1 163 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 3 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 ͥͥͥͦ͟ ̀ŹͥΝΜ ͦ = ͧ ͦŹͥ΢Ν ͦ ͦͥ΢Νͦ ͥͥͥͦ͟ The hat changes to a vector & the ͟|ͥͥ||ͥͦ| ̀ŹͥΝΜͦ = ͦ ͦ̂ͥ΢Νͦ ̀ͥͦ = ͦ ͦͥ΢Νͦ distance squared changes to ͦͥͦ distance cubed . means the force of charge 1 on charge 2 ̀Źͥͦ means the magnitude of the force between charges 1 and 2 ̀ͥͦ means the vector from the center of 1 to the center of 2 ͦŹͥͦ 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 ͟ = ͨ_T ͤ = ͨ_& Cͦ N ∙ mͦ t 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 .v due to gravity in freefall was . ͕Ź= −͛̈́̂ 4 21.1 Determine the units of the Coulomb constant by looking at &|,u||,v|. ͟ ̀ͥͦ = v -uv 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. ͦŹͦͧ 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) ͥͥ = +1.00 μC ͥͦ = −2.00 μC ͥͧ = −3.00 μC ͭ 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 ͥͦ ̀Źͥͧ ̀Źͦͧ ͥͥ 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 1 (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 3.46 Ɩ 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). Ɩ 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. 2ͥ ͥ a) Predict the direction of the force if the sensor charge is placed midway between and . ͥ 2ͥ 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. 2͕ ͬ ͬ 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 .