for the inner conductor. Hint: one of these is zero. b) Determine the electric field in the region between the conductors. c) Determine the potential difference between the inner and outer conductor. d) Determine the capacitance of the coaxial cable. e) Determine the capacitance per unit length of the coaxial cable. Extra practice: Assume reasonable numbers for all constants. Assume the inner cylinder is grounded (held at 0 volts). Determine the electric potential as a function of and plot it. Compare the slopes of the vs plot to the values on the vs plot. Parting note: remember, no cylinder is infinitely long in real life. These results are only valid if is really big compared to the radii involved. I suspect if is 20 times larger than any radius the calculations probably have less than 5% errors but I haven’t done the math.

25.3½ Challenge: Consider the last problem in the previous chapter (two long parallel wires). Use the results of that problem to determine the capacitance per unit length. Note: capacitance is a positive quantity. As such, feel free to use when doing these types of problems to get rid of any minus signs.

Simple Parallel Capacitor Circuit: Equivalent circuit:

C1 C2 C12

ℇ ℇ

Simple Series Capacitor Circuit: Equivalent Circuit:

C12

ℇ ℇ

Combining capacitors in PARALLEL Combining capacitors in SERIES 1 1 1

Ceq = C1 + C2 + ... = + + ... Ceq C1 C2

The voltages across C1 & C2 EQUAL the The voltages across C1 & C2 ADD UP TO the

voltage across the Ceq ! voltage across the Ceq !

The charges on C1 & C2 ADD UP TO the The charges on C1 & C2 EQUAL the charge on charge on the equivalent cap! the equivalent cap!

In both cases the stored energy in C1 & C2 ADDS UP TO the stored energy in the equivalent cap!

In all cases:

Big mistake in previous equations: people try to always use the voltage of the battery (that is only true for the capacitors in parallel with the battery!!!!). To get the correct charge on a capacitor you need the voltage across only that capacitor!

25.4 Q= Q= V= V= ∆ ∆ Q= U= U= V= 5 µF 15 µF ∆ U= 24 µF

8.25 µF

Q= ∆V= U= + 1.5 V Get C eq in series C’s have same Q ,

for the so Q 3.75 =Q 5=Q 15 …use to Q= two in find ∆V5 and ∆V15 (check: ∆V= series on U= those should add up to Q= the top ∆V3.75 ) ___ µF ∆V= U= 24 µF

8.25 µF Q= ∆V= U=

+ 1.5 V

Get C eq in parallel C’s have same ∆V , for the Q= Q= two in so ∆V3.75 =∆V12 =∆V8.25 …use to find Q 3.75 and Q 8.25 (check: ∆V= ∆V= parallel those should add up to Q 12 ) U= U= ___ µF 24 µF

+ 1.5 V Get C eq in series C’s have same Q , for the Q= so Q 8=Q 12 =Q 24 …use to two in find ∆V’s (check: those ∆V= series U= ___ µF should add up to

∆V8=∆Vbattery )

+ 1.5 V 81

25.4¼ Consider the circuit at right with two capacitors in parallel with an ideal battery of potential difference . Capacitor while .

First take a guess for parts A through D first without thinking too much. Then do parts E & F to check your intuition. A Is the equivalent capacitance more or less than ? B Which capacitor carries more charge? C Which capacitor has larger potential difference? D Which capacitor stores more energy? After doing your guess do the following: e) Draw the equivalent circuit. Determine as a simplified fraction times . Determine the equivalent potential difference in terms of the ideal battery voltage . Determine the equivalent charge as some fraction times . Determine the equivalent energy stored as decimal number times . f) Determine the charge, potential difference, and stored energy associated with capacitors & . Write all answers in terms of and . Simplify fractions for ’s & ’s while using decimals for the ’s.

25.4½ Consider the circuit at right with two capacitors in series with an ideal battery of potential difference . Capacitor while .

First take a guess for parts A through D first without thinking too much. Then do parts E & F to check your intuition.

A Is the equivalent capacitance more or less than ? B Which capacitor carries more charge? C Which capacitor has larger potential difference? D Which capacitor stores more energy? After doing your guess do the following: e) Draw the equivalent circuit. Determine as a simplified fraction times . Determine the equivalent potential difference in terms of the ideal battery voltage . Determine the equivalent charge as some fraction times . Determine the equivalent energy stored as decimal number times . f) Determine the charge, potential difference, and stored energy associated with capacitors & . Write all answers in terms of and . Simplify fractions for ’s & ’s while using decimals for the ’s.

25.4 Consider the two circuits shown at right. Circuit A Circuit B a) In circuit A are capacitors 1 & 2 in parallel, series, or neither? b) In circuit A are capacitors 2 & 3 in

parallel, series, or neither? c) In circuit A are capacitors 1 & 3 in parallel, series, or neither? d) In circuit B are capacitors 1 & 2 in parallel, series, or neither? e) In circuit B are capacitors 2 & 3 in parallel, series, or neither? f) In circuit B are capacitors 1 & 3 in parallel, series, or neither?

25.4 Imagine the following circuits using ideal batteries. Case 1 Case 2 Case 3 a) In which case(s) are the caps in series? b) In which case(s) are the caps in parallel?

25.4 You have a circuit with a battery and a capacitor connected in series. You have a several additional capacitors of identical capacitance. The voltage across the battery is and the capacitor stores charge . a) To increase equivalent capacitance, should I put an extra cap in series or in parallel with the first cap? b) Is it possible to insert a single extra capacitor without changing the charge on the first cap? If yes, state if the extra cap must be placed in series or parallel with the first cap? If no, explain why it is impossible to insert a single extra cap without changing the charge. c) Suppose you want to use more than one extra capacitor. You want the new circuit to store the same amount of total energy ( ) but each capacitor must have potential difference less than . Is this possible? If yes, draw a possible circuit schematic. If no, explain why it is impossible.

25.4 A capacitor network is built using an ideal battery as shown in the figure. The capacitances are , and . The stored charge on is . a) Determine the equivalent capacitance. Answer as a simplified fraction (or decimal # with 3 sig figs) times . b) Determine potential difference across the battery in terms of and . c) Determine the energy stored by . Answer in terms of and .

25.4 Easy to set up, but hard to finish…everyone should do the set-up: A capacitor network is built using an ideal battery as shown in the figure. The

capacitances are . The potential difference of the battery is . Capacitor has stored energy . Determine the required capacitance for . Answer as a simplified fraction times .

25.4 A capacitor network is designed as shown at right. Assume each capacitor has capacitance X X Y Z . One can connect to the circuit at nodes , , or . Note: a point labeled in circuit is usually called a node. a) Determine the equivalent capacitance of the network between nodes X & Y. Z b) Determine the equivalent capacitance of the network between nodes X & Z.

25.5 Here is another capacitor circuit. For this problem assume that C1=C, C2=2 C, C3=3 C, and C4=C. Determine the equivalent capacitance. Your answer should be written as some fraction times C (just like I would require on an exam). C2 C3 If you finish that up you can try to find the charge on each capacitor. Sometimes on an exam I would ask you to find the ℇ charge, voltage or stored energy for a particular capacitor of my choosing.

25.6 Here is another capacitor circuit. For this problem you may C1 assume that all capacitors have capacitance C. Determine the equivalent capacitance. Your answer should be written as some fraction times C. If you finish that up you can try to find the charge on each capacitor. Sometimes on an exam I would ask you C2 C3 C4 to find the charge, voltage or stored energy for a particular capacitor of my choosing. ℇ

25.7 An ideal battery is connected to a capacitor network. The network has C and 2 C in parallel with this combination in series with C. Assume the maximum energy allowable on the 2 C is U. Sketch a picture of the circuit. Determine the maximum allowable battery voltage that one can connect to such a circuit. Answer in terms of U and C. Hint: do the problem just like always where you assume the battery has potential difference . At the end of the problem you will know the energy of each capacitor. Use the energy equation for the known capacitor,ℇ in this case 2C, to solve for . ℇ 84

25.8a This time two capacitors are in parallel. Both capacitors have capacitance C when no dielectric is present.

The switch is closed and as such both C1 and C2 are fully charged. Draw the equivalent circuit and determine the equivalent capacitance, the charge on, voltage across, and stored energy of each capacitor (in both the equivalent and original circuit).

C1 C2

ℇ

25.8b Now a physics student wants to make you miserable. She first opens the switch AND THEN inserts a

dielectric material with constant κ (the Greek letter kappa) into the capacitor C2. Draw the equivalent circuit and determine the equivalent capacitance. In this case we know that the charge on the equivalent capacitor is unable change! HOWEVER, since C1 and C2 still form a circuit, some of the charge on C1 can flow over to C2 (or vice versa). In this case, the net charge is the exact same as Qeq from above, but the charges Q1 and Q2 are not the same for both cases! Use this information to figure out the charge on, voltage across, and stored energy of each capacitor after the dielectric is inserted.

C1 C2

ℇ

25.8c Use this to determine the change in the stored energy in the circuit ( ∆U = Uf Ui). Did the energy go up or down? How do the voltages on each capacitor compare? − Do a simulation online to discuss more…

25.9a This time two capacitors are in series. Both capacitors have capacitance C when no dielectric is present. The switch is closed and as such both C1 and C2 are fully charged. Draw the equivalent circuit and determine the equivalent capacitance, the charge on, voltage across, and stored energy of each capacitor (in both the equivalent and original circuit).

ℇ

25.9b Now a different physics student wants to make you miserable. WHILE THE BATTERY IS STILL

CONNECTED, he inserts a dielectric material with constant κ into the capacitor C2. Draw the equivalent circuit and determine the equivalent capacitance. In this case we know that the potential difference across the equivalent capacitor is unable change (it still must equal ∆V of the battery)! Use this information to figure out the charge on, voltage across, and stored energy of each capacitor after the dielectric is inserted.

ℇ

C2 86

Sometimes physics questions will have two capacitors, fully charged, disconnected from any circuit. These capacitors are then connected together. If they are connected together positive plate to positive plate and negative plate to negative plate. The following things happen: 1) A circuit is formed. 2) The voltage change around an entire circuit must be zero. We will learn later this is special way of saying energy is conserved. 3) From 2 we know the potential differences (after the caps reach steady state) are . 4) Excess charges can flow freely between the two upper plates (and between the two lower plates). 5) Since excess charges are not created or destroyed the two caps must split up the sum total of charge on each positive plate. We know .

If the plates are connected positive plate to negative plate only step 5 needs modification. We would instead use Think about it. Some of the excess electrons form a negative plate will move over to the positive side and reduce the excess charge on all plates. The two caps split up the difference between the charges instead of the sum.

25.10 Assume a capacitor is connected to a battery with potential difference . It is then Stage 1: Each cap is charged by some battery then disconnected disconnected from the battery. The charge on is . A different battery is used to charge to This cap is then disconnected from the battery. We are now at Stage 1 in the figures at

right… a) Determine the potential difference of the 2 nd battery. Answer as a simplified fraction times . Stage 2: Instant caps are connected and . Charges have b) Sketch Stage 2. Assume the caps are not had time to move yet. Draw the figure yourself. connected + to + and – to – but the charges have not had time to move. c) Determine the charge on each plate after the caps reach steady state. d) What if? Suppose we had instead connected the plates Stage 3: The caps reach steady state. Compute charge on each plate and sketch the final steady state picture. Draw the figure yourself.

25.11 What if the previous problem is redone but we connect the caps + to – and – to positive. Basically, assume is flipped upside down before the caps are connected to each other. Redo everything including drawing a new set of 3 pictures.

A note on the next 6 problems: On tests I prefer to use algebraic questions. That said, sometimes doing a few problems with numbers helps build intuition/understanding before doing the algebraic work. These will also help you understand my way of communicating for these types of problems.

25.11¼ Capacitor is charged using a ideal battery. We Before switch closed know the initial charge is and the initial stored energy is . Capacitor is then connected to the circuit shown at right while it is still charged. The plus and minus symbols indicate which plates of the capacitors are positively or negatively charged. Capacitor is initially uncharged. a) After the switch is closed (and equilibrium is reached), what are the After switch closed & charges on each capacitor? Said another way, determine ? equilibrium is reached b) What is the final potential difference across ? What is ? c) What is the final combined stored energy? What is ? d) Challenge: Even in this ideal system with zero resistance, some energy is lost. With zero resistance there is no energy loss due to heating of the ideal wires. How is energy lost from this ideal system even though the same amount of charge is still present and the wires don’t get hot?

25.11½ Capacitor is charged using a ideal battery. Capacitor is charged using a ideal battery. While each capacitor is still Before switch closed fully charged, they are connected in the circuit shown at right. Notice the positive plate of will be connected to the positive plate of upon closing the switch. After the switch is closed the system is allowed to reach equilibrium. a) Determine the initial total energy stored. b) After reaching equilibrium, which plates of the capacitors will be positive: top, bottom, or neither (both caps uncharged)? After switch closed & c) Determine the charge, potential difference across each capacitor after the equilibrium is reached system reaches equilibrium? d) Determine the energy loss expressed as a %.

25.11¾ Capacitor and are wired in parallel and Before switch closed charged using a ideal battery. The capacitors are then removed from this circuit and rewired as shown at right. The process was done carefully so no charge was lost from either capacitor while they were being rewired. Notice the positive plate of will be connected to the negative plate of upon closing the switch. After the switch is closed the system is allowed to reach equilibrium. a) Determine the initial total energy stored. b) After reaching equilibrium, which plates of the capacitors will be positive: After switch closed & top, bottom, or neither (both caps uncharged)? equilibrium is reached c) Determine the charge, potential difference across each capacitor after the

system reaches equilibrium? d) Determine the energy loss expressed as a %.

25.11 Capacitor and are wired in series and charged Before switch closed using a ideal battery. The capacitors are then removed from this circuit and rewired as shown at right. The process was done carefully so no charge was lost from

either capacitor while they were being rewired. Notice the positive plate of will be connected to the negative plate of upon closing the switch. After the switch is closed the system is allowed to reach equilibrium. a) After reaching equilibrium, which plates of the capacitors will be positive: top, bottom, or neither (both caps uncharged)? After switch closed & b) Determine the charge, potential difference across each capacitor after the equilibrium is reached system reaches equilibrium?

25.11 A circuit is connected using , & . Fully charged, switch re-opened The ideal battery has potential difference . The switch is closed and the capacitors are allowed to fully charge. After the caps are fully charged, the switch is

reopened. After the switch is reopened, is filled with a dielectric of . Assume will indicate after the dielectric is in the system. a) Determine the charge on and potential difference across each capacitor before the dielectric is brought into the system. Also determine the total

stored energy of the system before the dielectric is in place. b) Make this slightly more fun by speculating about the following questions before computing: Should equivalent capacitance go up or down? Will While switch open , across go up, down, or remain constant? What about the charges dielectric placed in on each capacitor… up, down or constant? What about ? c) Determine equivalent capacitance after the dielectric is in the system. d) Determine the charge on & potential difference across each cap after the dielectric is inserted. e) Determine the final stored energy .

25.11 An ideal battery has potential difference . A circuit is connected Fully charged, switch closed using , , and . Note: the value of capacitance for already accounts for the dielectric of which completely fills . The switch is closed and the capacitors are allowed to fully charge. After the caps are fully charged, the switch remains closed. Now the dielectric is removed from . Here indicates after the dielectric is removed. a) Determine the charge on and potential difference across each capacitor

before the dielectric is brought into the system. Also determine the total stored energy of the system before the dielectric is in place. Hint: same as previous question’s part a if you want to skip to the fun parts. While switch closed , b) Speculate before computing: Should equivalent capacitance go up or dielectric removed from down? Will across go up, down, or remain constant? What about the charges on each capacitor? What about ? c) Determine equivalent capacitance after the dielectric is removed. d) Determine the charge on & potential difference across each cap after the dielectric is remove. e) Determine the final stored energy .

25.12 Two caps ( C and 2 C) are connected in parallel with an ideal battery with ∆V= E. They are disconnected from the battery and each other. Then they are reconnected to each other, with the negative plate of C connected to the positive plate of 2 C. Also, the positive plate of C connected to the negative plate of 2 C. Which of the following situations are true/false after the unusual reconnection: a) The charges stored on the capacitors will stay in place on each capacitor. Each capacitor has the same charge on it as when they were fully charged by the battery. In other words and . b) The charges will move around in the wires. We should find that the total initial charge splits up amongst the two capacitors. . c) The charges will move around in the wires. Some of the initial excess charges will be cancel out on each capacitor. The remaining charge splits up amongst the two capacitors. . d) We can view the capacitors as being in series and use the series rules for finding the charge on and voltage across each capacitor. We can apply any series equations for finding an equivalent. e) We can view the capacitors as in parallel; each has same potential difference as battery. . f) We can view the capacitors as being in parallel with each one having the same potential difference each other. However, the potential difference across each capacitor is no longer E. Here: . ≠ g) We can view the capacitors as in parallel and apply parallel equations for finding equivalent capacitance. h) It is appropriate to use and . i) It is appropriate to use and . j) It is appropriate to use and . k) It is appropriate to use and . 25.13 Capacitor C is charged with a battery with potential difference E. A second capacitor, 2 C, is separately charged with a battery with potential difference E/2. They are each disconnected from their respective batteries. Then they are reconnected to each other. They are connected with positive plate to positive plate and negative plate to negative plate. Which of the following situations are true/false after the reconnection: a) We can view the capacitors as in parallel, each having the same potential difference as C. . b) We can view the capacitors as in parallel, each having the same potential difference as 2 C. . c) The charges will move around in the wires. We should find that the initial charge on each capacitor will be canceled out completely. . d) The charges stored on the capacitors will stay in place on each capacitor. Each capacitor has the same charge on it as when they were fully charged by the battery. e) The charges will move around in the wires. We should find that the total initial charge splits up amongst the two capacitors. In other words . f) We can view the capacitors as being in series and use the series rules for finding the charge on and voltage across each capacitor. We can apply any series equations for finding an equivalent. g) We can view the capacitors as being in parallel with each one having the same potential difference each other. The potential difference is neither E nor 2 E. . ≠ h) We can view the capacitors as in parallel and apply parallel equations for finding equivalent capacitance.

i) It is appropriate to use and . j) It is appropriate to use and . k) It is appropriate to use and . l) It is appropriate to use and .

25.14 Two capacitors with capacitance C1=C2= C are connected in series with an ideal battery with ∆V= E. They are disconnected from the battery but the capacitors still connected to each other as shown in the figure at right. At this point, someone changes the capacitance of one of the capacitors by decreasing the plate spacing on one of the capacitors. Which of the following situations are true/false after the change: a) We can view the capacitors as being in parallel, each one having the same potential difference as in the first case. . b) We can view the capacitors as being in parallel with each one having the same potential difference each other. However, the potential difference across each capacitor is no longer E. Here: . ≠ c) The charges stored on the capacitors will stay in place on each capacitor. Each capacitor has the same charge on it as when they were fully charged by the battery. In other words and . d) The charges will move around in the wires. We should find that the total initial charge splits up amongst the two capacitors. . e) The charges will move around in the wires. We should find that some of the total initial charge will be canceled out on each capacitor. Remaining charge splits up amongst the two capacitors. . f) We can view the capacitors as being in series and use the series rules for finding the charge on and voltage across each capacitor. We can apply the series equations for finding an equivalent. g) We can view the capacitors as in parallel and apply parallel equations for finding equivalent capacitance.

h) It is appropriate to use and . i) It is appropriate to use and . j) It is appropriate to use and . k) It is appropriate to use and . l) It is appropriate to use . m) It is appropriate to use .

WHAT IF? Suppose that, instead of disconnecting the battery and changing the plate spacing someone disconnected the battery and then connected a wire between the top and bottom points of the figure above. What, if anything, would be different if the plate spacing was changed (again with battery disconnected) before connecting the wire?

25.15 Two capacitors with capacitance C1=C2= C are connected in parallel with an ideal battery with ∆V= E. They are disconnected from the battery but the capacitors still connected (+plate to +plate, –plate to –plate). At this point,

someone decreases the plate spacing on one C2. Which of the following situations are true/false after the change: a) The charges stored on the capacitors will stay in place on each capacitor. Each capacitor has the same charge on it as when they were fully charged by the battery. In other words and . b) The charges will move around in the wires. We should find that the total initial charge splits up amongst the two capacitors. . c) The charges will move around in the wires. We should find that some of the total initial charge will be canceled out on each capacitor. The remaining charge splits up using . d) We can view the capacitors as being in series and use the series rules for finding the charge on and voltage across each capacitor. We can apply the series equations for finding an equivalent. e) We can view the capacitors as being in parallel, each one having the same potential difference as in the first case. . f) We can view the capacitors as being in parallel with each one having the same potential difference each other. However, the potential difference across each capacitor is no longer E. Here: . ≠ g) We can view the capacitors as being in parallel & apply parallel equations for finding equivalent capacitance. h) It is appropriate to use and . i) It is appropriate to use and . j) It is appropriate to use and . k) It is appropriate to use and . l) It is appropriate to use . m) It is appropriate to use .

25.16 Suppose you have two identical air filled capacitors with area and plate spacing . Now, a dielectric of constant fills the bottom half of . In the same dielectric fills the left half only. Which one has more capacitance? Are they the same? Do your results makes sense when ?

Do several more capacitor network problems from a book. I should write up the infinite capacitor chain as well…That is fun and actually relates to something in real life…

Also redo all these problems without looking at the solutions. 93

Resistivity and Current Electric current is a scalar given by Technically current is not written as a vector but we all think of it like one. We usually figure out how many amps of current flow in a wire and also which way charges flow. Strictly speaking, however, is not a vector.

26.1 A wire contains several moving charges. Each charge below is ±1 nC. In one second, the charges shown all pass through the cross-sectional area indicated by the shaded area. How much current flows in the wire?

+ + _ _ + +

Current density is a vector. It relates to current using

If current is uniformly distributed through a wire, we know where is the cross-sectional area of the wire.

26.2 Assume a metal pipe has length , inner radius and outer radius . Current , flowing from left to right in the pipe, is uniformly distributed. Determine the current density in the wire.

26.3 Assume a metal pipe has length , inner radius and outer radius . The pipe has a current density given by . Determine the total current through the pipe.

26.4 A metal pipe has length , inner radius and outer radius . Current flows radially from the inner sidewall of the pipe to the outer sidewall of the pipe. Total current through is . An end view of the pipe is shown. Where is current most dense: near the inner side wall or outer sidewall? Hint: current (not current density ) is the same through any cross-section in this scenario.

We don’t do a ton of stuff with current density but it does come in handy every once in a while…

Relating current to drift speed of electrons Suppose a known current flows through the slab of metal shown at right. How fast, on average, are the electrons moving? To figure this out, do the following:

First determine how many electrons are available to move in the material. One way to do this is to make a guess. For instance, I will assume one electron is available to move from each atom of the metal. Now I need to determine the number of atoms in the slab. From chemistry we know In this equation is Avogadro’s number. We can relate the mass of the slab to the mass density of the slab using ! This gives . Again, we assumed only 1 electron per atom is actually free to move throughout the metal.

WATCH OUT! Typically, physics books like to discuss electron number density ( ) instead of the number of electrons ( ). Divide both sides by volume to find electron number density. For aluminum and giving = Remember each aluminum atom has 13 total electrons but we assumed only 1 of those 13 is available to conduct.

Now relate current to average speed.

Here is the cross-sectional area of the slab through which current passes, is the magnitude of the charge on an electron, is number density (electrons available per unit volume) and is the drift velocity (average speed of electrons). People are often quite jealous of this equation…see how it is? For a sheet of aluminum foil with , and I found = 95

Resistance ( ) and resistivity ( ) are related but not the same thing. • Resistivity ( ) is a property of a material o any chunk of gold has resistivity Ω o say “ohm meter” not “ohm per meter” 44 ∙ • Resistance is a property of an object o Resistance depends on the length, cross-sectional area, AND resistivity of the wire o Resistance is measured in units of ohms using the symbol Ω

Analogy: Gold has a mass density. Any chunk of gold has the same mass density but every chunk of gold has a different mass. To get the mass of a particular chunk of gold you need some dimensions and the density (or a balance).

Every chunk of gold has the same resistivity but a different resistance. To determine the resistance of a chunk of gold you need not only the density but also the dimensions (or a DMM). Where the analogy breaks down: the resistance of a particular chunk of gold will also depend on which direction you try to push current…

Important equations for resistance & resistivity Equation Comments

This symbol is conductivity (not surface charge density) = This is Ohm’s law! In some materials conductivity varies depending on direction traveled through the material. Such materials are anisotropic = (pronounced “not on the test”).

Current density version of Ohm’s law = is resistance while is resistivity

is crossectional area and is length = Ohm’s law using current is potential difference (not potential) Δ Δ is the voltage across a resistor Δ = is current through a resistor is power delivered to a resistor Δ Some say is power dissipated by a resistor = is resistivity at , temperature in Celsius temperature coefficient of resistivity ° C = + − ° C Most materials = have …semiconductors have > < Here is resistance at at = + − ° C = ° C Sometimes handy. In this equation Δ Δ Δ Don’t forget…in this equation = = + Δ = − ° C

The Chacho Method I like to call this the Chacho method for three reasons • Chacho was one of the first students to tell me he liked this trick. • I needed some kind of name for it. • You try pronouncing Piuliye (Chacho’s given name). Note: no one else in the world calls this “the Chacho Method” but they aren’t having nearly as much fun as us.

The pictures are used to minimize algebra mistakes when using , and . = = = Consider the first circle. Suppose a problem asks you to find when both and are given. Cover up in the circle. Notice what is uncovered in the figure looks like . Notice from algebra . = In general, cover up the symbol you want to determine in any of the circles. Whichever symbols are left will appear in the correct orientation (fraction or multiplication).

It is not rocket science, but if it helps reduce mistakes I’m all for it. Some students find this helpful. And don’t forget what Uncle Chacho always says: Twinkle, twinkle, little star Voltage equals times . 97

26.5 At cylindrical gold wire has length , radius and . Ω a) °CDetermine the resistance of the wire. Answer = in engineering notation = with appropriate = 44prefix. ∙ b) By what factor must you change the length to decrease the resistance by 5%? c) By what factor must you change the radius to decrease the resistance by 5%? d) Suppose you heat or cool the wire wire and maintain the wire as some other temperature. What temperature must be used to decrease the wire’s resistance by 5%?

26.6 Consider two cylindrical resistors made from the same material. The first wire has diameter and length . The second wire has a third of the diameter and half the length. The first wire is connected to an ideal battery (negligible internal resistance) using ideal wires (negligible resistance). The battery has potential difference and the first wire carries current . a) What is the resistivity of theℰ material? b) What potential difference should be applied to the second resistor to drive the same current as the first wire? Again, assume ideal battery with ideal wires. c) Determine the ratio of power delivered to the second resistor to the first resistor.

26.6½ A common unit used to rate batteries is . One particular AA battery is rated at . V a) What does this unit represent? Energy, voltage, ∙ charge, current, or something else? Please convert ∙ to an SI unit. Hint: recall current is defined as . b) This battery ∙ is used to run a device for 90.0 minutes. =How much charge is left in the battery? W c) Assume one can use all of the battery’s stored energy at a constant rate. How long can the battery operate the device? Note: real life operation time will be less than this. W 26.6¾ When you get an electric bill, typically you are charged by the . For instance, assume some electric W company charges at a rate of $0.12 per . k ∙ W a) What is in SI units? k ∙ W b) Supposek you plug∙ in your phone charger and leave it in all month (30 days). For 90 minutes a day you are actually charging your phone consuming power at a rate of about . The rest of the time your charger W is still using power but at the much lower rate of about . How much does it cost per month to W 4 charge your phone? What percent of the total cost is vampiric4 (see note below)? The power used by devices plugged in but not in use is called vampire power, vampire draw, or ghost load…

26.7 A light bulb has a power rating of 60W. This is rated assuming it is used with a 120 Volt source. Although it is rated for an AC source which we have not discussed, our power equation is still valid. Assume the length of the filament is 58 cm while the diameter of the filament is 0.045 mm. The filament is tungsten which has a resistivity of 5.6x10 -8 at 20°C. At 3000 °C, the resistivity of tungsten is about 105x10 -8. a) Determine the temperature coefficient for the resistivity of tungsten using the two numbers listed. b) Now determine the resistance of the bulb while it is in operation. Note: the temperature of the bulb while it is operating is unknown. c) Determine the resistivity of the bulb while it is operating. d) Estimate the temperature of the bulb while in operation. Compare this to the melting point of tungsten at about 3400°C. e) As the bulb is left on, some of the tungsten actually vaporizes. What should happen to the radius of the filament, the resistance of the bulb, and the brightness of the light over time? The vaporized metal eventually becomes a thin solid layer on the inner surface of the bulb. This process is called deposition. This also tends to make the bulb less bright. Vapor deposition can be used to get a thin layer of silver on a glass plate (to make a mirror) or in making the tiny wires in semiconductor chips!

Side note: Evidently in Europe around 1900 they used osmium filaments. It was so expensive you could recycle broken bulbs! Other uses of osmium are rare (very special applications).

26.7¼ A cylindrical wire has resistivity R (µΩ ) . An experiment is set-up 35 Ω to measure the resistance between the two = ∙ 30 ends of the wire. A plot of resistance versus length is shown at right. 25

Determine the diameter of the wire. 20 Hint: if , what is equal to the slope 15 of the graph? = 10

0 L (mm) 0 200 400 600 800

26.7½ A cylindrical wire is used as a resistor by connecting I (mA) to each end. The resistor has length and diameter 140 (about twice the diameter 8of an average human hair). Room temperature is standard . The wire is μ ℃ 120 connected to a variable power supply and the potential difference across the resistor is slowly increased from 100 to . The current for each applied voltage is V V recorded and plotted as shown at right. Notice the curve 8 80 bends at high voltage as the wire gets hot. We are assuming Ohm’s law is valid for the wire ( ). 60 Δ a) Determine the wire’s resistance at standard = temp ( ). 40 Δ ≲ V b) Determine 4 the resistivity of the wire. c) Determine the wire’s resistance at . 20 Δ V d) For , the temperature of the resistor is Δ V = . Determine the temperature coefficient of 0 ∆V (V) ℃ = resistivity. 0.0 2.0 4.0 6.0 8.0 e) Determine the temperature of the wire when operating at . Δ V f) For non-ohmic regions = of a plot, differential resistance is . Determine the differential resistance for the high temperature region ( ). = Δ ≲ V While unimportant to us now, differential resistance is a useful4 concept for semi-conductor device physics. Note: for an ideal (perfect) ohmic device, differential resistance is always equal to the resistance (regardless of applied voltage). In general, and in real life, the slope of an -plot at some arbitrary applied voltage does NOT give resistance. 26.7¾ Imagine an experiment similar to the previous one germanium with instead of the metal nichrome. Again, assume the temperature increases for . Sketch a plausible plot. How does the magnitude of current Δ change? How does the slope change for if we assume comparable temperature changes? Δ ≳ 4V ≳ 4V 100

26.8 In some special instances the resistance of a wire can be designed so that its resistance is not affected by temperature. One way to do this is to attach a cylindrical piece of carbon wire and to a cylindrical piece of nichrome wire end to end (see figure below). In this case we say that the carbon wire and the nichrome wire are resistors in series. In series the total resistance of the two wires is just the sum of the two resistors (opposite the rule for combining capacitors).

In this case the combined resistance is R1 + R2 = 10.0 Ω and is independent of temperature. The length of the carbon

wire is L1 and the length of the nichrome wire is L2. The radius of the each segment of wire is r = 2.0mm. The resistivity of carbon is 3.5 ×10-5 Ω m with temperature coefficient of resistivity -0.5 ×10 -3 (°C) -1. The resistivity of -8 -3 -1 nichrome is 110 ×10 Ω m with temperature· coefficient of resistivity 0.4 ×10 (°C) . Note: there are several different· types of nichrome used in making wires and some of these have different resistivities than the one listed for this problem.

L1 L2

Determine L1 and L2. Hint: consider the fact that the net change in resistance is zero for all temperatures (not true when the wire gets extremely hot but ok for the purposes of this problem). Bonus: suppose you wanted a similar wire (10 Ω of resistance independent of temperature using carbon and nichrome wires in series). Suppose you wanted this wire to be 100 times shorter. How could this be done? What problems might you have with this type of wire?

26.8½ Consider a resistor made from a solid cylinder of carbon coated with a thin Side view of resistor shell of nichrome. Connections are made to each face of the cylinder such that current flows in the direction shown. We will learn later this system is essentially two resistors in parallel with net resistance given by Direction of current flow

= + If you care, this can be rewritten as

= Assume is the unknown resistance of the solid+ cylindrical carbon core while is the unknown resistance of the nichrome shell. The total resistance is when the length is . The resistivity of carbon is while the resistivity of nichrome is . Assume the temperature is . The temperature coefficient of resistivity for carbon is ℃ while for nichrome assume = − < ℃

We are told the shell & core have equal resistance. = +4 > ℃ a) Determine the resistance of the core. b) Determine the diameter of the core. c) Determine the shell’s thickness. d) As current runs in the resistor it heats up. What happens to the resistance of this resistor as it heats up? Does resistance increase, decrease, or remain constant?

101

26.8 ¾ When you plug in an incandescent light bulb, it is initially room temperature. Within a short period of time, the bulb rapidly warms. The entire time the bulb has approximately constant voltage across the filament. Sketch a plausible plot of filament current in the bulb versus time as it reaches operating temperature. Keep in mind plots are always drawn y vs x; whatever comes first in the sentence in on the vertical axis.

26.8 Suppose you take a nichrome wire and use it as a heating element. You connect the wire to a variable output power supply and gradually ramp up the voltage from zero to . Note, when voltage is applied across the wire, it gets very hots and glows…Sketch a plausible plot or current versus applied voltage for the wire. Keep in mind plots are always drawn vs ; whatever comes first in the sentence in on the vertical axis.

-8 . -3 -6 26.9 ρAl =2.82x10 Ω m, αAl =3.9x10 (temp coef. of resistivity), αAl =24x10 (linear expansion coef.), c Al =900 J/(kg .°C) is specific heat of aluminum. Given a cylindrical wire of length 0.10 m at 20°C. Diameter of wire is about 0.500 mm. Wire is made of aluminum. Density of Aluminum is 2.7 g/cm 3. Wire is connected to 9.00 V battery. As a result a large current flows and the wire gets heated to 120.0°C.

a) Determine Resistance of wire at 20°C. b) Determine current that flows through wire at 20°C. c) Determine Resistance of wire at 120°C. d) Determine current that flows through wire at 120°C. e) Determine the average current by averaging the current at 20°C and at 120°C. f) Use this average current to find the Power ( ) dissipated by the resistor using =I ∆V. g) Now determine the mass of the wire.

h) Determine the heat required to increase the temperature to 120°C. Use Q=mc Al ∆T. i) Estimate the time required for the wire to heat up. Hint: . Try checking the units! ? ?

102

Calculating resistance using geometry and, if needed, calculus Suppose we build up the resitance of a shape using

= In this equation you must know • You should only use this formula if either the cross-sectional area or resistivty is non-uniform • Your shells must be designed such that current flows through the area of the shell • In this equation the thickness of the shell is like the length of a resistor • Usually is uniform. If is non-uniform, usually you are given a formula to plug in.

26.10 A metal pipe has uniform resistivity , length , inner radius and outer Case 1 Case 2 radius . In case 1 (side view shown) current flow longitudinally through the pipe. In case 2 (end view shown) current flows radially through the pipe. a) Determine the resistance for case 1. b) Use to determine the resistance in case 2. c) For what length= are the two resistances equal. Answer in terms of .

26.11 Determine the resistance between two concentric spherical shells. Assume resistivity , inner radius and outer radius .

26.12 A piece of metal has constant resistivity ρ and is shaped such trapezoidally as shown in the figure. Height only varies from the left end to the right end. Note: the change in height is exaggerated in this picture b (compared to real-life proportions) for ease of drawing. a) Determine resistance between the front (light grey) and back faces. b) Determine resistance between the left and right (dark grey) faces. a c

d 26.13 Suppose now that the height was actually varying parabolically as shown in the figure (pure side view). The depth into the page is still c. y Assume the parabola’s minimum occurs at the left face. a) Determine an equation for the height y and as a function x using the coordinate system shown. b) Set-up the integral for determining the resistance between the left and b right faces of this piece of metal (resistivity still ρ). c) What happens to your equation as a goes to zero? Does this make a sense? How much resistance should there be if a ≈ 0…a little or a lot? d) Suppose you again tried to determine the resistance between the front x and back faces. Would you need to us calculus or not? Explain and set d up the problem. e) Explain, in general, how you would handle a problem that had a non-uniform resistivity.

103

What do the ratings on light bulbs (and other appliances) imply? Notes on the discussion below: strictly speaking the household socket uses AC power which can require different mathematics. For problems using only resistance (no capacitance or inductance) the mathematics of household circuits is identical to those with DC sources (batteries). That said…

26.14 Suppose you now have a 40 W light bulb and a 100 watt light bulb. Both are designed to be used with a 120 V socket. When you place the bulbs in series and connect them to the 120 V socket, which bulb will appear brightest?

26.15 Suppose you have a 40W light designed to be used in a standard household socket. You know that the standard household socket operates at 120 V. This means the bulb will use 40 W of electrical power when the voltage across the bulb is 120 V. What is the resistance of the bulb? Does this affect your reasoning/answer to the previous question?

26.16 Consider two liquid filled cylinders acting as resistors in a parallel circuit. Then ions are added to one of the cylinders which have the effect of lowering the resistivity of the fluid in that chamber. Assume the resistivity is lowered by 25%. Determine the ratio of power delivered to that chamber after the solution is added to the power before the solution is added.

104

105

Resistor Circuits: show the basic variations (also show how batteries in series/parallel affect things)

Simple Series Circuit Equivalent Circuit

R1 + + R12 - -

Simple Parallel Circuit Equivalent Circuit

+ + R12 - R1 R2 -

Combining resistors in PARALLEL Combining resistors in SERIES 1 1 1

= + + ... Req = R1 + R2 + ... Req R1 R2

The voltages across R1 & R2 EQUAL the The voltages across R1 & R2 ADD UP TO the

voltage across the Req ! voltage across the Req ! The currents through R1 & R2 ADD UP TO the The currents through R1 & R2 EQUAL the current through the equivalent resistor! current through the equivalent resistor!

In both cases the power delivered to R1 & R2 ADDS UP TO the stored energy in the equivalent cap!

In all cases:

and

Big mistake in previous equations: people try to always use the voltage of the battery (that is only true for the resistors in parallel with the battery!!!!). To get the correct current through a resistor you need the voltage across only that resistor!

106

27.1 I= I= V= V= ∆ ∆ I= P= P= V= 5 Ω 15 Ω ∆ P= 24 Ω

30 Ω

I= ∆V= P= + 1.5 V -

Get Req I= in series R’s have same I, for the so I 20 =I 5=I 15 …use to find ∆V= two in P= ∆V5 and ∆V15 (check: I= series on those should add up to ___ Ω ∆V= the top V20 ) P= ∆ 24 Ω

30 Ω I= ∆V= P=

+ 1.5 V -

Get Req in parallel R’s have same ∆V , for the so V20 = V30 = V12 …use to I= I= two in ∆ ∆ ∆ parallel find I 30 and I 20 (check: those ∆V= ∆V= should add up to I 12 ) P= P= ___ Ω 24 Ω

+ 1.5 V - Get Req in series R’s have same I,

for the so I 24 =I 12 =I 36 …use to find I= two in ∆V’s (check: those should ∆V= series P= ___ Ω add up to ∆V36=∆Vbattery )

+ 1.5 V - 107

Assume each light bulb is actually a resistor with resistance R. Assume the power supply has negligible internal resistance and potential difference E. For each circuit (other than the first one), draw the equivalent circuit. Determine the current through, voltage across, and power delivered to each bulb in terms of E and R. Use this information to determine the total power delivered by the battery in each circuit as well. 27.2 Circuit 1:

Bulb 1 + -

27.3 Circuit 2: Initially the switch is open. Determine the power delivered to each bulb.

Bulb 1 + - Bulb 2

How should the brightness of bulb 2 change when the switch is closed? By what factor will the brightness of bulb 1 change after the switch is closed?

27.4 Circuit 3: Initially the switch is closed. Determine the power delivered to each bulb.

Bulb 1 + Bulb 2 -

How should the brightness of bulb 2 change when the switch is opened? By what factor will the brightness of bulb 1 change after the switch is closed?

108

27.5 Circuit 4: Initially the switch is open. Determine the power delivered to each bulb.

Bulb 1 + Bulb 3 - Bulb 2

How should the brightness of bulbs 1, 2, and 3 change when the switch is closed? By what factor will the brightness of bulb 1 change after the switch is closed? By what factor will the brightness of bulb 3 change after the switch is closed? Does the battery deliver more power when the switch is open or closed? Determine the ratio power delivered with the switch open to power delivered with the switch closed. Answer as a simplified fraction.

27.6 Circuit 5: Initially the switch is closed. Determine the power delivered to each bulb.

Bu lb 1 Bulb 3

+ -

Bulb 2

How should the brightness of bulbs 1, 2, and 3 change when the switch is closed? By what factor will the brightness of bulb 1 change after the switch is closed? By what factor will the brightness of bulb 2 change after the switch is closed? Does the battery deliver more power when the switch is open or closed? Determine the ratio power delivered with the switch open to power delivered with the switch closed. Answer as a simplified fraction.

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27.7 Circuit 6: How should the brightness of bulbs 1 and 2 compare the bulb in Circuit 1 ? How does the power delivered by each battery compare to the power delivered by the battery in Circuit 1 ?

CAUTION: if doing a demo AND using power supplies, be sure you float one of the power supplies to avoid a dangerous ground loop issue…

+ Bulb 1 -

+ Bulb 2 -

27.8 Circuit 7: How should the brightness of bulbs 1 compare the bulb in Circuit 1 ? How does the power delivered by each battery compare to the power delivered by the battery in Circuit 1 ?

CAUTION: I do not recommend doing this demo using power supplies. Why? Even a small dissimilarity between the power supply voltages can cause problems with the left loop. More after we know KVL…

+ + Bulb 1 - -

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27.8¼ Up to now we have always assumed batteries are ideal (no-internal resistance). A better model for a battery is to assume an ideal battery plus a small internal resistance in A series. Assume our non-ideal battery is placed in the circuit shown at right. a) While the switch is open, how much current flows through resistor ? Don’t overthink it. b) If one were to measure the voltage across the battery terminals A & B while the ℰ switch is open , what voltage would be observed? I care about the size of , not the sign. c) Now assume the switch is closed. Determine the current in the circuit. d) If one were to measure the voltage across the battery terminals A & B while the B switch is closed , what voltage would be observed? I care about the size of , not the sign. In particular, is this result bigger or smaller than part b? e) Determine the power delivered to resistor . f) What value of load resistance will receive maximum power from the battery? g) Make plots of current vs , potential difference vs. R and power delivered versus . Use values for between and assuming & . Think about the best choices of prefix to make the data look as clean as possible. = ℰ = Note: if the load resistor is tiny (think short-circuiting the battery) we get max current in the circuit but most of the voltage is across the internal resistance and the battery burns up. If the load resistor is huge the internal resistance is negligible but not much current flows. If the load resistor is “just right” we get some voltage across each resistor and a large current giving maximum power delivered to the load resistor. Adjusting load resistance to optimize power transfer is an example of “impedance matching”. Impedance matching shows up in acoustics, optics, and collisions in mechanics and that is why I thought it was worth mentioning.

27.8½ Wheatstone Bridge The circuit shown at right is called a Wheatstone bridge. This type of circuit is used when someone wants to accurately determine the unknown resistance in terms of the other resistances (which are known). Resistor has a sliding contact which allows one to change its resistance. In practice, one adjusts the value of until points B & C are at the same A potential (same node voltage). In lab one verifies nodes B & C are at equal potential by connecting a sensitive ammeter which measures zero current.

Assuming points B & C are at equal potential, show the value of is ℰ B C = If you are clever you need not do massive amounts of computation. Hint: If you don’t see any clever tricks, first compute and get the D current through each resistor like always (pretend is known). After you have results, eliminate the unknown parameters ( and the currents). This way takes a while but it is good practice. Both soℰlutions are shown and are good practice.

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27.8¾ Resistor ladder circuits (infinite chain of resistors)

Suppose you have an infinite array of identical resistors as shown. The circuit shown at right is useful in Analog to Digital (ADC) conversion. A Similar circuits relate to models used in telecommunications transmission lines. Personally, I just think they are cool.

a) Determine the equivalent resistance when & . b) Determine the equivalent resistance when = .= = = B Hints for solving: • The standard trick is to compare the resistance of the entire network to the resistance of only the portion to the right of nodes A & B. If you don’t see anything, try the next hint. • If you want to use a computer, assume . Start with a non-infinite chain (say the first two loops) and compute . Then add one more rung = and recompute. Repeat this procedure until your answer converges (stops changing in the 3 rd sig fig). In theory you could continue in this manner until you obtain whatever arbitrary precision is required for your result.

Side note: a web search for “infinite resistor chain” came up with a bunch of different ways to solve this problem I had never thought of. Some used special techniques you would learn in higher level courses.

27.8 A variable resistor is made from a long skinny wire with the total length providing resistance . One can tap into the long skinny wire some distance from the top end of the wire. This effectively spits the wire into two resistors. a) Recall resistance of a wire is proportional to length. Determine expressions for the resistances of the top and bottom segments of the ℰ wire shown in the figure at right. In the solutions I call them & respectively. Answer in terms of , & .

b) Determine equivalent resistance of the circuit in terms of , & R. c) For the special case of are the resistors in parallel, series, or something else? = d) For the special case of are the resistors in parallel, series, or something else? e) Determine the power delivered to resistor . I wrote my final answer in terms of , & to keep = things manageable and easy to read. ℰ f) What value of delivers max power to ? Hint: consider the special cases for . Does it depend on the size of the resistances used? g) Assume , , and . Make a row of constants in a spreadsheet = program. Tabulate = in ℰincrements = of 0.5 mm. = Make a second column for , a unitless parameter that will make computation easier. Then use your table to compute , , the= voltages across each resistor, the total current, the current through each resistor, the total power delivered by the battery & the power delivered to each resistor. Make plots showing the currents and the powers delivered versus . Tip: use a prefixes as to make the data look as clean as possible (get rid of leading/trailing zeros).

h) Now try changing the constants to . Now try . How does the plot change in each case? Which scenario allows you to =vary the power delivered = to with minimal power delivered to and ? This might be a good time to do a web (or video) search for “rheostat”, “potentiometer”, or “rheostat versus potentiometer”. You can see pictures of what the devices look like in real life and see practical applications.

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KVL states that the sum of the ∆V’s around an entire loop must be zero (basically conservation of energy). The only tricky part comes in assigning signs for each ∆V and being patient with the algebra in a long problem. Physics KVL and KCL sign conventions:

+ + i1 i2

i3 i4 Loop direction exits on + Loop direction exits on - terminal of battery, ADD terminal of battery, the ∆V of the battery SUBTRACT ∆V of battery Current in = current out i1 + i2 + i4 = i3

i i INs – OUTs = 0 i1 + i2 – i3 + i4 = 0

Loop direction matches the Loop direction opposes the direction drawn for the current , direction drawn for the current , SUBTRACT ∆V = iR for that resistor ADD ∆V = iR for that resistor

27.9 Notice that i3 is exactly the opposite of the direction we know it will flow…that is ok. Upon completing the i = math for this problem we should find out that 3 -1A. The negative sign in the final answer implies that the current i3 is opposite the direction drawn. Can you tell that i1=2A and i2=1A without doing KVL? 10 Ω 5 Ω

i2 i1 i3 L2

10 Ω L1 20 V

27.10 Write down KVL for the two obvious loops. R 2R

i3 i2

i1 L1 R E

L2 2E

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27.11 Consider the circuit shown at right. Assume and are known. ℰ Before writing down any equations answer the following: a) How many unknown currents appear? b) How many loop equations should you use?

c) How many junction equations should you use? ℰ ℰ d) Determine the current in each branch of the circuit.

27.12 Consider the two circuits shown at right. Assume the ideal battery Circuit 1 voltage is and the resistances & are known. ℰ = = a) We currently have two techniques to determine currents

(series/parallel rules vs KVL/KCL). KVL/KCL works for all circuits while series/parallel rules (which are generally easier) only work for certain cases. For which circuit is it easy to series/parallel rules.

b) Determine the power delivered to resistor for circuit 1. c) Determine the power delivered to resistor for circuit 2.

At this time you may find it interesting to do a web search for “delta y transformation”, “delta star transformation”, or “delta why”. This Circuit 2 transformation converts a circuit like circuit 2 into one like circuit 1 and allows you to solve without KVL/KCL! That said, I do not recommend learning this for your physics courses…

ℰ

27.13 Update soln? Perhaps one wants to charge a battery using the circuit shown at right. The max current one can use to charge higher the battery is known as . Any charging current will destroy the battery. Model the charging battery (left battery) as an ideal battery with unknown potential difference with small resistance in series . Model the dead battery as ideal battery of potential difference with large resistance in series. Assume . a) Determine an= algebraic expression for the potential difference for the charging battery. b) Assume . Determine numerical values for each current and potential difference ℰ . = = = c) As the circuitℰ runs, we expect the internal resistance of the dead battery to decrease.

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27.14 What about a problem with variable resistor that is tapped in the middle? Done in 27.8 7/8.

27.15 What about the infinite chain to model cable subscribers… Done in 27.8 ¾.

27.16 Assume a string of Christmas lights is wired with all bulbs in series. The lights are designed such that whenever a single bulb burns out, a piece of metal will bypass that bulb to keep the entire string lit. Assume the sting of lights runs off a standard household outlet. For this chapter, it suffices to say the circuit can be accurately modeled as a 120 V source. At operating temperature, the power of the entire string of lights is 130 W. a) Determine the resistance of a single bulb. b) Suppose a single bulb burns out. By what percentage does the power to the entire strand change? Also, by what percentage does the power delivered to each remaining bulb change? c) Suppose the entire strand has a fuse set to blow if the current exceeds 1.5 A. How many bulbs must burn out for the fuse to blow?

27.17 All resistors in the network shown have resistance R except for the i2 resistor connected to power supply V3 which is labeled with resistance r. i3 Write down three loop equations for the loops shown at left. Your answers i1 should be in terms of the currents shown, R, r, and the power supply V1 L2 V L1 2 voltages V1, V2, and V3. Determine the appropriate number of junction equations needed to i4 determine the currents shown. Write down those junction equations. i5 Write the equations in matrix form. In theory one could use that matrix to solve for the currents. That is not what I intend to test you on but you are welcome to use matrix methods for solving the problems (but not your L3 graphing calculators). V3

i6 r 116

RC transient circuit: An RC transient circuit uses DC voltage (a battery) in a circuit with resistance and capacitance. When the switch is opened or closed there is a transition period (usually very brief) where the current through and voltage across the various R’s and C’s change as a function of time. Equation for RC transient circuits: where Above equation is valid for determining current = and +voltage − but not energy or power!= Energy or power (as a function of time) can be found by first determining the current or voltage and then plugging that equation into the equation for energy or power.

C C C

ℇ ℇ ℇ R R R

t = 0- (switch open long time) t = 0+ (instant after switch is closed) t = ∞ (after switch closed long time)

C initially uncharged C preserves voltage from t = 0- C is now fully charged

When C is uncharged it acts like a When C is fully charged it acts like an short circuit (acts like a closed switch) open circuit (acts like an open switch) and allows maximum current to flow! and allows no current to flow!

27.18 Now consider some time t between the t = 0+ and t = ∞ pictures shown above. At this time t the capacitor is partially charged up. As a result it allows some current to flow but not the maximum amount. This means some voltage drop is across both C and R. C a) Write down KVL going around the loop for this arbitrary time . b) Find current in the resistor as a function of time and sketch a graph of the function. c) Find charge on capacitor as a function of time and sketch a graph of the function. d) After a time equal to five time constants ( t= 5RC), what is the current in the ℇ resistor. R e) How long for the current to reduce by 75% from the maximum value? Note: this is the same thing as saying “How long for the current to drop to 25% of the maximum value”? Watch out for this type of wording!!! 117

27.19 A circuit is initially connected in position A for a long time. Then the switch is thrown from position A to

position B. Draw the t= 0+ and t= ∞ pictures. In each picture label the voltage across each element in the circuit. Think about that 2 R resistor and how much it really matters… Determine the energy stored in the capacitor as a function of time.

AAA BBB

ℇ R

27.20 A circuit is initially connected in position A for a long time. Then the switch is thrown from position A to

position B. Draw the t= 0+ and t= ∞ pictures. In each picture label the voltage across each element in the circuit. Will the 2 R resistor matter this time? How so? Think about doing KVL at a random point in time t between the

t= 0+ and t= ∞ pictures. Or think of it this way: from the capacitor’s perspective (between the t= 0+ and t= ∞ pictures) the R and 2 R are in ______.

AAA BBB

R ℇ

R 2

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27.21 A circuit is initially open for a long time. Then the switch is closed. Draw the t= 0+ and t= ∞ pictures. In each picture label the voltage across each element in the circuit.

R ℇ

At this point we are about to exceed what is a reasonable expectation of students in an introductory physics class.

To be clear, you can reasonably be expected to determine the state of the system for the t= 0+ and t= ∞ pictures and use that information in a problem. However, it is unreasonable to expect you to determine the time constant in this system. You might think it is obvious (especially if you’ve had circuits) but it is non-trivial in general to determine the time constant for this type of circuit. If you are interested in learning more look up Thévenin equivalents or try to visualize how the resistors appear to the capacitor. In this case the R and 2 R are in parallel from the capacitor’s perspective which makes the time constant τ = 2/3 RC ! Note: I may do simple and obvious cases like the previous problem on exams.

27.22 ACTUALLY SOLVED IN SOLUTIONS Consider the circuit at right with the switch initially open for a long time. The switch is closed at time then allowed to reach steady state. a) What is the charge on and voltage across each capacitor the instant after the switch is closed? ℰ b) What is the current through and voltage across each resistor the instant after the switch is closed? c) What is the current through and voltage across each resistor once the circuit reaches steady state? d) What is the charge on and voltage across each capacitor once the circuit reaches steady state?

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27.23 Suppose you have the circuit shown in the right figure. Initially the circuit is open and has never been connected and the capacitor was initially uncharged. C Assume that each resistor has resistance R. All answers for this question will be in terms of R, C, and . ℇ R ℇ 1 a) Redraw the circuit for the instant after it is closed. Clearly label this new figure with the following items: R2 1. The currents through and voltages across each resistor. 2. The stored charge on and voltage across the capacitor. b) Redraw the circuit for a point in time a long time after the switch was closed. Clearly label this new figure with the following items: 1. The currents through and voltages across each resistor. 2. The stored charge on and voltage across the capacitor. Use parts a & b to create determine functions for current through the battery and voltage across the capacitor as functions of time. Sketch plots of each. One of these will seem a little different than usual… c) Now redo parts a & b if the circuit was initially closed for a long period of time and then suddenly re-opened.

Using the case described in part c, determine the current through R2 as a function of time. Plot the current through

R2 as a function of time.

Still using the case described in part c, also determine the power delivered to R2 as a function of time. Plot power

delivered to R2 versus time. Note which curve, or , falls off more rapidly with time. 2 Finally, use this result to determine the time at which resistor R2 has power /10 R delivered to it. d) Now compare and contrast the situation described in a/b with the situationℇ described in c. In particular, imagine that each resistor was a light bulb. How would the brightness of each bulb vary in time after the switch changes from open to closed (in a/b) or from closed to open (c)?

27.24 A battery has potential difference . It is connected in series to a resistor and capacitor with . In parallel with the capacitor is a special flash bulb. The bulb is similar to a spark gap. The bulb acts like a = Bulb short circuit any time the potential difference across it exceeds . The ℇ = circuit is design so the capacitor will charge, when the cap hits the bulb will flash. Assume the time to flash is completely negligible compared to the charging time constant of the capacitor. You may assume the capacitance of the spark gap bulb is negligible compared to . a) What capacitance should be used to make the bulb flash 3 times per second? b) Determine the power delivered to the resistor as a function of time. c) Determine the total energy usage per second. d) Suppose you want bulb to sit on a sign to warn motorists of some hazard. The bulb needs to flash non-stop for 2 weeks. How much charge (in ) is required of your battery? e) Suppose we want 2 flashes per second. One option is to change the capacitance. Another option is to change the resistance. Either option should require less total charge and make for a cheaper battery. Is there any advantage to changing capacitance instead of resistance (or vice versa)?

Disclaimer: I made up these numbers. After running the math, they are probably a bit unrealistic but I’m too lazy to rework this question. That said, the problem is good practice even if my bulb flash is probably too dim to be seen.

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Ch28 Magnetism

Show Phet of magnets and show magnetic levitation demo, large magnet with a box of paper clips and chain Still have like poles repel and opposite poles attract.

The earth has a magnetic field (bring two dip compasses) showing earth has component of magnetic field parallel to surface AND perpendicular to surface.

Mag fields are measured in Tesla and Gauss.

Earth’s field is approximately 0.5 Gauss near the surface, MRI about 1 Tesla ( ). 1 = 1 Biggest fields on earth can be 20 or 30 T in special small spaces at low temperatures.T G

Just like everything is conducting (if you use enough voltage), everything exhibits some type of magnetism…you just have to use massive magnetic fields to see it (example: a frog has been levitated by a magnet…).

Magnetic forces are exerted on other magnets and charges MOVING PERPENDICULAR (or partly perpendicular to the mag field B).

Mag force on a moving charge is = × Right hand rule 1a determines direction

RHR1a – Determine Direction of force on moving charge (relates , , and ) 1) Point fingers of RIGHT hand in direction of velocity (direction charge moves) 2) Curl fingers of right hand towards (might need to try various hand orientations, flip hand over) 3) Thumb of right hand points in direction of (for positive charge) 4) IF NEGATIVE CHARGE, thumb points OPPOSITE direction of force

Magnetic fields also exert forces on wires Mag force of a uniform field on a current carrying wire is = × points in direction current flows (strictly speaking current is not a vector). Right hand rule 1b determines direction

RHR1b – Determine Direction of force on moving charge (relates , , and ) 1) Point fingers of RIGHT hand in direction of current 2) Curl fingers of right hand towards (might need to try various hand orientations, flip hand over) 3) Thumb of right hand points in direction of force

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Solutions use black for blue for and green for 28.1a) An electron moves to the right through a 28.1b) A proton moves into the page through a region with a magnetic field. Here the dots region with a magnetic field. Here the arrows indicate the magnetic field points out of the plane. indicate the magnetic field points to the right. The The dotted arrow indicates velocity. Sketch the dotted × indicates the direction of the velocity. direction of the force on the electron. Sketch the direction of the force on the proton.

× + -

28.1c) A proton moves up and to the right in the 28.1d) An electron moves downward (dotted direction shown with the dotted arrow. A arrow) in a region where a magnetic field points magnetic field points into the board as indicated by upward (solid arrows). Sketch the direction of the the ×’s. Sketch the direction of the force on the force on the electron. proton. -

28.1e) A proton moves into the page (shown by 28.1f) An electron moves in the presence of a dotted ×). The proton experiences a magnetic magnetic field that points downwards (solid force of deflection that is upwards (solid arrow). arrows). The electron experiences a magnetic Sketch the direction of a magnetic field that could force of deflection out of the page (indicated by the be causing the force. Could there be others? ). Sketch the direction(s) the electron could be ··· moving.

-· × + 28.1g) An proton moves into the page (shown by 28.1h) A neutron moves to the right (dotted dotted ×). The proton experiences a force of arrow). You are told the neutron experiences a deflection to the left (solid arrow). Sketch the magnetic force of deflection upwards (solid arrow). direction of a magnetic field that could be causing What magnetic field could cause such a force? the force. Could there be others?

× + N

Notice: a magnetic force is always to both & . ⊥ On the other hand, & need not be to each other. ⊥ We require some component of to (or some component of to ). ⊥ ⊥ 123

This page is screaming out for people to do some simulations… Now look at the motion of the charged particles moving in the presence of an external magnetic field. 28.2a Boring case: is parallel (or anti-parallel) to . No force; no deflection. Particle moves in a straight line. 28.2b Case 2: is to . X’s indicate and solid arrow indicates . Draw the trajectory of the charge! ⊥

28.2c Particle has components both perpendicular and parallel to the B-field.

B-field +

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28.3 A negative ion enters a region of space with a uniform external field (shaded box). We want the ion to travel in a half circle then exit the field as shown. A coordinate system is given in the figure. a) For the ion to travel in such a path would we use a uniform external electric or magnetic field? Use Use Impossible to determine without more info = = b) What orientation should be used for the field? Answer with a unit vector including sign. 28.4 A positive ion has charge & mass . It moves to the right with speed . The charge is moving between two parallel plates separated by distance . Between the plates, an external magnetic field is present with magnitude directed out of the page. We are told the charge travels horizontally through the plates (it experiences no deflection). Assume gravitational forces are negligible.

a) What direction is the magnetic force on the ion? b) If the magnetic force was the only force present, we know the ion would deflect. By applying voltage to the parallel plates, we can cause an electric force to balance the magnetic force. Which plate must be at higher potential (to balance the magnetic force)? c) Determine the potential difference between the plates required to balance the forces on the charge moving at speed . d) Suppose we want a negative charge ( , same ) to travel to the right and pass undeflected through the plates. Would we need to flip the polarity of plates? e) If a slower charge (still & ) passed through the plates, would it be deflected? If so, which way? f) Does the ion mass affect either of the trajectories parts d or e? If so, which cases or cases and what role does mass play?

28.5 Predicted Hall voltage for a sheet of aluminum foil: Consider a small piece of aluminum foil carrying current exposed to an external magnetic field. The piece of aluminum foil has thickness , length , and height . Current flows through the sheet from left to right. The average speed of electrons in the metal is given by the drift velocity ( ). Assume current density is uniform. The external magnetic field has magnitude and points into the page as indicated by the X’s in the figure. a) Which direction are electrons flowing through the sheet of aluminum? b) At the instant current begins to flow, no Hall voltage is present. Gravitational forces on the electrons are negligible. In which direction are the flowing electrons deflected by the external magnetic field? c) Excess charge rapidly accumulates on the top and bottom of the foil. This excess charge creates an electric force . In what directions do the electric force and the electric field point? d) In a very short amount of time, the electric force is sufficient to balance the magnetic force. Electrons are no longer deflected and flow horizontally through the wire. Determine potential difference between the top and bottom faces of the foil. This potential difference is the Hall voltage. Hint: the potential difference relates to the electric field and one of the plate dimensions… e) For a sheet of aluminum foil with , and I found We 1 = = = = have a magnet with a magnetic field magnitude of about 0.10 T. Compute the expected Hall voltage.

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Stage 1 Stage 3

Stage 2

Ion v v Source + + + v0≈0

Computer Screen

# of impacts CCD

r of impact

28.6 The above figure shows a mass spectrometer. Suppose the ion source produces positive ions moving very slowly. In Stage 1 the positive ions enter a region with electric field only. The external electric field will accelerate the ions the right. In Stage 2 a different external electric field is present as well as an external magnetic field. In Stage 3 the same external magnetic field is present but no electric field is present. The ion has charge & mass . In Stage 3 the ion moves with speed . In Stage 2 the plate potential difference is with plate spacing . In Stage 2 the plate potential difference is with plate spacing . a) In what direction is the external magnetic field oriented? Hint: consider Stage 3. Note: assume the field is aligned with one of the principle axes to minimize the required field strength. b) What direction is the magnetic force on the ion in Stage 2? c) For the ion to pass undeflected through Stage 2, which plate must be at higher electric potential? d) For the ion to be accelerated to the right in stage 1, which plate must be at higher potential? e) Determine the speed of the ion just after leaving Stage 1. Assume . f) Determine the external magnetic field magnitude. Hint: consider Stage 2. g) Determine the radius of curvature of the ion in Stage 3. h) What happens to the radius of curvature if charge increases by a factor of 2? i) What happens to the radius of curvature if mass increases by a factor of 2? j) What happens to the radius of curvature if charge & mass both increase by a factor of 2? k) Suppose we wanted to analyze a negative ion instead. The screen remains in the same location. What would we need to do (if anything) to the orientation of the magnetic fields and the polarity of the plates? Stage 1 is called an accelerator. Stage 2 is called a velocity selector (or region of crossed fields). Assuming thousands and thousands of ions are produced, a small number of ions will have just the right velocity to pass through the velocity selector undeflected. Different ions will have different amounts of charge and mass. The computer screen, therefore, displays a different number of impacts (# of counts) for each radius. Just like fingerprints, every material will end up with its own unique pattern. This allows people to determine what materials are present in a sample as well as the relative abundance of each type of material. 126

28.7 The ion implanter: The front end this device operates in similar fashion to a mass spectrometer. To make the next generation cell-phone chip a designer ion +1 Region of comes to you for help. She asks you the following: B-field I want to implant boron into my silicon wafer. The boron atom has charge ( ) and mass of ( ). The+ boron = atoms are ejected from a filament at and = enter region with an external magnetic field. The radius of the quarter-turn is .

a) Which direction does the external magnetic field point?

b) How much work does the external magnetic field do on the ion as it Accelerator Applied Voltage Voltage Applied travels through the quarter the turn? c) How much voltage must be applied in the accelerator stage to speed up Wafer

the ions to velocity to be implanted in the silicon? Note: assume the accelerator is a set of parallel plates. d) How strong is the external magnetic field? e) What external field is required if the implanter instead used phosphorus ions ( )? Note: by forcing the ions to travel through a quarter turn, the implanter can filter ions which do not obey = −. If these ions are pre-filtered using a velocity selector, one ensures only ions with the appropriate charge to mass = ratio are implanted in the wafer!

28.8 What would happen if electric field and B-field are parallel to each other and a proton at rest is dropped in? Code this using Python. Note: I give you a starter code in the solutions (if the link works…)

28.9 What if , , and a positron is released from rest at origin? Can you explain why the path below makes some sense. = The shape actually ends up being a cycloid. There is probably some relationship to the brachistochrone problem as that also produces cycloids. Code this using Python. Note: I give you a starter code in the solutions (if the link works…)

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Forces on current-carrying wires in the presence of an external magnetic field

Here is a tiny length of wire and the direction is in= t he direction of current flow. Note: in certain special cases (e.g uniform ) magnetic force on a wire segment result simplifies to The magnitude of the force on the wire segment (in a =uniform external magnetic field) is

where is the angle between the direction of current flow= in the segment and the direction of the external field. 28.10 Suppose a -field is given by the equation . Notice the strength of the field depends on the location left or right of the origin but the direction points= into the page (see figure)! A current-carrying wire has current I to the right as shown. Write an integral that describes the net force on the wire of length . Determine the direction of the force with the right hand rule. A trig sub using tangent works here…or an integral table.

28.11 Determine the net force on each segment of the semi-circular loops below. Also determine the net force on each wire. Assume the external field is uniform with magnitude . Current in the loop is and the radius of the circle is . In Case A the external field points to the right while in Case B the external field points out of the page. For the circular portion the direction of current is . Tip: always do the cross-product before integrating. For extra fun: also consider if the wire is under+ compression or tension. Will the loop tend to twist?

Case A Case B

28.12 A rectangular loop of wire lies in the -plane. The loop carries current in the direction shown. An external magnetic field is present given by where is a positive constant. Side note: this type of external magnetic is created in real life by placing a wire on the -axis which has current running downwards. Several distances ( ) are noted in the figure. a) Determine the units appropriate for . b) Determine the direction of force on each segment of the loop. Answer as a unit vector with sign. c) Determine the force on each segment of wire. Think: which segments do not require calculus? d) Determine the net force on the loop. Think: why isn’t the net force on this loop zero as in the previous two examples? 128

28.13 A magnetic field can also cause a TORQUE on a LOOP of wire. Consider a single rectangular loop of wire of width and height . The a b loop is connected to an axle through the center of the loop as shown in the figure. The loop carries current while . a) What direction is (to make a right-handed coordinate system)? b) Using the right hand rule determine the direction of the force experienced by each segment of wire in the loop pictured below. c) Determine the torque for segments bc and ad. d) Determine the net torque on the loop.

d c Magnetic moment vector The magnetic moment vector of a loop is

where is the number of turns of wire used to make the loop, is current in the loop and is the area vector of the loop. The area vector points perpendicular to the plane of the loop. Most loop are flat (exception discussed later). For flat loops, curl the fingers of your right hand around the loop in the direction current flows. Your right thumb then points in the direction of the area vector. The magnitude is simply the area of the loop.

The magnetic moment vector is used to simplify calculations of potential energy and torque for loops of wire.

= − Tip: In physics objects tend to approach energy minima over time. For instance, if you put a marble in a bowl it will roll down and eventually settle into the bottom of the bowl. If you release a pendulum from some angle it will eventually settle into an equilibrium position hanging straight down. The analogous situation for loops of wire is the tendency of the magnetic moment vector to align with the external magnetic field.

28.14 Reconsider the single loop of width & height carrying current . a) Determine the magnetic moment vector of the loop. b) Determine the torque on the loop in the presence of the uniform external field. c) What is the potential energy associated with the loop in the presence of the external field? d) Assume the loop is held in place while we move around the magnets causing the external magnetic field. What orientations of external magnetic field give maximum & minimum potential energy?

28.15 A current carrying loop (square of side ) is oriented as shown in the figure. Current direction is indicated by the arrow on the loop. a) Verify the coordinate system is right-handed.

b) Determine the direction of the magnetic moment vector. c) Suppose we wish to turn on an external magnetic field such that we cause a torque on this loop in the direction. Determine all possible orientations of the external magnetic field which could cause such a torque.

d) To make devices as cheaply as possible, we typically want to apply large torques with the smallest possible magnets. What is the optimum orientation of the external magnetic field? 129

28.16 Consider a semi-circular loop of wire that is bent into the shape shown. Assume the loop carries current and the circular arc portions have radius . A uniform external magnetic field is present given by = . Assume the loop is held in place by a Class II zombie. = Note: I have no clue what a Class II zombie is…but it sure sounds nice. a) Determine the magnetic moment vector of this loop. b) Determine the potential energy associated with this loop in this external magnetic field. c) Determine the magnetic torque acting on this loop in the presence of the external field.

28.17 A loop with turns carries current & lies in the -plane with hinge along -axis. The long side of the loop along the - axis has length while the short side has length . A constant external magnetic field lies in -plane. The -field is strong enough to keep the loop from swinging downwards due to gravity. a) Correctly label the axes based on the problem wording. b) Determine the size of in terms of the other variables and standard physics constants.

28.18 A long piece of wire with mass and Front view of coil Side view of coil Side view after total length of is used to make a rectangular before placed in -field before in -field placed in -field coil with sides of & . The coil is hinged along a horizontal side and the coil carries a hinge hinge current . The coil is placed in an external, constant, vertical magnetic field . Somewhere on the hinge a battery is hidden which causes current to flow in the loop. a) In what direction is the current

running at the bottom of the loop (in the far right picture)? b) Determine the number of turns of the wire in terms of , , and . c) Write an expression for the magnetic moment VECTOR of the coil. Recall the magnitude is & the direction is found by curling your right hand fingers in the direction of current (thumb points in direction of ). d) Determine the angle the plane of the coil makes with the vertical when the coil is in equilibrium. e) Find the torque acting on the coil due to the magnetic force at equilibrium.

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28.19 A triangular loop of wire has height and length (figure not to scale). Current runs clockwise in the loop. The loop is orientated such that in lies in the - plane as shown (top view). We are able to place a hinge on this loop such that it may rotate. There are two possibilities for the hinge: along the -axis or along the -axis. This loop will be placed in a uniform external magnetic field. We are also able to control the orientation of the external magnetic field. a) Which orientation of the uniform external magnetic field causes no magnetic torque on the loop about the -axis? Circle the best answer.

in -plane in -plane in -plane Any orientation of No orientation of Impossible to determine causes torque about -axis causes torque about -axis without more info b) Which orientation of the uniform external magnetic field causes no NET force on the loop? Circle the best answer. Any orientation of aligned with aligned with aligned with causes zero NET force No possible orientation of in -plane in -plane in -plane causes zero NET force c) Assume the external magnetic field initially causes maximal torque on the loop about the -axis. Without reorienting the loop , the external magnetic field is re-oriented to cause maximal torque about the -axis. The magnitude of the external magnetic field is unchanged. Is the magnitude of maximal torque about the -axis ( ) greater/less than/equal to maximal torque about the -axis ( ). Circle the best answer. Impossible to determine

without more information

28.20 An inclined plane is built such that the top face of the plane is essentially a giant magnetic north pole. The plane has length and incline angle . A cylinder of radius , height and mass is placed on the plane. Wrapped around the center of the cylinder is a coil of copper wire with turns carrying current . At the instant shown, the cylinder has just been released from rest with the coil perpendicular to the ground (not perpendicular to the incline). A side view of the situation is shown at right. Assume the situation remains at rest after initial release. a) Determine the magnetic moment vector associated with the coil. b) Determine the minimum magnetic field magnitude (caused by the magnets in the plane) is required to prevent the cylinder from rolling down the incline. Assume the magnetic field just above the inclined plane is roughly uniform. c) Assume the situation remains at rest after initial release. After a few seconds the coil warms slightly. Will the cylinder tend to roll uphill, downhill, or remain at rest? Is it impossible to determine without more information? You may assume the potential difference used to generate the current remains constant.

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28.21 A permanent magnet has its north pole facing Slanted side view straight upwards. Just above this permanent magnet a circular loop of wire carries current . The coordinate system is defined in the bottom image. Also notice, the external field magnitude is the same at all points on the loop. The external field direction is different for each point on the loop. This is a non-uniform magnetic field and it could cause a net force on the loop. N a) Write down the external magnetic field in terms of the magnitude , the angle and the unit S vectors & ? b) Determine the net force on the loop. Pure side view Top view c) What direction is the magnetic moment of the loop? d) Think: compare the magnetic moment direction to the permanent magnet’s field direction. Does the loop act like a magnet with the north N

pole on top or on bottom? Based on this intuition, should the loop be attracted or S repelled by the magnet? Does your answer to the previous part agree with this reasoning?

28.22 This problem is totally unrealistic but is designed to help you practice set-ups. A single semi-circular loop of wire (with radius ) lies in the -plane carrying current in the direction shown. A non-uniform external magnetic field is present given by

where are positive constants. a) Set-up an integral for determining the force acting on the straight wire segment. Do not do the integration!!! b) Without doing the integration, which components of the magnetic field should cause zero contribution to the force on the straight wire segment? c) Set-up an integral for determining the force acting on the curved wire segment. Do not do the integration!!! d) Without doing the integration, which components of the magnetic field should cause zero contribution to the force on the curved wire segment? e) Think: in this case, should we expect the net force on the entire loop of wire to be zero? f) Think: if you start at the origin and move to the right , how is each component of the field affected? g) Think: if you start at the origin and move to the up , how is each component of the field affected? h) Without doing any computation, speculate the direction of the net force on the loop.

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In electricity: Charges cause electric fields & charges feel electric force.

In magnetism: Moving charges cause magnetic fields & moving charges feel magnetic force. Moving charge is essentially current . Currents cause magnetic fields & currents feel magnetic force.

Show how several compasses are affected by current through a long straight wire. Show the B-field created by a coil with iron filings.

For THE CENTER of a circular coil of wire with current I & N loops:

RHR2a – Find Direction of B caused by a coil (curl RIGHT fingers in direction of current, thumb in dir of B)

For anywhere inside an extended cylindrical coil (a solenoid) with current I & n=N/L =# of turns per unit length

RHR2a – Find Direction of B caused by a coil (curl RIGHT fingers in direction of current, thumb in dir of B)

For a distance r away from an infinitely long (or really long) straight wire current I: RHR2b =– Find Direction of B caused by straight wire (R thumb in direction of current, fingers curl in dir of B)

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Procedure for doing Biot-Savart with straight line segments 1) Draw perpendicular bisector from point of interest to wire 2) Draw coordinate system with origin at point where perpendicular bisector hits wire 3) Draw 4) Label displacement vector distance from source ( ) to point of interest 5) Figure out , , and 6) Shove into

= 7) Usually the calculus steps can be skipped if you use the following memorized formula: a. Here is the distance from the point to the segment − along the bisector. ⊥ b. Use the angles of the segment endpoints from the bisector. ⊥ c. Tip: if you screw up the initial and final angles, no big deal. That simply switches the sign of the field. You can correct for this by checking field direction with the right hand rule. d. WATCH OUT: angles to the left of the bisector are negative , while angles to the right of the ⊥ ⊥ bisector are positive . It is possible in some problems to have both angles positive or both angles negative. 8) Do cross-product before thinking about symmetry of integrating 9) When putting in limits, think carefully about signs 10) Use integral table correctly

29.1 A wire segment has current flowing to the right. The segment has total length . Assume that the figure is not drawn to scale. The shortest distance from the wire to the point P is . a) Use the Biot-Savart Law to determine the magnetic field vector at the point P (caused by the current in the wire). Answer in terms of . b) What happens to the angles as and tend towards infinity (infinitely long straight wire)? c) What is the field magnitude created by an infinitely long straight wire?

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29.2 Derive the magnetic field due to a circular coil of wire with turns a distance on-axis from the Slanted Side View Pure Side View center of the coil of wire. The current in the coil of wire is flowing clockwise when viewed from above. The coil has radius . P P a) Consider d at the right side of the loop for simplicity. • Label a coordinate system for each other view. • Show direction of current at d as a dot or × • Write down an expression for d in terms of and . • Draw . Write down an expression for in terms of and . • Draw d (the infinitesimal field created by just Top View that tiny segment of wire at the very top of the loop). It is a small vector located at P. Use the

right hand rule since d . b) Think about symmetry of the problem. To do this, consider what would happen to all the above expressions for the wire segment at the left side of the loop. What should the d vector look like for the left side segment? c) Write down an integral which you will use to calculate the magnetic field. Solve this integral and write out B (both magnitude and direction). Answer is hidden in equation sheet on back of workbook. d) Use step b to simplify the integration. Which part of the -field will cancel out due to symmetry? Which term, after doing the cross product, should survive the integration? You can ignore the other one! e) Do the integral and determine at the point P. Check your result with the right hand rule for a circular coil of wire. Check your result for units! f) Get the special result for , the magnetic field at the center of a circular coil.

29.3 A wire is bent into the configuration shown in the figure. The wire carries current . The straight segments are so long they may be considered I to go on to infinity for the purposes of this problem. The curved section is a R portion of a circle with radius and . a) Determine the magnitude of the magnetic = field at the point P. θ P b) Determine the direction of the field at the point P.

Hint: Rather than go through all the nasty integrals, use symmetry. Find the direction of the field from each segment. Determine if they should all add up or if some are positive and some are negative. Then identify that you have two halves of an infinite wire and 1/6 of a circle! Just add up those parts and state the direction with the right hand rule. Note: this only valid for certain special cases.

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29.4 This again uses the result of the BS law without having to do all the calculations. Three very long wires run current into and out of the page as shown. ××× From this viewpoint we can only see the cross-sections of the wires. Determine the net -field at the point P. Assume is twice the magnitude of . Instructor tip: I used blue for 1, green for 2, and red for 3 in solutions.

P 29.5 Consider the two cases shown at right. Assume the two cases are, in reality, spaced very far apart such that case 1 doesn’t affect case 2 in terms of magnetic field. Also assume the straight wire segments each extend to infinity. a) What direction must current flow in the upper straight segment (for each case) if the magnetic field at each dot to points into the page? b) Determine the ratio of currents required (case 1 to case 2) if the magnitude of the magnetic field at each dot is identical.

29.6 Consider a loop with turns in the funky shape shown. The dimensions of the shape are indicated in terms of the radius of the semi-circular segment. The magnitude of the magnetic field at the dot is and points out of the page. a) Determine direction of current in the loop. b) Determine an expression for the current in the loop in terms of , , and . c) Rank the wire segments in terms of the contribution to the magnetic field at the dot from

smallest to largest clearly indicating any ties.

29.7 Two wires carry DC current. The magnitude of the current is unknown but you are

told the magnitude of current in each wire is the same. Wire 1 carries current into the page

while wire 2 carries current out of the page . The magnitude of the magnetic field due to wires 1 & 2 at point P is . Distance is known. a) Determine the current in a single wire. Answer in terms of , and . b) Determine the direction of the magnetic field at P. Include a small sketch showing the approximate direction to clarify your answer.

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29.8 A wire is bent into the bizarre shape shown in the figure. To be clear, assume that the straight wire portions extend to infinity on the left and right. The wire segments are labeled 1 through 5 for ease of communication. Segment 3 is a circular arc of radius . A magnetic field is created at point P due to the current in the wire. Assume the current direction in each segment is shown by the arrows in the figure. Figure not to scale. a) Determine the radius of the circular arc segment in terms of . b) Determine the direction of the magnetic field at P. P c) Which segments, if any, do not contribute to the magnetic field at P?

d) Determine the magnitude of the magnetic field produced at P. Answer as a number with 3 sig figs times . 29.9 Consider the three possible coil designs shown below. Assume each coil carries current . The direction of current in each coil is indicated with an arrowhead. The radii of the circular arc segments are , , and . Each figure shows a black dot. a) Which coils produce a magnetic field out of the page? If none of them do, state “None”. b) Rank the magnitudes of the fields (at the dot in each picture). Rank the magnitudes from smallest to largest clearly indicating any ties.

Case 1 Case 2 Case 3

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29.10 Force between parallel wires running current: Suppose you have two wires; both are very long. Wire 1 (the left wire) will create a magnetic field. This field will exert a force felt by wire 2 (the right wire). This problem derives the force on wire 2 caused by the B-field created by wire 1. Assume . By making this assumption we can estimate the field created by either wire is approximately equal to the field of an infinite wire. a) Sketch the direction of the B-field caused by wire one. It will be a pattern of dots and ×’s. Indicate the strength of the field by the size of the dot or ×. Indicate the field strength at points on both sides of both wires and in between the wires. b) Which direction is the force on wire 2 due to the B-field

created by wire 1? c) Determine an algebraic expression for the force in terms of the currents, the indicated lengths, and physics constants.

d) If I double I1 and reverse its direction what will happen to the force on the right wire? e) Do like currents attract or opposite currents attract? f) Assume , , and . What force is exerted on each wire? Alternatively, this demo can be done using car batteries = connected = to bare copper wires (no resistors in loop). I did this for years but the currents get so large and the wires get so hot the switch used would occasionally weld shut!!! The weld was always super weak so we just broke it apart and kept using the switch.

29.11 CHALLENGE Imagine, instead of infinitely long parallel wires, infinitely long crossed wires. Assume they carry the same current (one to the right, the other upwards). y Is it possible to find the net force one exerts on the other? HA HA HA…(evil cackle). x

28.12 A long straight wire carries current I vertically. Distance x from the wire a ball with positive charge q y and mass m is released from rest. a) At the instant just after the charge is released, what direction is the magnetic force acting on the charge? x z b) Instead of being released from rest, the charge was thrown with initial speed v0? In what direction or directions, if any, could the charge be thrown so that it experiences no initial magnetic force? c) Now suppose the charge is with speed v0 angled 30° above the horizontal at the instant shown. The only forces acting on the charge are gravity and the magnetic force. Determine the magnitude of the acceleration of the charge. y d) Challenge: I wondered what the path of travel would look like for such a charge with such initial conditions. x Do you think it should be circular or not? Explain. What if gravity was negligible? As always, for fun one can z think about flipping the sign of the charge, the sign or direction of the velocity, or some combination of those v0 elements! So many fun problems…so little time. Perhaps the easiest way to analyze this scenario is to code it in Python.

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29.11 on axis through centroid of equilateral triangle Now consider the triangular loop of wire carrying current shown at right. Use the Biot-Savart law to compute the magnetic field at some point on the -axis. The loop lies in the -plane. The -axis runs through the centroid of the triangle. The point of interest is distance above the -plane.

29.11¼ A solenoid is created by wrapping wire around a cylinder of radius and height . The wire has turns of wire; the number of turns per unit length is . By slicing the solenoid into coils (in a fashion similar to what was done in the section on continuous distributions of charge) show this solenoid produces a magnetic field

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29.11½ Three arrangements of three straight wires carrying equal current pointing into or out of the page are shown. a) Rank the arrangements according to the magnitude of the external magnetic field at wire A due to the currents in the other wires.

b) Below I zoomed in a bit on the figure for Case 3 . Sketch the direction of the net force on wire A. I am looking for two things: 1) is it in the correct quadrant and 2) is it closer to the - or - axis? Hint: first determine the direction of the external B-field. In this case, the external B-field is caused by the other two wires .

29.11¾ Two circular loops of wire each carry current . Slanted side view Pure side view One circular loop has radius while the other has radius . The larger loop is fixed in the -plane. The other loop can be rotated by up to . The figure shows

. a) Is there an angle which gives zero magnetic field at the origin? If yes, what angle is it? If no, explain why not. b) Determine an expression for the net magnetic field (at the origin) for an arbitrary angle of rotation. c) Plot the net magnetic field magnitude at the origin vs angle. Think: what angle should give the minimum magnetic field? What angle should give the maximum? d) Review Q: what angle(s) will allow the larger loop to exert the max/min torque on the smaller loop?

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29.11 Two coils, each with radius , are spaced distance apart. = One coil is distance to the left of the origin while the other is the same distance to the right of the origin. Both coils are centered on the horizontal axis (which I chose to call the -axis). Assume the current in each coil is running down on the front side. a) Show one finds:

+ +

+ + + + + − b) Simplify this expression for the case of . This special case is called “Helmholtz coils”. c) For the case of , determine the magnetic field at the origin. d) Also determine the derivative (with respect to ) of the magnetic field at the origin. Remember to take the derivative first, then plug in . 29.11 Review: A current carrying rectangular loop of wire has length and height . The loop is placed next to a very long straight wire. Both the loop and the wire carry current . The wire carries current to the left but the current direction in the loop is unknown . The segments of the wire are numbered for ease of communication. The top edge of the loop (segment 1) is distance from the wire.

a) Suppose the magnetic force on the loop (due to the wire) is upwards . What 1 direction does current run in the loop? Circle the best answer. 4 Impossible to Clockwise Counter-clockwise determine without 2 more info 3

b) What direction is net torque on the loop (due to external mag field from the wire)?

No net torque c) Determine+ the −magnetic force+ (magnitude) on each segment.

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Magnetic Flux is a way to describe the amount of magnetic field that penetrates a loop of wire

If the external magnetic field is uniform, this simplifies to =

Here is the area vector pointing perpendicular to the plane of the loop of wire, is the external magnetic field, and is the angle between direction of and direction of . Notice: flux can be zero even when the external field is non-zero!!! Think: under what circumstances can one have non-zero magnetic field but zero magnetic flux?

29.12: Suppose a rectangular loop of wire with long side and short side is placed next to a straight current carrying wire of length . You may assume is much greater than either or . Assume the current in the straight wire is . In each case the distance from the wire to the closest part of the loop is . a) Which orientation below gives rise to the maximum magnetic flux through the loop of wire?

b) For case 1 only, determine an expression for the magnetic flux in terms of c) If you were to redo your calculation for case 2, what would be different compared to the previous part? d) In cases 3 & 4, is the flux zero? Explain why or why not without doing any calculations.

Case 1 Case 2 Case 3 Case 4

c c c c

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29.13 First consider the figure for case 1. A solenoid has radius R and length L>> R. Case 1 Case 2 Current I runs in the solenoid counter-clockwise as seen from the top. At the end of a solenoid lies a square loop of wire. You are told that at the end of a solenoid the magnetic field has a magnitude exactly half of the magnitude in the middle of the solenoid. You are also told that the diagonal of the square loop of wire is exactly the same as the diameter of the solenoid. Determine the magnetic flux for case 1. Now consider case 2. Assuming everything else is the same except the orientation for the square loop. Determine the magnetic flux for case 2. Bonus: for each case, think about the net force and torque on the square loop. Think about the magnetic moment vector of the loop in each case. Do you see that in general the torque on a loop tends to align the magnetic field of the loop with any external field (in this case the external field comes from the solenoid)?

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Ampere’s law problems…

Remember Gauss’s law? We used it for cases of perfect symmetry to relate enclosed charge to electric fields. Ampere’s law is similarly used to relate enclosed current to magnetic fields.

Gauss’s law uses charge enclosed by a surface (an area) which, in cases of ideal symmetry, gives the equation

= Ampere’s law uses current enclosed by a loop (a perimeter) which, in cases of ideal symmetry, gives the equation

Notice that instead of using a Gaussian surface area and enclosed volume we now use Amperian loop length and enclosed area. 29.14 Consider the case of a very long straight wire carrying current. Assume the current is directed into the page. The Case 1 and Case 2 figure at right show a cross-sectional view of such a wire. Assume that a uniformly distributed current I runs out of the page. Case 1: Amperian Loop Case 1: outside the object ( r>R ) 1) Draw the Amperian loop outside of the wire. Label that radius with little r. 2) At several points on the Amperian loop, sketch the direction of . 3) Explain why in this case. I am looking for you to mention something about the magnitude of as well as the directions of the differential arclength R vector and at various points on the loop. 4) How much current is enclosed (how much current penetrates the loop)? 5) Use to determine the magnitude of B for r > R. Hint: now you need to find the length s of the Amperian loop (not the area). 6) Don’t forget to check the units on your answer. To do this most simply, compare it to a known equation for B. For instance, at the center of a ring of radius a the field

is given by . We see the should be on top times units of A/m. Case2: 1) Now use an Amperian loop inside the object so that only some, not all, of the current is enclosed. Case 2: Amperian Loop 2) We expect now that our equation will look like inside the object ( r

The term J is the current density. = What are= the units on J? R 3) Recall for this example the current is uniformly distributed. Use this fact and the units of J to write down the correct expression for J in terms of I and R. 4) The term d A implies a tiny portion of the area enclosed by the Amperian loop. In this case we will be building the area of the Amperian circle out of rings. In your figure show one such ring of radius and thickness . Note that for this case. Also, write down the correct expression for d A in terms of < and < .

5) Finally, put it all together to to determine the magnitude of B for r < R. 6) Think: you’ll check the units…but what else can you check now? 145

7) Plot the magnitude of B versus r. Be sure to indicate the special radius R on the horizontal axis as well as the value of B at that radius on the vertical axis. Pay attention to the shape/concavity of the line.

Sketch the B-Field at various points, assume the spacing between adjacent points is R/2 146

A made a really nice, color coded reference for Ampere’s law in cylindrical symmetry. Check the Chapter 29 solutions.

Solenoids A solenoid is essentially a cylindrical coil of wire. When current is run through the coil we say the coil is energized. The region inside the coil is called the core. Typically we encounter either air cores or iron cores. Iron cores greatly increase the magnitude of the field created.

I like to think of the air solenoid as the magnetic equivalent of the oppositely charge parallel plates in electric fields. In electrostatics, parallel plates are the practical way to create (roughly) constant electric fields. Similarly, the air core solenoid has a (roughly) constant magnetic field in the core. Another similarity between the two cases, the fields outside large plates and outside a long solenoid is essentially zero. Let’s see why using Ampere’s law…

Consider the circular loop in the cross-section view. Assume the solenoid is a cylinder of radius R. To be clear, this is supposed to be a circular loop that has a radius r > R.

How much current penetrates the area of the circular loop? That is to say, in this case, how much current is flowing vertically? What then Cross-section view is the B-field outside the solenoid? Note: we are simplifying things a little bit to get through the problem. I will discuss in greater detail the x x ··· B-field outside both singly wound and doubly wound solenoids on the x ··· next page… x ··· x ··· x Segment 2 ··· Now consider the rectangular loop. Assume the lengths of segments 1 x ··· and 2 are y while the lengths of segments 2 and 4 are x. x ··· x ··· Segment 3 The B-field is zero on either segments 1 & 3 or segments 2 & 4. On x ··· x Segment 1 ··· which segments is the B-field zero. Use symmetry to defend your x ··· answer. x ··· x ··· x ··· One of the remaining segments should have essentially no field from x ··· Segment 4 our previous discussion of the circular loop. Which one?

Now use Ampere’s law for the entire rectangular loop.

Here the left side will simplify to only the contribution from = the single remaining segment with non-zero B-field. The right side is not simply I but will relate to the number of turns ( N) enclosed. Compare to the given result for solenoids we’ve already been using and make sure it makes sense to you. 147

Doubly wound versus singly wound solenoids Cross-section Cross-section The figure at right shows the cross-sectional views of a singly wound solenoid view, Singly view, Doubly and a doubly wound solenoid. In the singly wound solenoid the wire comes in Wound Solenoid Wound Solenoid at the bottom, wraps around a cylinder a bunch of times, and then exits at the top. x ··· x x ··· ··· x ··· x x ··· ··· For the doubly wound solenoid the wire comes in at the bottom, coils around x ··· x x ··· ··· x ··· x x ··· ··· the cylinder on the way up, then coils around it again on the way back down, x ··· x x ··· ··· x ··· x x ··· ··· and finally exits at the bottom. x ··· x x ··· ··· x ··· x x ··· x ··· x ··· ··· x ··· x x ··· ··· Below these two figures are the same two solenoids viewed from a much x ··· x x ··· ··· x ··· x x ··· ··· greater distance. x ··· x x ··· ··· x ··· x x ··· ··· x ··· x x ··· ··· x ··· x x ··· ··· What does the singly wound solenoid look like and what should be the x ··· magnetic field outside of the solenoid? Strictly speaking is the field outside the singly wound solenoid zero? Would you expect this to be large or small compared to the field inside a tightly wound solenoid with many turns per unit Singly Wound Doubly Wound length? What if the solenoid was only loosely wound (gaps between each Solenoid viewed Solenoid viewed turn)? from afar from afar

Now consider the magnetic field outside of the doubly wound solenoid. Why is the magnetic field outside the doubly wound solenoid much closer to zero?

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29.15 Cylindrical shell with uniform current density. Suppose the inner radius is R and the outer radius is 3 R. Assume the total current is I and it is distributed uniformly. Assume the current runs out of the page.

Determine the current density in terms of R and I. Determine the equation of the B-field for all important regions. Plot B versus r indicating all important values on the r-axis and B-axis. Sketch the direction of the magnetic field at various locations (say every 90 degrees) both inside and outside the shell.

29.16 Cylindrical shell with non-uniform current density. Suppose the inner radius is R and the outer radius is 3 R. Assume the total current is I but is non- uniformly distributed according to the equation .

Determine the units of α. Determine an expression for α in terms of R and I. Determine the equation of the B-field for all important regions. Plot B versus r indicating all important values on the r-axis and B-axis.

29.17 Now consider a cylindrical cable of radius R with non-uniform current density given by

You are told that at the point P the magnetic field points to the left. On exams I’d provide an integral table with some decoy integrals, probably like the one shown below. a) Determine units of and such that the current density is dimensionally correct. b) Determine the total current in the wire in terms of . c) Determine which direction current is flowing in the wire. d) Determine the magnitude of the magnetic field for points both inside ( ) and outside ( ) the wire. P e) Plot . f) At what radius (in terms of ) will the current density start to decrease? Note: this assumes that the radius is large enough to include this special radius. Note: current in any given differential section of a cylindrical wire is given by . Current through a cross-sectional ring is not the same as the current density in that same = cross-sectional ring.

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29.18 What about nested cylindrical shells? Assume the radii of interest are , , , and for the figure shown at right (figure not to scale). Assume that in 3 the inner shell a uniform current runs into the page. In the outer shell a 4 uniform current runs out of the page. a) Determine an expression for the magnetic field in terms of and . You should do this for all five regions of interest. b) Plot the magnitude of the field versus distance from the center of the concentric shells. Alternatively, you could plot the field (not just the magnitude) if you assume terms are positive and terms are negative.

r < a a < r < 2 a 2a < r < 3 a 3a < r < 4 a r > 4 a

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Induced Voltages and Inductance (Demos: ring launcher and light bulb, wind generator with large magnet and oscilloscope, magnet down tube, eddy current pendulum, pedal generator, etc)

Reminder: Magnetic Flux is a way to describe the amount of magnetic field that penetrates a loop of wire (or anything else)

If is uniform this simplifies to Here points perpendicular to loop of wire and is angle between direction of and direction of . Notice flux can be zero even in the presence of a large magnetic field. Think: what must be true of the angle?

Faraday’s Law: A changing magnetic flux will INDUCE an EMF (EMF is another way of saying voltage). If a coil of wire has N loops we can find the voltage induced (the induced EMF) using: ε = Lenz’s Law: Current caused by induced EMF’s travels in a direction that creates a magnetic field with flux opposing the change in the original flux. That is the minus sign in Faraday’s law.

Ways to have a changing magnetic flux through a coil of wire or metal rod:

1) Bring a magnet closer to the coil (increase ΦB)

2) Bring a magnet farther from the coil (decrease ΦB)

3) Bring the coil closer to/further from the magnet (increase/decrease ΦB) 4) Rotate the coil or the magnet (this changes the angle θ and thus the flux!) 5) Increase or decrease the area of a wire (this is done by putting part of the loop on a sliding rail)

If a rod is moving perpendicular to a -field an induced EMF is created but no current flows (a single rod doesn’t make a circuit). The induced EMF on a rod moving perpendicular to a mag field is .

If a coil of wire is used the induced emf sets up an induced current (I’ll call it ). The induced current creates a -field (I’ll call it ). This is confusing because a changing flux caused by a -field creates an additional -field! The direction of OPPOSES THE CHANGE IN FLUX . To be clear, the direction of only sometimes opposes …but it always opposes . This subtlety takes some getting used to. Do problems 3.1 through 3.9 and you should have no further trouble.

30.1 Initially no current is running in a loop of wire in the presence BEFORE AFTER of a 4 T magnetic field into the page (far left figure). The magnetic field is suddenly changed to 1 T. This change in magnetic field INDUCES a voltage (and thus a current) in the loop of wire. This induced current creates a B-field that opposes the change in flux. In this case a flux into the page was decreased. Therefore the induced current will try to increase the flux into the page (create and induced field into the page). Which direction will the induced current flow?

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30.2 Initially no current is running in a loop of wire in the presence BEFORE AFTER of a 2 T magnetic field into the page (far left figure). The magnetic field is suddenly increased to 4 T. This change in magnetic field INDUCES a voltage (and thus a current) in the loop of wire. This induced current creates a B-field that opposes the change in flux. In this case a flux into the page was increased. Therefore the induced current will try to decrease the flux into the page. Note: decrease the flux into the page is the same as increase the flux out of the page. Which direction will the induced current flow?

30.3a Initially a counter-clockwise current is running in a loop of BEFORE AFTER wire that is in the presence of a 2 T magnetic field into the page (far left figure). The magnetic field is suddenly changed to 2 T but directed out of the page. Will the induced current flow tend to increase or decrease the already present current flow? Or will there be induced current? 30.3b Which of the above three examples will have the largest INDUCED currents? Explain.

BEFORE AFTER

30.4 Initially an almost complete metal ring is in the presence of no external field. A moment later the external field is increased to 4T directed out of the page. In what direction is the induced current?

30.5a Four metal rods are formed into the shape of a square. The metal BEFORE AFTER square is in the presence of a uniform external field. A current is running counter-clockwise in the square. Due to the current, the square is very rapidly heated and changes in size (exaggerated in the after picture). Will the induced current tend to increase or decrease the current already present in the square? 30.5b Based on the initial picture only, is there a force exerted on the square? If so what will the force on the wire tend to compress or expand the wire? 30.5c Based on the initial picture only, is there a torque exerted on the wire? If so, will the torque tend to twist the loop clockwise or counterclockwise as seen from the top?

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30.6a I claim that an induced current is running clockwise in the loop of wire (far left figure). Presently there is an external 4T magnetic field pointing out the page. If this is not possible, defend your answer in coherent English by stating which laws are violated. If it is possible, show a specific set of 2 pictures (before and after) that demonstrate how the current could be produced. CLAIM BEFORE AFTER

Is this claim impossible or plausible?

30.6b Suppose you had used a loop of wire with 10 times more windings in it than the loop used in the previous part. How would the induced current in this new loop compare to the old scenario?

30.7 This is now linked to the demonstration! A solenoid (grey inner circle) is connected to a battery with a switch. A metal ring (black outer circle) is placed around the solenoid. The left two figures below are as seen from the top. The right two figures show the side view of the solenoid (tall grey cylinder) and the metal ring. Initially no current runs through the solenoid and no current runs in the ring. The switch is connected causing current to flow in the solenoid (let’s assume it flows CW as seen from the top). a) For each view (in the after pictures), draw the direction of the -field created by the solenoid. b) In each after picture, sketch the direction of induced current in the ring (thick black circle). c) Compare the directions of the magnetic fields created by the induced current in the ring and the solenoid. Will the ring be moved upwards or downwards? d) Obviously, to increase the jump height we could use more input power or a solenoid with more windings. What changes could we make to the ring to increase the jump height?

Switch Open Instant after Switch Closed Switch Open Instant after Switch Closed

e) Just after closing the switch (a few microseconds later), the rate of change of the solenoid current rapidly decreases. What happen to the induced current in the ring? Explain using Faraday’s law. f) Now suppose the current in the solenoid switches direction rapidly many times. This can be done by connecting the solenoid to AC power (instead of using DC power and a switch). How would such an input signal affect the induced current in the ring? g) What if one used a coil of wire connected to a light bulb instead of a metal ring…what should happen? Will it still levitate? Will the bulb light up? Will both happen? Note: technically the ring is an circuit. We will analyze this more correct model later in 31.7 …

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30.8 Drop magnet down a tube The following set of pictures shows two instants in time as a magnet Before After falls down through an aluminum tube. Notice I drew several dotted line circles on each tube. The goal of this problem is to understand any induced currents at the locations of these dotted line circles as the magnet falls down through the tube. I will refer to these circles Upper as the upper and lower loops from now on. loop N a) Draw three field lines exiting (or entering) each face of the magnet for both the before and after pictures.

b) For each loop, determine the direction of each induced N current (at upper and lower loop). Draw these induced S currents in the after picture. Lower c) Determine the direction of the magnetic field created by the loop S induced current in each loop. Draw the magnetic field created by the induced current in the after picture. d) For each loop, determine if the field created by induced current exerts an upwards or downwards force on the magnet.

30.9 Eddy Current Pendulum Just before disc enters region of A solid copper disc swings into a region of external magnetic field. Assume the external magnetic field is directed into the page and is approximately uniform. Note: since the copper disc is at the end of long pendulum at the bottom of its swing, we may assume the disc is moving essentially horizontally.

a) Determine and draw the direction of induced currents for each “just Just after disc enters region of after” picture. b) Determine the direction of magnetic force on the leading edge of the disc as it enters the region of external magnetic field. c) Determine the direction of magnetic force on the trailing edge of the disc as it exits the region of external magnetic field. d) Going further: we are assuming the disc is constrained to swing in Just before disc exits region of the plane of the page. In real life, the pendulum is pushed and pulled slightly into and out of the page during each swing. As the disc swings to the right and enters the region of external magnetic field, is the plane of the pendulum’s oscillation slightly pulled into the page or slightly pushed out of the page? How about as it exits the region of external magnetic field? What about on the return swing (if the Just after disc exits region of disc was moving to the left )?

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Motional EMF: If a rod is moving perpendicular to a -field an induced EMF is created but no sustained current flows (a single rod doesn’t make a circuit). The induced EMF on a rod moving perpendicular to a mag field is

Note: one could argue there is a transient current as the rod electrically polarizes. This current is tiny and lasts for a tiny amount of time (depending on the rod, I’m guessing it is probably lasts a few nanoseconds to microseconds).

30.10 A neutral metal rod falls through an external magnetic field. Assume the external magnetic field is aligned with one of the principal axes of the given coordinate system. a) What direction is the magnetic field pointing if the negative charges in the rod re- orient themselves as shown?

b) As the rod falls, it speeds up. How does this affect the rod?

c) What would be different about this problem if the moving rod was made of glass?

30.11 A conducting rod of negligible resistance is free to slide on frictionless, conducting rails of negligible resistance. The rails are connected at the left end by a resistor. The conducting is pulled with unknown force to the right at constant speed . The distance between the rails (and the length of the conducting rod) is . A uniform external magnetic field is directed out of the page as shown. NO BATTERY IS IN THE CIRCUIT. a) As the rod is pulled to the right, a motional EMF is created. Determine the motional EMF. b) This motional EMF is a voltage that causes current to flow in the circuit! Determine the current that flows in the circuit due to the motional EMF. c) Determine the direction of the induced current caused by the motional by considering Lenz’s law. Is it CW or CCW? d) Now determine the magnetic force magnitude on the wire due to the EXTERNAL field. e) What direction does this force point? Will it tend to speed up or slow down the conducting rod? f) What is the force magnitude ? Explain.

30.12 Another classic version of this problem has the rod falling vertically as gravity pulls it down. Assume are givens. Parts a through c use the Case 1 figure. Case 1 a) What direction will the induced current flow in the loop? b) What is the direction of the magnetic force on the falling rod? c) If the rod is released from rest it will eventually reach some terminal velocity due to the “magnetic braking”. Determine an algebraic expression for the terminal velocity in terms of . d) How do your answers to parts a) and b) change if the circuit looks like Case 2?

e) How do your answers to parts a) and b) change if the circuit looks like Case 1 but the external field points the opposite direction? f) While the rod is approaching terminal velocity it is obviously speeding up and current Case 2 is increasing in the rod. Does the current in the rod increase with a constant rate, increasing rate, or decreasing rate? Is it something else? g) Determine velocity as a function of time for the original scenario ( Case 1 ). Assume the rod was released from rest.

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30.13 A current carrying rectangular loop of wire has length and height . The loop is placed next to a very long straight wire. The wire carries current to the left . The segments of the wire are numbered for ease of communication. The top edge of the loop (segment 1) is distance from the wire. a) Determine magnetic flux through the loop.

b) You can pull the loop to the right or down. Which direction(s) induces 1 significant current in the loop? 4

Neither way Both ways Impossible to Pulling Pulling 2 induces induce determine right down 3 current in current in without more only only loop loop info

c) If you pull the loop down with constant speed, will the amount of induced current increase, decrease, or remain constant over time? Circle the best answer

Remains constant at Induced current is Induced current is Induced current is zero , there is no non-zero, remains non-zero and non-zero and induced current constant while decreases over time increases over time pulling down pulled down

d) Determine induced EMF in the loop as a function of time assuming the loop is pulled down with constant speed .

30.14 A loop of wire lies in the -plane as shown in Figure Taco . In each of six trials, the loop is exposed to different magnetic fields directed parallel (or anti-parallel) to the -axis. Figure Frog shows the -component of the magnetic field as a function of time for the different trials. To be clear, in the first trial, the magnetic field as a function of time experienced by the loop is shown by plot 1, in the second trial, plot 2, etc. Note: plots 2 and 6 have slope while plots 3 and 4 have slope . a) Which trials or trials induced no current in the loop? b) Which trials or trials induced CLOCKWISE current in the loop? c) Which trial or trials induce currents in the loop with the largest magnitude ?

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30.15 A rod is attached to a frictionless pivot at one end. The rod rotates with constant rate in the direction shown. A uniform external magnetic field with magnitude points into the page. Beware: all points on the rod rotate at the same rate but they do not translate with the same speed… a) First consider an arbitrary tiny segment (differential segment) of the rotating rod of length . Write an expression for the motional EMF of such a segment in terms of . b) Determine the motional EMF generated between the center and the end of the rotating rod. c) Consider an aluminum helicopter blade that is 9.00 meters long rotating at 420 RPMs in the presence of the earth’s magnetic field of about . Determine a numerical result for the EMF induced between the ends of this rotating helicopter= blade. d) As the helicopter flies around in the real world, the blades will rarely, if ever, rotate perfectly in a plane perpendicular to the magnetic field. Would the real-world induced EMF be larger or smaller than what was computed in part c?

30.16 The figure shows a wire (light grey line) bent into an unusual shape with outer radius and inner radius . A straight rod (black line) slides along the wire making electrical contact with both the inner and outer circular portions of the wire. Assume both the wire and the rod have the same resistivity and cross sectional area . Assume the rod begins at and accelerates from rest

with angular acceleration . A uniform magnetic field with magnitude points into the page. Note: figure not to scale. Beware: notice points on the rod do not all move with identical speed… a) Determine length of the loop formed for arbitrary angle (in radians). b) Determine the area of the loop for arbitrary angle . c) What direction does induced current flow in the rod : radially inwards or radially outwards? d) Determine the induced EMF for this loop as a function of time .

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30.17 a) A magnetic bullet is shot through a short solenoid. A snapshot of the bullet approaching the solenoid taken at time is shown above. Sketch a

S N plausible plot of meter voltage = (across the solenoid) versus time for the bullet. Assume the bullet moves at essentially constant speed and that the bullet is roughly the same length as the solenoid. 0 Video solution provided… - +

COM hi

30.17b) How would the plot change if, instead of passing through the solenoid, the bullet instead passed over it oriented as shown in the second figure? Specifically, S N would the max EMF increase or decrease compared to the first scenario? Would the COM - max EMF be zero in this new case? Would the signs on the meter change? Defend this

answer for credit. hi

Video solution provided…

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30.18 A conducting rod of mass , length , and resistance slides down a pair of conducting rails angled above the horizontal. The rails are connected at the bottom by a wire with negligible resistance. The rails also have negligible resistance compared to . A uniform magnetic field (magnitude ) points to the left at all points between the rails. Assume friction is negligible between the rod and the rails. A figure is shown below. Assume the rod length is essentially the same as the rail separation. A coordinate system is shown in the pure side view for ease of communication. The magnitude of freefall acceleration is . Slanted Side View Pure Side View

a) As the rod slides downwards, which direction does induced current flow in the rod? Note the coordinate system shown in the pure side view picture? Circle the best answer.

b) As the rod slides downwards, in which direction does the magnetic field exert a force on the sliding rod? Circle the best answer.

c) Assuming the rails are long enough, what terminal velocity is reached by the rod as it slides downwards?

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Show simulation for a generator. How does a motor compare to a generator?

30.18½ A loop of wire with turns has area . The total loop resistance is . The loop rotates in a uniform external magnetic Front View Top View field (dots in the figure). The loop has constant angular speed about the axis shown (dotted line in the front view).

a) Determine an expression for magnetic flux at some arbitrary instant in time in terms of the parameters discussed above and the angle shown in the top view.

b) Determine the induced EMF in the loop as a function of time and plot it. c) What is the largest value taken by the induced EMF as the loop spins? Call this value .

d) Going further: come up with aℰ set of parameters that generates a sine curve with spinning 60 times per second. Think: this is equivalent to the output of a wall socket!!! ℰ =

What is back EMF? When you turn on a motor you input current and the motor begins to spin. As it spins the motor becomes a loop of wire spinning in the presence of a magnetic field (or a magnet spinning near a loop of wire). As a result, the spinning loop of the motor serves as a generator and creates an induced EMF in the coil of wire. This induced EMF opposes the change in flux as the loop (or magnet) spins. The net effect is to create an EMF that opposes current flow in the motor. Therefore, as the motor spins, the induced EMF opposes the motion of the motor!!! For this reason we call the induced voltage opposing the motion of a motor back EMF. Note: as the motor increases speed, the back EMF increases. Eventually the back EMF is a major factor in limiting the ultimate speed of a motor.

For thought/web search: How do guitar pickups work? How do GFCI’s work (those plug-ins with the reset buttons)? How does a rechargeable toothbrush work? How does wireless charging work for a phone? How does an induction stove work? How does a shake flashlight work? How do transformers work…discussed on the next page!

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30.19 Transformers: A transformer is a device consisting of two coils of wire in close proximity (usually two oils wrapped on the same chunk of iron or laminated steel). One coil can be thought of as the input; the input coil is call the primary. The output coil is called the secondary. To use the transformer, an alternating electrical signal is connected to the primary. As the magnetic field created by the primary changes , flux through the secondary changes causing an induced EMF in the secondary. In this manner electrical energy is transferred from the primary to the secondary without any electrical contact!

The equation relating coil voltages (for a lossless transformer) is

where is primary voltage, is secondary voltage, is number of turns in the primary, is number of turns in the secondary, and is called the “turns ratio”. a) Why do transformers require an AC signal (think function generator or wall socket) connected to the input instead of a DC signal (think battery or DC power supply)? b) A step-up transformer has a secondary voltage greater than the primary voltage. Which coil, primary or secondary has more turns? Is the turns ratio greater than or less than 1? c) Based on the previous question, what is the logical name for a transformer with turns ratio less than 1? d) Assume energy losses in a transformer are minimal. We know power relates to energy via . Use this information to determine a relationship between the currents in the primary & secondary. Note: the expression above relates primary and secondary voltage for ideal (lossless) transformers. In practice, simply placing two coils of wire next to each will transfer some energy but far less than the ideal theoretical predication. By wrapping the primary and secondary on top of each other, more flux from the primary penetrates the secondary…energy losses are reduced. This is called an air-core transformer. By wrapping the loops around a soft iron rod (easily magnetizable), some flux is trapped in the core and increases flux through the secondary…power losses are reduced. By using a loop core instead of a rod core, more magnetic flux is trapped and forced to permeate the loops.

A shell core (shown at right), is typically made from many layers of silicon-steel (steel laminate core). This improves performance by reducing eddy currents while more effectively A trapping flux in the core. The primary and secondary are both wound on the central limb of C the steel structure. In the figure shown the electrical signal would connect to the 6-turn D primary (thin black lines) at points A & B. The 3-turn secondary coil (thick black lines) are B accessed by connecting at points C & D. Enameled wire (insulated wire) is used to ensure no short circuits accidentally connect the primary to the secondary. In this case, because the secondary has ½ as many turns than the primary, we expect the secondary voltage to be half of the primary (step-down transformer). Since almost all energy is conserved for such a transformer, we expect input and output power to be nearly identical. Therefore we expect twice as much current in the secondary coil (once it is connected to a circuit)!

30.20 Explain why the use of transformers with AC power helps power companies reduce energy waste along transmission lines. Hint: each transmission line has some small resistance . Consider resistive losses in each line for Case 1 (no transformers) versus Case 2 (shown below).

Power Devices Plant in your home Step-up Step-down Transformer Transmission Transformer lines

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Inductors and Self-Inductance Def’n of self-inductance, typically ’s are used in circuits.

transient circuit: An transient circuit uses DC voltage (a battery) in a circuit with resistance and inductance. When the switch is opened or closed there is a transition period (usually very brief) where the current through and voltage across the various ’s and ’s change as a function of time. Capacitors store electrical potential energy, inductors store MAGNETIC potential erg. A cap stores charge but an inductor cannot store current because current is flowing… Eqt’n good for many & transients: where Above equation is valid for determining current and voltage but not energy or power! Energy or power (as a function of time) is found by first determining the current or voltage and then plugging that equation into the equation for energy or power.

ℰ ℰ ℰ

(switch open long time) (instant after switch is closed) (after switch closed long time) = = = initially NOT energized preserves current from is now fully energized = Why? Any coil opposes changes in flux. When is fully energized it acts like a Changing current in a coil changes the flux short circuit (acts like a closed switch) through itself! and allows max current to flow!

30.21 This question refers to the circuit shown above. The solution uses red for and blue for .

a) Determine the current in the inductor (in the above circuit) as a function of time (and graph it). b) Determine the voltage across the inductor as a function of time (and graph it). c) At what time (in terms of and ) will the current reach 99% of its the maximum value?

d) Determine the power delivered to the resistor as a function of time (or the energy stored in the inductor as a function of time).

30.22 A circuit is connected in position A for a long period of time. Assume the battery and wires are ideal with negligible resistance. At time the switch is closed.

a) Just after , what current flows through = resistor ?

b) A long time = after the switch is closed, what current flows through resistor ? c) How long does it take for the inductor to energize to 65% of its steady state value? ℰ

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30.23 Suppose the circuit shown has the switch in the open position for a long period of time. All resistors have resistances of while the battery (with negligible internal resistance) has potential difference .

a) While the switchℰ is still open, what is the current in the resistor ? b) Now the switch is closed and remains closed for a long period of time. What is the final current in resistor ?

30.24 Delicious Problem Consider the circuit at right with and . Assume the switch is initially open for a long time. a) Just after the switch is closed, how much current flows through each branch? b) What is the time constant as the inductor is energizing? c) What energy is associated with in the inductor in steady state? d) What is the magnetic flux in the fully energized state? e) Now assume the switch is re-opened while the inductor is fully energized.

What is the time constant for current decay? f) Describe the energy associated with the inductor as a function of time (after the switch is reopened). g) What is the voltage across the inductor just after the switch is re-opened?

Inductors and circuits are much funkier/trickier than capacitor and circuits. Large voltage spikes and sparking (arcing) will often occur in circuits with large inductances involved. Motors, transformers, relays, and generators all have circuits with large coils of wires. Read about inductive sparking and the use of “flyback diodes” after this problem. Perhaps you have noticed a large spark if you pull a power cable out of the wall for a vacuum cleaner (other device running a large motor)? Do not try this at home…ask your instructor to demonstrate this in some other way.

30.25 Limitations of our model Consider the circuit at right. Assume equal resistors and equal inductors. Assume mutual inductance between the inductors is negligible. Assume the switch is initially open for a long time. a) How much current flows in each branch when the switch is first closed?

b) How much current flows in each branch in steady state? c) Now assume the switch is re-opened. How much current flows in each branch upon reopening? What’s going on here? The current will ramp down very quickly in each branch!

If I am not mistaken, this happens at a rate faster than predicted by in . = Our model = is not+ appropriate − for handling this. Take a circuits class to learn more. I will avoid giving this kind of confusing problem on a test. 165

Oscillator Hints: • Remember these circuits involve a battery and a switch, not a function generator!!!

• energy stored in capacitor is • energy stored in inductor is • = • Charge oscillations are written as sin cos o The plus minus sign depends on± the polarity you choose to assign to the capacitor o If the capacitor is initially uncharged use sin , if it is initially fully charge use cos o Voltage oscillations can be found using • Current oscillations are written as sin = ± = cos o The plus minus sign depends on± the polarity you choose to assign to the capacitor o If the capacitor is initially uncharged use , if it is initially fully charge use sin cos • relates the current and charge oscillation amplitudes . • If =current is 70% of max = 0.70 • WATCH OUT!!! If current decreases by 70% = = 0.30 PHASE ANGLES: In general we may write

= cos sin + The symbol represents a phase angle. The phase = − angle reduces +the amount of thinking about the plus minus signs. Notice what happens when you set in the above equations. 0

= = cos

sin By choosing the appropriate phase angle one ensure both the current and charge oscillations of modelling equations match up to your real life scenario.

WATCH OUT! Phase angles appear in the equations of mechanical mass spring oscillations, travelling waves, oscillator circuits, damped oscillator circuits, and AC circuits including the series circuit. Some books will write equations with instead of . Sometimes textbooks change the sign used in conjunction with the phase angle within a single chapter!!! Furthermore, once a phase angle is employed it doesn’t really matter if you model an equation with sine or cosine. Often books will switch these around on you as well.

What is the point? Phase angles are commonly used for many types oscillations and waves. When using phase angles, be sure to note both the type of trig function (sine or cosine) and the sign used in conjunction with the phase angle. The type of trig function and the sign used affect how the phase angle relates to the initial conditions.

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31.1 oscillator 1: In the figure , , and . Assume .0 = 0.0 = 0.0 the battery is ideal with . The switch is left in position A for a long time. At = .0 time the switch is changed to position B. = 0 a) Determine the maximum charge on the capacitor. b) Determine the oscillation frequency of current (or voltage) oscillations for . 0 c) Determine the amplitude of current oscillations for . 0 d) Determine an expression for the current as a function of time for . Plot 0 current versus time. Clarify the initial direction of current in the inductor. e) At what time (after ) does the current magnitude first reach 90% of its = 0 maximum value? f) Determine the energy stored the magnetic field of the inductor as a function of time for . 0 g) Determine the energy stored in the electric field of the capacitor as a function of time for . 0 h) Determine the total energy stored in the oscillator as a function of time for . 0 i) Explain how this problem is analogous to a person on a swing or a mass on a spring. j) In real life, why do the current and voltage oscillations not go on forever? Where does the energy go?

31.2 oscillator 2: In the figure , , and . Assume = .0 = 0.0 = 0.0 the battery is ideal with . The switch is left in position A for a long time. At = .0 time the switch is changed to position B. Note: a special sliding switch is used to = 0 ensure connection to position B prior to disconnection from position A. This prevents

inductive sparking to position A. Furthermore, assume the switching happens rapidly

enough to ignore any charging of the capacitor by the battery during the switching. a) Determine the amplitude of current oscillations for . 0 b) Determine the amplitude of voltage oscillations on the capacitor.

c) Compare the oscillations in this case to the previous case: should the oscillation amplitudes be identical or not? Defend your reasoning. d) Determine the oscillation angular frequency of current (or voltage) oscillations for . 0 e) Determine expressions for voltage on the capacitor and current in the inductor functions of time for . 0 Plot them. Think about how the capacitor’s polarity and the initial current direction to determine signs of your equations. f) Determine the energy stored in the magnetic field of the inductor as a function of time for . 0 g) Determine the energy stored in the electric field of the capacitor as a function of time for . 0 h) Determine the total energy stored in the oscillator as a function of time for . 0 i) At what time (after ) does the solenoid first transfer 75% of its energy to the capacitor? = 0

Note: the last line of the previous problem statement applies as long as switching time is much less than the charging time constant (in this case 0.24 msec). Another way to start an oscillation with magnetic energy is to first place a large magnet near an inductor in an circuit. One could then rapidly remove the magnet causing a large induced current in the inductor. This gets the oscillation going on the inductor side of things.

Side note: one could perhaps envision using a function generator square wave circuit to give an inductor initial energy during as well…

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AC circuits What does AC mean? Graph of current versus time Compare average current, peak current and rms current Compare instantaneous power to average power 2 2 Power (avg) used by resistor is I RMS R (or V RMS /R) Or still use poison ivy

, , or circuits with AC source = sin = sin − = Notice this phase angle uses a minus sign while the oscillator used a plus sign…I chose this convention to match up with common textbooks.

C only AC circuit: VC will lag I (or I leads V) by 90 degrees (voltage hits peak ¼-cycle after current hits peak)

L only AC circuit: VL will lead I (or I lags V) by 90 degrees (voltage hits peak ¼-cycle before current hits peak)

R only AC circuit: VR will be in phase with current (voltage hits peak a same time as current hits peak)

In LRC series use the following: “Current leads voltage” (or “Voltage lags current”) implies negative phase angle “Current lags voltage” (or “Voltage leads current”) implies positive phase angle.

Pneumonic device: the woman for inductor implies positive phase angle implies voltage leads current ( comes before in ). For frequencies ABOVE resonance the inductor dominates, voltage leads current, thus current lags voltage, and we know the phase angle in is positive. = sin − Equations for a SERIES circuit with an source are shown below. A table explaining how to use these equations is sh own on the next page.

= + − n = = =

= = = − = = =

= cos =

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Equations & Terms used for the Series Circuit Relating peak-to-peak source voltage ( ) to max source voltage ( ) =

Relating peak source voltage to peak current = Total Impedance (note: has units of ) = + − Phase angle

Typically given in degrees… − − Convert to radians to do anything useful = n

Capacitive Reactance (note: has units of ) = = Inductive Reactance (note: has units of ) = = WATCH OUT!!! In the previous two formulas, the symbol stands for source frequency (the frequency of the function generator). It IS NOT resonance frequency . =

Source voltage = Source voltage as function of time = sin Current = Current as a function of time = sin − WATCH OUT!!! In many resources you will see & . = sin = sin + Shifting to the left by (relative to ) is the same as shifting to the right by .

WATCH OUT!!! In the previous two formulas, the symbol stands for source frequency (the = frequency of the function generator).

WATCH OUT!!! People often say source frequency for even though it is actually source angular frequency . Peak voltage across the capacitor =

Voltage across the capacitor as a function of time Hint: ELI the ICE man… must lag current…thus the extra = sin − − −

Peak voltage across the inductor =

Voltage across the inductor as a function of time Hint: ELI the ICE man… must lead current…thus the extra = sin − + +

Alternative version of the phase angle triangle.

Notice this triangle is obtained by multiplying each side of the previous − triangle by . 169

31.3 A series circuit uses a function generator as a source operating with voltage . Here = sin and . Resistance is 1.00 k Ω, capactiance is 20.0 µF and inductance is 3.00 mH. = 9.59 = 00.0 ols a) Determine the resonance frequency of the circuit. b) Determine the phase angle in the circuit. c) Determine the total impedance of the circuit at the operating frequency. d) Determine the amount of power delivered to the resistor at the operating frequency. e) Plot the voltage of the source and current versus time on the same plot.

31.4 A series circuit with AC source has resonance frequency 636.6 Hz . The operating frequency of the voltage source is unknown. A student finds . The resistor has resistance . = .0& = 8.00 0.0 a) Is the phase angle (between current and source voltage) positive, negative or zero? Does current lead or lag source voltage? b) Calculate the values of and for this circuit. c) What is the average power delivered to the resistor if source voltage is RMS? 0.0

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31.5 Full-Width at Half Max (FWHM) iRMS (A) One way to characterize the frequency spread of a resonance 1.00 peak is to determine the FWHM. The graphs at right show resonance peaks for two different series circuits. Both circuits use the same function generator operating which produces a 10.0V RMS output. In both cases the circuits use a 0.75 100.0 mH inductor and 10.0 µF capacitor. The circuit used to make the top graph uses while the other circuit = 0.0 uses . = 0.0 0.50

a) Verify the resonant frequency is 1000 rad/s. Think: does it make sense that changing the resistance has no effect on the resonant frequency? 0.25

b) Verify the max RMS currents should be 1.00 A for the top graph and 0.250 A for the bottom graph. 0.00 ω (rad/s) 0 500 1000 1500 2000

c) What value of will make the current half of the max value? iRMS (A) Answer in terms of . 0.25 d) What value of will make the power half of the maximum value? Answer in terms of . 0.20 e) Determine the frequencies that will cut the power in half. Show you get the quadratic equation 0.15

− − = 0 with roots that can be written as 0.10

= ± + 0.05 The width is the difference between the two roots given by 0.00 ω (rad/s) 0 500 1000 1500 2000 ∆ = + Going further, the -factor is given by = = = ∆ + + The -factor describes how effective the circuit is at storing energy delivered to it. High circuits are narrow and tall (like radio circuits) while low circuits are broad and short. Notice that as increases increases (the peak is wider) while the max power, max current, and -factor all decrease.

Note: some people compute the half-max using the current. Others compute the -factor differently. The point of this exercise is to get you used to dealing with the mathematics of the series circuit algebraically instead of just using numbers all the time. Also, FWHM and -factors are terms used in many branches of physics and engineering. It is good to be exposed to these terms.

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31.6 A series circuit contains an unknown resistance and capacitor in series with a function generator. At frequency the RMS current in the circuit is . If the frequency is doubled, the RMS current increases by 10%. a) Determine the resistance in the circuit. Your answer will look like # . = b) Determine the % change in the phase angle as the frequency is doubled. Is it the same as the % change in current as frequency doubles? c) Assume . Plot impedance ( ), phase angle ( ) & RMS = 00, = .00k& = .00 current ( ) versus from 1 to 600 Hz. Identify the spots in the graph corresponding to and . = 31.7 In the levitating ring experiment, an aluminum ring is suspended on an iron core that extends from the center of a solenoid. The solenoid is connected to a time varying current. The solenoid induces a voltage in the ring given by . Note: this induced voltage is out of phase with the current in the solenoid. The inductance = sin 80° of the aluminum ring is and its resistance is . The ring is in equilibrium due to weight pulling down and a levitation force associated with the out-of-phase magnetic fields associated with the ring and the solenoid. a) Determine an expression for in terms of the given variables.

b) Determine an expression for current in the ring in terms of the given variables. c) Determine the phase angle in the ring in terms of the given variables. d) Depending on the experimental set-up, typical values might be and . The ≈ 60 ≈ 50n experiment is usually run off a standard outlet operating at . The current in the ring can peak at 1.0 60 kA…yes, that’s kilo Amps! Determine the induced emf in the zring. e) Assume and . Plot versus for all frequencies between 0 and 60 Hz. ≈ 60 ≈ 50n f) Assume . Plot current amplitude versus . Said another way, plot versus . = 0.63 g) Plot versus . Note: it is possible to use a function generator connected to a power amp to produce enough power to levitate the ring while still allowing us to use a variable frequency.

31.8 Challenge: Determine the phase angle between the current in the solenoid and the current in the ring. Determine the time averaged force on the ring. You’ll probably need to research on the web but you might be able to follow this resource “Measurements and mechanisms of Thomson's jumping ring”, Paul J. H. Tjossem and Victor Cornejo, Am. J. Phys. 68, 238 (2000). It is readable for a motivated individual at the sophomore level.

Using calculus to determine RMS voltage of a waveform I like to think of RMS voltage using the following formula

= where is some periodic function for voltage with period . 31.8½ Assume . In this equation angular frequency relates to period using the standard = sin relationship ( ). = a) Determine the RMS voltage . b) Suppose the source voltage was instead where is some phase angle. How would = sin + this phase angle affect the result for . V(kV) 20 31.8¾ What about a triangle wave with voltage as a function of time given by 10 the plot shown at right? Hint: split the voltage function into two pieces and inte3grate each piece separately. 0 t (msec) 0 2 4 6 8 10 -10 -20 172

31.9 Look at this complicated set of plots shown below. This information models a series RLC circuit. a) Determine the source voltage amplitude. b) Determine the angular frequency of the source voltage . c) The resonance frequency (not source frequency) is 2.93 rad/sec. Based on the units, do I mean frequency or angular frequency ? d) Think about the phase angle: i. Is it positive or negative? ii. Is this circuit dominated by inductance or capacitance? iii. Estimate the numerical value of the phase angle. e) In the second plot we see four voltages as a function of time. Which voltage corresponds to which circuit element (source, resistor, inductor, cap)? f) I assumed the source voltage had amplitude of 3.50 V, resistance was 0.700 Ω, angular driving frequency was 3.14 rad/s, phase angle was +60.0 degrees, and inductance was 3.00 H. I made my plots and am getting annoyed with an error. Something is wrong with these plots; what is it? Hint: first figure out , , and . Then determine equations for the voltage across each circuit element as a function of time. Do you see what I did incorrectly? 4.0 4.0

3.0 source voltage 3.0 I

2.0 current 2.0 (A) (V)

V 1.0 1.0

0.0 0.0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 -1.0 t (s) -1.0 -2.0 -2.0

-3.0 -3.0

-4.0 -4.0

V (V) Series1 25.0 Series2 20.0

15.0 Series3

10.0 Series4

5.0

0.0 t (s) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 -5.0

-10.0

-15.0

-20.0

-25.0

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31.10 Filter Circuits: It is very useful to consider how a circuit containing ’s, ’s, and/or ’s will behave at various frequencies. By designing circuits appropriately, one can use appropriate combinations of components that filter out only high frequencies (low-pass filter), only low frequencies (high-pass filter), or both low and high frequencies (band-pass filter). Furthermore, diodes, which only allow current to flow with positive voltages, can be added into the mix. Often diodes in a special arrangement are used with capacitors very effectively create a roughly constant voltage from an AC signal (AC-DC converter). Let’s start with the basics… a) At very low frequencies, capacitors will have enough time to become fully charged. When a capacitor is fully charged, how much current flows towards or away from it? b) Using the answer to the above question, do you think a capacitor in a low frequency circuit will act like an open switch (a “break”) or a closed switched (a “short”)? c) At very high frequencies, the capacitor has no time at all to allow charge to build up on it. At high frequencies, do you expect the capacitor to act like a short or a break? d) Now consider inductors. We know that the voltage across an inductor is proportional to the rate of change of current running through it ( ). Using this information, determine if the voltage across an = − inductor will be large or small for a low frequency voltage source. e) Using answer to the previous question, determine how inductors behave with a low frequency voltage source (like a break or like a short). Hint: consider how much voltage is across a closed switch versus an open switch in a DC circuit. f) Now consider an inductor in connected to a high frequency voltage source. Would you expect the voltage across the inductor to be large or small? For high frequencies, will the inductor behave like a break or a short? 174

31.11 A summary of the previous pages results. I have also added two interesting results. These results discuss how a series combination (or parallel ) combination behave at resonance. To truly understand how these last two results come about one should consider complex number theory. This is done later in the class if we have the time (see the next section). Notice there are four boxes you should fill in.

At low frequencies At high frequencies At resonance Component(s) (when ) (when ) (when XL=X C at ) =

Can only have resonance if both and in circuit

in series Think: what should it be? Think: what should it be?

in parallel Think: what should it be? Think: what should it be?

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31.12a Now consider a complete circuit. Sketch what the circuit would look like at low frequencies, high frequencies, and (if applicable) at the resonance frequency. Below each picture, write a statement about how much current flows in the circuit at that frequency. For example, “almost no current” or “max current”. Original Circuit Circuit for low Circuit for high Circuit at

31.12b Use the above information to sketch a plot of total impendence ( ) as a function of angular frequency ( ). Be sure to label the interesting point on the angular frequency axis. = Also, label on the vertical axis. Think: you should already know how relates to at ?

31.12b Now sketch a plot of what current versus angular frequency might look like. Be sure to label the interesting point on the angular frequency axis. Also, label the interesting point on the current axis. = Think: how does relate to and at ?

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31.13a Now consider a different circuit. Sketch what the circuit would look like at low frequencies, high frequencies, and (if applicable) at the resonance frequency. Below each picture, write a statement about how much current flows in the circuit at that frequency. For example, “almost no current” or “max current”. Original Circuit Circuit for low Circuit for high Circuit at

31.13b Use the above information to sketch a plot of total impendence ( ) as a function of angular frequency ( ). Be sure to label the interesting point on the angular frequency axis. Also, label on the vertical axis. Think: how does relate to at very low or very high ?

31.13c Now sketch a plot of what current versus angular frequency might look like. Be sure to label the interesting point on the angular frequency axis. Also, label the interesting point on the current axis. Think: how does relate to and at very low or very high ?

31.13d Lastly, compare the I vs ωωω graphs for the two previous cases. Which combination acts as a band-pass filter? Said another way, which combination allows current to flow in the resistor for only a narrow range of frequencies. To go further, consider a capacitor in series or parallel with a resistor. Which acts as a high pass filter or low pass filter for the resistor? What about an inductor in series or parallel with a resistor?

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 b  1) j = −1 2) z = a + jb = re jθ 3) r = a2 + b2 4) θ = tan −1    a  jθ − jθ jθ − jθ j e + e e − e 5) re θ = r(cos θ + j sin θ ) 6) cos θ = 7) sin θ = 2 2 j 8) “The REAL part of ( a + jb )” = Re( a + jb ) = a 9) “The IMAGINARY part of ( a + jb )” = Im( a + jb ) = b

Complex #’s: Warm-ups/Memory joggers 31.14a Hint: remember to combine imaginary terms with imaginary terms and reals with reals

(a+ jb )+(c+ jd ) =______-3 + j4.2 + 12 – j2.4 = ______

31.14b Hint: remember that a real times an imaginary gives an imaginary while imaginary times imaginary gives…

(a + jb ) ⋅ (c + jd ) = ______(−3 + j )2.4 ⋅ 12( − j )4.2 = ______

______

jθ 31.14c Rewrite the following in polar ( re ) form (hint: eqt’n 4 gets you θ, eqt’n 3 gets r):

0.5 + j0.866 = ______-5 – j12 = ______

31.14d Bonus: Derive eqt’ns 6 and 7 from eqt’n 5 [hints: set r=1, cos(−θ) =cos(θ) & sin( −θ ) = − sin( θ ) ] 31.14e Bonus: Show, using a Taylor series expansion (say the first 6-12 terms) that equation 5 is valid.

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Using complex number theory for ¨® circuits with AC voltages (aka Using Phasors) Here is a brief synopsis of how one does this:

1) Write source voltage as it relates to complex polar form (e.g. ). 2) Write the impedance of each component in complex form. 3) Use standard series-parallel combination rules to determine complex impedance . 4) Real world impedance is the magnitude of the complex impedance. In equations we say where is the complex conjugate of . 5) The complex value of the= current is given by . Tip: when doing this, write in polar form ! 6) Current in the real world is given by Complex =Number Representation Voltage Source Resistive Impedance Capacitive Impedance Inductive Impedance

R − 31.15 Consider the series circuit shown.

a) Show total impedance (written as a complex #) is given by R

~ b) The magnitude of any = complex + number − is given by the square root of the real part squared plus the imaginary part squared. Show that this − gives rise to

= + − c) Show the polar form of complex impedance is

d) Notice total current in the real world is

= = = −

31.16 Now consider the circuit at right. Show we obtain = − − = + − R ~ 179

31.17 Phase angle practice 1: Answers in solutions. Consider the following plot showing numerous curves with different phase angles. The equation was used to generate the curves. Assume the cos horizontal axis represents time in msec and the vertical axis represents charge in mC, a) What charge amplitude and angular frequency were used? b) What is the phase angle for each curve?

q (mC) 15.00

10.00

5.00

0.00 t (msec) 0 2 4 6 8 10 12 14 16 18 20

-5.00

-10.00

-15.00 q1 q2 q3 q4 q5

31.18 Phase angle practice 2: Answers in solutions. Consider the following plot showing numerous curves with different phase angles. The equation was used to generate the curves. Assume the horizontal sin axis represents time in msec and the vertical axis represents current in mA. What is the phase angle for each curve? i (mA) 15.00

10.00

5.00

0.00 t (msec) 0 2 4 6 8 10 12 14 16 18 20

-5.00

-10.00

-15.00 i1 i2 i3 i4 i5

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Electric charge creates electric fields Coulomb’s law was used for point charges. The point charge formula was used to determine electric fields created by continuous charge distributions. • Build up rods and arcs out of point charges o Use o points from the source (the charge) to the point of interest (where you want to determine ) o for a rod along x-axis or for an arc or ring o If uniform …if non-uniform expect some formula for o For an arc, where must be expressed in radians o We assume rods are 1D objects and your slice is also 1D. we assume arcs/rings are simply 1D rods bent into a circular shape and are thus, as far as charge density is concerned, 1D. • Build up thin disks out of rings o Use / o is the radius of the ring (radius of your ring shaped slice of the disk) o is the distance from the center of the ring to the point of interest o …notice the area of a ring shaped slice of radius is 2 2 o Notice a thin plate is 2D and our ring shaped slice is now considered as 2D!?!?!? o The dimensionality of a slice always matches the dimensionality of the object sliced… Gauss’s law tells us

Φ = ∙ = • For isolated conductor, excess charge resides on outer surface. • For conductor, external electric field points perpendicular to surface ( ). Within conductor . = = 0 • Outside long cylinders, long cylindrical shells, or infinite line charge expect . = • Outside spheres or spherical shells expect . = • For plates see the table below. Type of plate on single surface in terms of in terms of Insulator with on a single surface = = = 2 2 Conductor with spread out over both surfaces = = = 2 2 • For perfect symmetry the electric flux may be simplified as o For long cylinders ; assume the end caps are negligible. ∙ = 2

o For spheres ∙ = o For slabs ; the 2 comes from using both sides of Gaussian surface. 2 4 • Inside insulating objects, ∙ determine = enclosed charge using . = o Inside cylinders we use . = 2 o Inside spheres we use . = o Inside plates we use . 4 • To determine total charge of an object = use • For cylindrical objects/line charges convert charge to linear density (or vice versa) using . • For parallel plates we use either where is the surface charge on a single plate or ∆ where d is the spacing between the plates and is the potential difference between the plates. ∆ 182

Electric current produces magnetic fields The Biot-Savart Law tells us

• Recall points from the source (the= current carrying wire) to the point of 4 4 interest (where you want to determine ) • points along the wire in the direction of current • In practice, first do the cross-product, then think about symmetry to simplify integrals. • Use the right hand rule to get a feeling for the direction of created by any small portion of the wire o Align fingers of right hand with o Curl fingers of right hand to o Right thumb aligns in direction of o Notice is always perpendicular both to and to Ampere’s law tells us

∙ = • Outside of cylinders or cylindrical shells . • For cylinders/cylindrical shells use = . 2 • Inside cylinders/cylindrical shells use ∙ = where = = 2 o Notice this is not the same from Gauss’s law! o In Gauss’s Law referred to the sidewall of the cylinder through which the electric field… • To determine total current use =

As far as we know magnetic monopoles do not exist . All magnets known to humans are dipoles (two poles: north and south). Every time people break magnets into smaller pieces, each piece has become a smaller magnet, again with two poles. While not strictly forbidden by physics, humans have never encountered any situations where magnetic monopoles exist.

The equation that summarizes this result is

Φ = ∙ = 0 • Notice it requires a closed surface. • Through any closed surface you have as many magnetic field lines coming in as going out.

Changing magnetic fields induce electric fields We know from previous chapters that a time varying magnetic flux through loop creates an induced voltage. These results can be summarized by

Φ ∙ = − • If a closed integral is used, that simply indicates we have gone around an entire loop (say a loop of wire). • In electrostatics . This equation doesn’t apply to electric fields created by induction. Said = − ∙ another way, this equations applies only if . = 0 • Note: a conducting rod moving in the presence of a uniform magnetic field experiences and induced voltage . Assumes is field perpendicular to rod and velocity. Derive this from ℰ = = ×

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Changing electric fields induce magnetic fields!!! This is something we have not yet discussed. Imagine a parallel plate capacitor. Attached to this capacitor is a voltage source that is manipulated in some manner to create a current that increases at a constant rate. Between the plates we know a uniform electric field would also be increasing at a constant rate as the plates charge. Just as the current in the wires attached to the plates creates a magnetic field, the changing electric field between the plates also creates a magnetic field. This has been experimentally verified. In equation form we say

Φ ∙ = • Notice there is no negative sign as there were for induced EMFs. • The constants appear in this equation. • The speed of light in a vacuum ( ) relates to these constants by or . = = • Comparing to Ampere’s law we see must have dimension of current. We give this term the name of displacement current. o is not a real current. We call it a current because it produces a magnetic field and has the same units as current. o We may modify Ampere’s law to reflect this current using

∙ = + Φ =

Maxwell’s Equations The four most fundamental equations out of all of the above information are as follows

Name Integral Representation Differential Representation

Gauss’s Law Φ = ∙ = ∙ = Ampere-Maxwell Φ Law ∙ = + = × = + Faraday’s Law Φ ∙ = − = − Gauss’s Law for

Magnetism 0 ∙ = 0 Φ = ∙ =

One other key equation is the Lorentz Force Law. This describes the force on a charge in the presence of external electric and magnetic fields. Lorentz Force

Law = + ×

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Comments on earth’s magnetic field • Currently the earth’s geographic north pole is the approximate location of a magnetic south pole • At various locations around the earth the mag field is described by declination and inclination o Declination describes left or right angle from due north geographically o Inclination describes angle up or down towards the earth’s surface o The poles of the earth’s magnetic field reverse over long time scales (10 5-10 6 years) according to observations of weakly magnetized rocks on either side of the Mid-Atlantic ridge o It is suspected that the earth’s mag field can be explained by the “geomagnetic dynamo” principle. At the time of writing this, it appears likely this explanation is correct. o Explanations behind pole reversals are also in the preliminary stages.

Magnetism and Quantum numbers You may be familiar with quantum numbers from chemistry. The four you usually talk about are , , , and . Quantum Number Name What does it represent Typical Values Principal Q.N. Energy = , 2, 3, , … Orbital Q.N. Angular momentum 5 0,, 2, …4 Magnetic Q.N. -component of angular momentum , , … , , Spin Q.N -component of spin 2 Why am I bringing this up? If we think of an electron as orbiting a nucleus we can think of that moving charge as a current loop that creates a magnetic field. If we think of an electron as a spinning ball of charge we can imagine once again we have a tiny loop of current creating a magnetic field. While these descriptions are not good models (for reasons we have no time to discuss), it does give a feeling for why we expect electrons to create magnetic fields. The magnetic fields are associated with spin and orbit for electrons are typically written in terms of magnetic moments. Lastly, nucleons (protons and neutrons) can also produce magnetic fields. It was surprising to observe that neutrons, an electrically neutral particle, could produce a magnetic field! This was late explained using quark theory. Relevant equations are

ℎ J = .27 × 0 9 4 T ≈ ±

= − J = = . 0 4 T J = = −9.66 × 0 T = − ∙

= × • The electron’s magnetic moments are much larger than the nuclear magnetic moments. • Electrons with large angular momentum in the -direction produce the largest magnetic moments • Negative magnetic moments imply the electron and neutron tend to anti-align with external mag field! • The spin angular momentum of nucleons is exploited by MRI’s to create images inside the human body. • Electron magnetic moments are used to explain why materials can be magnetic

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Three types of magnetism Diamagnetism • Extremely weak magnetism • All materials exhibit diamagnetism but diamagnetism is negligible if material exhibits either para or ferro • Appears only while exposed to external mag field • Dia materials are repulsed by the non-uniform field of a typical magnet!!! • Web search for “diamagnetic grapes” and “frog levitation” Paramagnetism • Weak magnetism (but not as weak as dia) • Occurs in transition, rare earth, and actinide elements • Appears only when exposed to external mag field • Increasing temperature reduces paramagnetism • Para materials are attracted by the non-uniform field of a typical magnet. • Web search “paramagnetic oxygen” Ferromagnetism • Strong magnetism • Occurs in iron, nickel, and other elements/compounds (NdFeB=neodym magnets, Alnico, magnetite, etc) • Ferro materials are attracted by the non-uniform field of a typical magnet. • Spins in one atom tend to align with spins in adjacent atoms creating regions called domains. This happens in spite of atomic collisions tending to randomize spins. • Once aligned by an external magnetic field, object can remain partially magnetized • Ferromagnetic alignment does not occur above a material’s Curie temperature • Web search “stroke nail magnet”, “degauss a ship”, “hysteresis”, “lodestones”.

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32.1a Which types of materials can be repelled by a uniform external magnetic field? Circle all that apply.

Paramagnetic Diamagnetic Ferromagnetic None of these

32.1b Which materials can remain magnetized after removal of an external magnetic field? Circle all that apply.

Paramagnetic Diamagnetic Ferromagnetic None of these

32.1c Rank the types of magnetism from weakest to strongest. 32.1d Which type of magnetism, strictly speaking, is present in all materials.

Paramagnetic Diamagnetic Ferromagnetic None of these

32.1e Which types of materials can be repelled by a non-uniform external magnetic field? Circle all that apply.

Paramagnetic Diamagnetic Ferromagnetic None of these

32.2 This is a good time to show the diamagnetism of grapes (or a web search for “magnetic fruit”).

32.3 Perhaps a good time to show magnetic suction (doorbell demo). What happens if you reverse the current direction…will the rod shoot out the other way? Does the behavior of the rod depend on how long the switch is connected? Try short connection times versus leaving switch closed. Perhaps copper/aluminum versus iron rods?

32.4 Maxwell’s equations are shown below. Several comments are shown on the right. Each comment relates directly to one and only one of the Maxwell’s equations. Which comment relates to which equation?

A changing electric field produces a Φ = ∙ = magnetic field.

For electrostatics (no moving charges), the

Φ = ∙ = 0 electric field inside a conductor is zero.

Φ No magnetic monopoles are known to exist. ∙ = − A changing magnetic field (that penetrates a Φ ∙ = + = loop) will create induced current in the loop.

32.5 A capacitor is discharging at the instant shown. Current in the wire travels in the direction labeled.

a) Which way are electrons moving in the wires? b) Which plate is at higher electric potential? P c) What direction does the electric field point between the plates?

d) In what direction is displacement current between the capacitor plates? e) Is the magnetic field at P into or out of the page?

32.6 Consider the sphere at right in the presence of a non-uniform magnetic field. Assume the sphere experiences a force (due to the external magnetic field) to the left. a) Is the sphere paramagnetic, diamagnetic, ferromagnetic or is it impossible to determine? b) How does your answer change if the force on the sphere is to the right?

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32.7 For this question, the orbital motion of an electron can be considered as a current loop. In each of the three cases below, a circular electron orbit is shown. In each case the electron orbit is in the presence of an external magnetic field. Notice only one of the cases has uniform magnetic field.

Case 1 Case 2 Case 3

a) Determine the direction of the orbital magnetic moment. b) Determine the direction of net force caused by the external magnetic field on each orbiting electron.

32.8 At one particular instant in time, an electric field vector points to the right. The circular line indicates the direction of the magnetic field induced by the electric field. Is the electric field magnitude increasing or decreasing?

32.9 A particular atom has a magnetic moment . We are able to turn on a uniform external = magnetic field with magnitude at the location of this atom. a) In what direction should the external magnetic field be oriented to give the atom maximum possible potential energy? b) If this orientation is used, and we assume perfect alignment, what torque is exerted on the atom? c) In practice no alignment is ever perfect (nor is any real-life external magnetic field perfectly uniform). Suppose the external magnetic field had a small positive component to the right in addition to your answer to part a. What direction would the molecule rotate? Try to answer with a unit vector. Hint: the direction of an angular velocity points along the axis… d) When the magnetic moment flips direction, does the atom gain or lose energy? How much? e) The energy of a photon propagating through empty space relates to wavelength using the equation . = What wavelength of photon is produced by such a spin flip?

32.10 This might be a good time to look up a simulation on how NMR or MRI works…

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Electromagnetic waves Water waves, string waves, earthquake waves, and other mechanical waves all require a medium to travel. They all travel through something . Electromagnetic waves can travel through nothingness – the vacuum of space. If an electron (or some other charge) wiggles, a changing electric field is created. This changing electric field induces a magnetic field as described by Maxwell’s equations. This induced magnetic field simultaneously changes and induces an electric field. As a result, the changing fields initially created by the charge propagate throughout space.

At right is a snapshot of an E & M wave at one instant in time . The electric field is indicated by the black arrows in the -plane. The magnetic field is indicated by the grey arrows in the -plane. Energy is carried to the right by this wave. A right hand rule is goes to goes to direction energy travels. Equation Comments This is called “wavenumber”. Check the units or context to determine if a problem 2 = is discussing wavenumber, spring constant, or one of the other ’s in physics… 2 Angular frequency relates to period and frequency in the usual way. = = 2 • Wave travels in direction of (in this case ). sin • Using in lieu of implies wave direction+ flips sign = sin • is electric field amplitude (biggest arrow in the wave) • is magnetic field amplitude (biggest arrow in the wave) Wave speed = speed at which energy is transported from point A to point B Wave speed for electromagnetic waves in a vacuum 3.00 × 0 s • Energy transported by a single photon (sometimes written ) • ℎ ℎ = 6.626 × 0 J ∙ s = . 0 ∙ s = ℎ = • ℎ ≈ 240 ∙ n 4 4 • Note: = .602 × 0 J Remember the velocity selector and crossed fields problems… = • o nin co i nis o × • The magnitude is intensity as a function of time = = • Direction of is direction wave travels (direction energy travels) • AVERAGE Intensity of wave = = = • Energy split equally between electric and magnetic fields 2 2 AVERAGE Intensity of a wave with power spread out over area = Force and radiation pressure if electromagnetic wave is 100% absorbed • Absorbing object has surface area = = • Electromagnetic wave has intensity incident upon entire area Force and radiation pressure if electromagnetic wave is 100% reflected • Absorbing object has surface area = = • Electromagnetic wave has intensity incident upon entire area

ℎ Momentum (magnitude) associated with a photon = = =

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Electromagnetic Spectrum I recommend doing a search for “electromagnetic spectrum animation”. I looked at several of the search results and found them both interesting and motivating.

Lo erg Hi erg Lo Hi Radio –microwave – IR – Red(650 nm) – Green (540 nm) – Blue(480 nm) – UV – X-ray – Gamma Ray Long Short

33.1 How far does light travel in 1.00 ns? You may assume that travelling through air and empty space are approximately equivalent. Also determine how long to travel 1.00 m.

33.2 An oscillator uses . What is the photon energy of electromagnetic radiation = 3.00μH& = 75p F produced by this oscillator? Answer in Joules & eV. Also, characterize what type of wave this is with a web search (e.g. radio, x-ray, blue, etc.).

33.3 Assume we have a microwave operating at frequency 2.45 GHz. Determine the wavelength of radiation produced by the magnetron of the microwave.

33.4 In general, electromagnetic radiation is produced by accelerating charges. Memorize this statement in case I ask about it on the test. a) Look at simulations of the patterns of radiation produced by different moving charges. a) Look at simulations of how radio transmission and reception occur. b) Consider mentioning rod antennas versus loop antennas. c) Antennas & radiated fields are worth mentioning to promote interest but not subjects on the test.

33.5 The intensity of a light wave is . Determine the amplitude of electric field oscillations. 555

33.6 The range of frequencies used for AM radio are to . The frequencies used = 535kHz = 605kHz by radio stations are spaced at intervals. = 0k Hz a) Before computing anything, which frequency corresponds to the longest wavelength AM radio wave? b) Before computing anything, which frequency corresponds to the highest energy AM radio wave? c) Possible frequencies used by stations are spaced at equal frequency intervals. Are wavelength intervals between adjacent possible wavelengths also equally spaced? d) Determine the largest and smallest wavelength difference possible for two stations.

33.7 Suppose you have red, green and blue light waves travelling through empty space. Rank the speeds of each color light wave from highest to lowest clearly indicating any ties.

33.8 An AM radio wave has a wavelength of 1 km. a) Determine the energy & frequency associated with this wavelength in vacuum. b) Determine the energy & frequency of a wave with 10 12 times the energy . c) Compare your answers from part b to a web search for “electromagnetic spectrum” to classify the wave.

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33.9 Two plane electromagnetic waves are described by the Case 2 cases below. For each case use the coordinate system Case 1 shown. Determine the propagation direction for each case.

33.10 A plane electromagnetic wave propagating through free space has an electric field described by

= 2.22 sin − 4.44 × 0 s a) Determine the wavelength & wavenumber ( ). b) In which direction does the wave propagate? c) Parallel to which axis does the magnetic field oscillate? d) What is the magnetic field amplitude? e) Determine wave intensity. f) Does the intensity of this wave increase, decrease or remain constant as it propagates through free space?

33.11 A laser beam emits light with wavelength 532 nm. The beam produces 10.0 mW of power distributed uniformly in a beam with circular cross-section of radius 3.00 mm. The beam travels in the positive -direction. Assume the beam travels through empty space. Assume light in a laser beam is modeled as a plane wave. Assume the light is polarized along the axis. This is simply another way of saying electric field oscillates parallel to the axis. a) What color is the laser beam? b) Determine the average intensity of the beam. c) For comparison, determine the average intensity of electromagnetic radiation as it leaves the surface of the sun using . = 3.90 ×0 & = 6.96 × 0 d) Determine max electric and magnetic fields in the beam. e) Determine the wavenumber and angular frequency of photons in the beam. f) Write down expressions for the electric field and magnetic field as functions of space and time.

33.12 A laser beam has a power rating of 5.00 mW. The laser is a Helium-Neon (HeNe for short) laser with a wavelength of 632.8 nm. a) Determine the energy of a single photon emitted by the HeNe laser. b) Determine the number of photons per second be emitted by the laser. c) Determine the momentum (magnitude) of a single photon. d) What color is the laser beam?

33.13 An astronaut in space holds a battery powered 10.0 W laser. The astronaut is motionless relative to her spaceship and has run out of thruster fuel. She is untethered and cannot simply pull her way back to the ship. She is 10.0 m from the ship. a) Describe how the astronaut could use the laser beam as a thruster to get back to her ship. b) What are the pros and cons of throwing the laser versus using the laser light as a thruster to increase the astronaut’s momentum towards the spaceship?

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33.14 Explain the physics of a solar sail. A solar sail may be conceptualized as a giant piece of aluminum foil floating in space that can reflect sunlight. a) Why might a piece of aluminum foil in space start accelerating away from the sun instead of being pulled towards the sun by gravity? b) Is the acceleration constant? Explain why or why not? c) Which is more effective: a black sail or a shiny sail? Explain why. 33.15 Suppose we want to make a dish for collecting radio signals from space. The diameter of the dish is (HUGE). The dish focuses all incident radio wave energy onto a sensor. 00 The sensor is sensitive enough to detect any signals larger than . At a particular instant, dish is receiving a .00 signal from a point source away. Note: ly = the distance light travels (in a vacuum) in one .00l = li earth year. a) When energy is leaving the point source, does the energy travel as a plane wave or does it spread out spherically? b) Consider the wave fronts impacting the earth. Are they best modeled as spherical or planar wave fronts? c) Determine the minimum power required for the point source signal to be detected.

33.16 A laser beam of cross-sectional area is pointed upwards is incident upon a spherical bead of radius . Assume the bead reflects the entire beam vertically downwards; this is approximately true if the > beam diameter is significantly smaller than the sphere diameter. Assume the beam has uniform intensity across the entire beam (not true in real life). The bead has mass . The bead is observed to be suspended motionless by the laser beam. This is called optical levitation. Note: the situation described is unstable. a) Determine the required power of such a laser beam.

b) Assume the bead is made of glass with density and diameter . 2200 25icons c) Suppose an absorbing bead with the same mass and radius is instead used. What is the magnitude and direction of the new bead’s acceleration when it is released from rest?

Look up how optical tweezers really work (the above problem is not an accurate depiction).

33.17 The magnetic field of a particular plane electromagnetic wave oscillates parallel to the -axis with a magnetic field described by the equation

= sin + a) In what direction does the wave propagate? b) What is the corresponding electric field equation?

33.18 What capacitance must be used in conjunction with a inductor (in an oscillator circuit) to produce 3.33 H light of wavelength ? Is this a large or small capacitor in your opinion? 600n

33.19 An astronaut and all her gear has mass . She is outside her spaceship at rest in deep space. Included in 25k her gear is a nuclear powered laser. Starting from rest she turns on the laser and points it to the left. She 20.0 then takes a nap and wakes up later. The laser was running the entire time (always pointing to the left). 5.55s How fast is she moving at the instant she wakes up?

33.20 A solar sail and the attached spacecraft have mass . The spacecraft is initially at rest distance = 2000k from the sun (radius of earth orbit). The power of the sun is and its mass = .50 × 0 = 3.90 ×0 is . What is the minimum size solar sail required for the radiation force magnitude to exceed the .99 × 0 k gravitational attraction to our sun. Does radius matter? What assumptions are you making?

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33.21 When exposed to extremely bright light the pupil of your eye might be as small as . Assume you stare 2 directly at a light bulb. The light leaves the source emanating hemispherically. When it impacts your pupil, 50 let us assume the light has traveled far enough such that the light incident upon your pupil can be considered as having planar wave fronts. Assume intensity of sunlight hitting the earth as about . a) How far is your pupil from the bulb for the bulb to appear as bright as sunlight incident upon the earth. b) Estimate how many photons per second are entering your eye. For comparison, assume a person with fully dark adjusted eyes can supposedly detect as few as 100 photons per second.

33.22 A cylinder of diameter and height is placed on its side on a = track. The coefficient of static friction between the track and cylinder is . A laser beam shines on the flat end of the cylinder. The cylinder completely reflects all incident light. Note: the light from the laser can be considered as plane waves. Assume the beam diameter is twice as large as the cylinder diameter. Also assume, somewhat unrealistically, the beam intensity is uniform at all points in the beam. The laser power is .

Determine the maximum density of the cylinder that allows it to be on the verge of slipping. Contemplate how realistic this situation is by considering a tiny cylinder ( ) with a very powerful laser = (say ) on a very slippery interface ( ). = 0 = 0.05

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Partial final exam review guide Chapters and content may have changed significantly so this list is not necessarily complete. Find the electric potential, electric field, electric potential energy, or electric force in terms of algebraic variables for positive and/or negative charges. Gauss’s law for cylinder/spheres, insulating/conducting, shells/solid, multiple shells of differing types, uniform/non- uniform, inside/outside List the field in terms of Q’s or λ's for all possible radii of interest. Make a graph of the electric field magnitude versus radius for all radii of interest as listed above. Label the radius axis with a, b, and c. Specifically show the value of the electric field at a, b, and c on the E-field axis of your graph. Memorize E for infinite sheets Memorize E for uniform cylinders, uniform spheres (both inside and out) Know how to derive equations for electric potential and electric field for thin rod and thin ring Know how to generalize thin ring to disk (as in text) Know how to generalize disk to cylinder (as in hmwk) Know all your charge densities Find the equivalent capacitance between points a and b of the circuit shown. Algebraically or with numbers. If a 6 V battery is connected across a and b, what is the total charge stored on the capacitors? What is the energy stored in the capacitors?

What is the voltage across C4?

What is the charge on C4? etc What if dielectric brought in/out with switch open/closed? Nasty 3 loop circuit with multiple batteries and resistors (do KVL) Use KVL to find I thru, V across or P dissipated by any R in the circuit Solve 3 by 3 matrix by hand ρL Plug and chug using R = or ρ = ρ (1+α∆T ) A 0 ρds dR = Integration using A RC circuit find V(t), I(t), Q(t), P(t)

Draw pics of RC circuit at t0-, t0+ , t=infinity Similar to above for RL circuits All those nasty RHR rules (force on moving charge, force on current-carrying wire, torque on current carrying loop, B-field created by long straight wire, B-field created by circular loop of wire, B-field created by a solenoid Biot-Savart law using segments of straight and circular wire (no integral table provided…know how to do straight wire segments using trig subs) Ampere’s law for concentric uniform cylinder and cylindrical shell (coax cable) Ampere’s law for non-uniform shell or cylinder Mass spectrometer or similar (velocity selector region and only B region) Charge moving in a circle due to B-field (forces and circular motion) Derive force per unit length on two long parallel straight wires LRC series circuits numerical LRC filter circuits conceptual/graphs of I vs ω or Z vs ω

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More final exam notes taken from previous semesters: There are too many problems to include and still have the test be doable in a two-hour time period. I will write the test and then start chopping stuff out to make it doable. Expect many short questions with maybe 2-3 meatier problems. Types of questions that come to mind are shown below. While there are no guarantees, ideally I would pick one question from each row from either column A or B (on some occasions both). Column A Column B Guaranteed some kind of question about E & M waves. Expect this problem to be blatantly copied from either the workbook or hmwk questions from Ch 33. This will be there for sure. As an example, one year I took a problem with numbers and simply made it algebraic. Other times I might change a number/prefix or the sign/direction of a quantity. Guaranteed conceptual/definition questions on diamagnetism, paramagnetism, ferromagnetism or Maxwell’s laws similar to hmwk. This will be there for sure. Expect this problem to be blatantly copied from either the workbook or hmwk questions from Ch 32. This will be there for sure. Point charges relating to Use Biot-Savart law with wires located on corners of , , Probably triangle, square, or circle geometry square, triangle, etc to relate to or force on a wire

Continuous distribution of charge (arc or rod) perhaps non-uniform…derive electric field or electric potential Biot-Savart law (see figure at right

as an example) 2 Gauss’s, spherical or cylindrical geometry (no slabs) Ampere’s law, cylindrical geometry (possibly non-uni) (insulator/conductor/or both, possibly non-uni) Simple cap circuit (think 2 in para plus 1 in series or 2 in Concept Q on caps in series versus para (charge voltage, series plus 1 in para…calculations) energy) Simple resistor network (think 2 in para plus 1 in series or Concept Q on resistors in series versus para (current, 2 in series plus 1 in para…calculations) voltage, power) Problem on & (think: rod on rails with , Conceptual Q’s on mag flux & induced currents (think Φ changing area with , is function of time changing magnet falls thru tube or changes through loop) flux through loop, generator/spinning loop ) oscillator series transient transient Force/energy on charge in electric field (think charge on a Force on moving charge or torque on loop of wire or forces string between parallel plates, charge travels between on charge moving in circular motion in magnetic field or parallel plates, two charged balls on strings) forces on wires carrying current Relate the resistance of a shape to geometry & resistivity. Think Conceptual questions on resistivity and resistance of resistance for uniform rectangular different shapes similar to hmwk slabs, uniform cylindrical wires, or uniform cylindrical shell Determine electric field from voltage by taking derivative OR determine potential difference by integrating electric Electric potential versus position with graphing field

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PHYS163 Test FinalA Fall 2018 Name:______No graphing calculators. CELL PHONES OFF! No food during the exam. Additional pages are provided as scratch paper at the end of the test. Clearly label scratch work just in case you are able to get some partial credit for it. When there is a box for your final answer use it or risk losing points.

⋅ 2 −2 C = .602×0 C ≈ 9 × 0 ≈ 3 × 0 0 = 8.854 × 0 N⋅2 T J ⋅ · J −7 ∙ ℎ = 6.626 × 0 s ℎ ≈ 240 n = .602×0 0 = 4 × 0 A 2 2 = = ∆ = + = + 2∆ 2 2 2 = 2 2 = = 2 = 2 2 ∆ ∙ = = ∥ = = = 0 0 20 = 2 3/2 = 2 /2 = − − = − ∙ +2 +2 ∆ ∆ ∆ ∆ = = 2 = = − 0 = + + ⋯ = + 2 + ⋯ = 2 = + 2 + ⋯ = + + ⋯ = = + ∆ 2 ∆ ∆ / = o

= × = × = × = 0 0 0 = − ∙ = = 2 ℎ = 2 0 × ∙ = 0 = ∙ = 4 2 Φ = ∙ 2 = − Φ = = = 2 ∆ 2 2 − = = − = + − n = ∆ = = = 0 sin = sin −

= = = = = = 0 = Resistivity at Temp. 2 Material 20° C Coefficient 2 = cos = (in SI units) (in SI units) Silver 1.62 × 10 -8 4.1 × 10 -3 = = = 2 = -8 -3 2 2 Copper 1.69 × 10 4.3 × 10 × 1 -8 -3 = = = = = Aluminum 2.75 × 10 4.4 × 10 0 2 0 2 0 2 0 Nichrome 1.00 × 10 -6 0.4 × 10 -3 = = ℎ = -5 -3 Carbon 3.5 × 10 -0.5 × 10 Germanium 0.46 -48 × 10 -3 . = = ℎ = =

−1 = = ln + ± / + + ± 1 / = = sin = ± + + ± 1 1 = tan ± = ± ± ln + ± + 2 2 Binomial expansion: 1 = ln + + 2 1 ≈ 1 ⋯