AP Physics – Electric Potential Energy

AP Physics – Potential Difference - 5

By convention, a point in an electric circuit is said to have zero electric potential (potential difference) if is grounded (connected to earth).

The other way to have zero potential difference is to have a point charge be located at an infinite distance from another charge.

We’ve been looking at a test charge placed in an electric field; we learned how to determine the potential difference on the test charge, the change in potential energy of charge, and the work needed to move the charge within the field.

We can look at the field itself and calculate the potential difference at a point within the field using this equation:

1 q V  4 0 r

V is the potential difference at a point that is a distance r from the charge q that created the field.

The Potential difference depends only on charge and the distance from the charge.

Superposition Principle: When there are two or more charges in proximity to one another, the change in potential energy is the sum of the potential difference for each charge. Potential difference is a scalar, therefore:

Total electric potential at some point near several point charges is the algebraic sum of the electric potentials from each of the charges

This can be written in general form as:

1 q V   i 4 0 i ri

1 q This shows that for a number of charges, the voltage is the algebraic sum of for 4 0 r each charge.

This is the equation you will be provided on the AP Physics Test.

55 q 1 q 3 r 1 r 3

r 2 q 2 Three charges are arranged as shown in the drawing to the right. To find the potential difference at the point in the center, you q q 3 calculate the potential difference for q1, q2, and q3. You can do 1 this if you know the value for the charge and the distance to the central point. Lastly, to find the total potential difference, you simply add up the three V’s that you found for the three points. r1 r3 Let’s put some numbers into the mix. Let q1 = 1.5 C , q2 = 2.0

C, and q3 = 2.5 C. For the distances, r1 = 2.0 cm, r2 = 1.8 cm, and r = 2.7 cm. 3 r2

1 qi V   q2 4 0 i ri

1q Nm 2   1  V1 8.99 x 109 1.5 x 10 6 C  6.7 x 10 5 V 1 2  2  4 0r 1 C  2.0x 10 m 

Nm 2  1  V8.99 x 109 2.0 x 10 6 C  10 x 10 5 V 2 2  2  C  1.8x 10 m 

Nm 2  1  V8.99 x 109 2.5 x 10 6 C  8.3 x 10 5 V 3 2  2  C  2.7x 10 m 

Now, having found each of the potential differences for the charges, we can find the total potential difference.

V V1  V 2  V 3

V=6.7 x 105 V + 10 x 10 5 V + 8.3 x 10 5 V = 25 x 10 5 V = 2.5 x 10 6 V

An important thing to understand is that potential difference is not a vector, it’s a scalar, so you add them up algebraically.

56  Two charges are situated as shown. The distances are given in the drawing. The charges have

the following values: q1 is 5.0 C and q2 is -2.0 C. (a) Find the potential difference at point P. (b) How much work is required to bring a third point charge of 4.0 C from infinity to P?

(a) Find the potential difference at P. P 1 q V  for q1 r 4 0 r 2 Nm 2  1  V 8.99 x 109 5.0 x 106 C 4.0 m 1 2    C  4.0 m   V11.2 x 103 V  1.12 x 10 4 V - 1 q 3.0 m q 1 2 We must now find r2:

2 2 r2 4.0 m  3.0 m  5.0 m

Nm 2  1  V8.99 x 109  2.0 x 106 C   0.360 x 10 4 V 2 2    C  5.0 m 

V   V1  V 2

DV =(1.12 x 104 V) + ( - 0.360 x 10 4 V) = 0.760 x 10 4 V = 7.6 x 10 3 V

(b) How much work to bring a third point charge of 4.0 C from infinity to P?

The work is equal to the potential energy change of the move.

6 3 J  U qV W  qV  4.0 x 10 C 7.6 x 10  C  W=30.4 x 10-3 J = 3.0 x 10 - 2 J

Velocity and Electric Fields: We can use all this impressive new knowledge about potential difference, potential energy, and work to solve impressive problems. In fact, you will be required to solve such problems to be successful on the AP Physics Test.

 Calculate (a) the energy of proton that is accelerated from rest through a potential difference of 120 V and (b) the speed of an electron accelerated from rest through a potential difference of 120 V.

57 Potential energy that the proton gains will be equal to the kinetic energy it will have after it has been accelerated through the field.

1 1 2qV U qV  W W  K  mv2 so mv 2  qV v  2 2 m

J  2 1.60x 1019 C 120 2qV  C  v     m 1.67x 1027 kg

m2 m m v=229.9 x 108 = 15 x 10 4 = 1.5 x 10 3 s2 s s

 Through what potential difference would an electron need to accelerate to achieve a velocity of 1.80 x 107 m/s?

We employ the same reasoning as in the previous problem: the work is the same as the potential energy that goes into the system. We can then set that equal to the kinetic energy that the electron ends up with after its been accelerated through the field.

1 2 U qV  W mv qV Solve for potential difference 2 2 31 7 m  2 9.11x 10 kg 1.80 x 10  mv   s V    9.22 x 102 V 2q 2 1.60x 1019 C

 An electron is 3.00 cm from the center of a uniformly charged sphere of radius 2.00 cm. The sphere’s charge is 1.00 x 10 –9 C. How fast will the electron be traveling when it hits the surface of the sphere?

Conservation of Energy:

Ui= K f k q q 1 e 1 2  mv2 r 2 2k q q v  e 1 2 rm

58 kg m  m2  2 8.99x 109  1.60 x 10 19 C 1.00 x 10  9 C s2 C 2    v    0.0300m 0.0200 m 9.11 x 1031 kg 

m2 m m v=315.8 x 1012 = 17.8 x 10 6 = 1.78 x 10 7 s2 s s

 An electron is fired into at the midpoint of a field between two charged plates. The initial velocity of the electron is 5.6 x 106 m/s. The plates are 2.00 mm apart. The V for the plates is 100.0 V. (a) Find the magnitude of the electric field. (b) Make a drawing of the two plates and put in some arrows showing the direction of the field. (c) Determine where the electron will hit on the upper plate.

(a) We need to find the magnitude of the field. We can use the equation for the electric field to find the field. We’re ignoring the minus sign as we’re only interested in the D magnitude of the field.

V100 V V E= = = 5.00 x 104 d0.0020 m m

(b) From the drawing, we see that the field is oriented so that the force on the electron is upward. This means that the field must point downwards. Recall that the direction of the field is the direction that a positive test charge would be deflected. A negative charge would go in the opposite direction.

(c) An upward force is exerted on the electron as it travels through the field. We can find the force from the field:

F 骣 4N - 19 - 15 E= F = Eq =琪5.00 x 10 1.60 x 10 C = 8.00 x 10 N q桫 C ( )

Since we know the force acting on the electron, we can use the second law to find the acceleration.

F F ma a  m kg m 8.00x 1015 2 m m as 0.878 x 1016  8.78 x 10 15 9.11x 1031 kg s 2 s 2

59 Note -- acceleration of gravity is insignificant compared with the acceleration form the field, so we can ignore it.

Next we find the time it takes for the electron to hit the upper plate. Since it’s in the center of the plates when it enters, it will be accelerated a vertical distance of 1 mm.

1 2 y at y is 1.00 mm or 10-3 m 2

3 2y 2 1.00x 10 m  t  0.2278 x 1018 s 2  0.477 x 10  9 s m a 8.78x 1015 s2

We know the time till the electron hits, we also know its initial horizontal velocity, and we know that it will travel horizontally at a constant speed. So knowing the time and the speed, we can find the horizontal distance it travels before the electron hits.

6m  9  3 x vx t 5.6 x 10 0.477 x 10 s  2.7 x 10 m  2.7 mm s  

Equipotential Curves:

Equipotential curves can be drawn for a charge. For example if we look at a positively charged conducting sphere, the lines of force would look like this:

Now if we draw a curve around the sphere that represents places that have the same potential difference, we get a sphere. It will be a circle on our two dimensional drawing. This is because the voltage only depends on the distance from the charge. The charge is, of course, evenly distributed on the sphere. The voltage drops off with distance, so the curves nearest the surface of the sphere would have the greatest potential.

1 q The equation for potential difference is simply: V   i 4 0 i ri

60 骣 1 q Which simplifies to: V = 琪桫4p 0 r

As the distance (r) increases, the potential difference decreases. So if we draw equipotential curves that represent a specific voltage difference – say each curve represents a decrease of 5 volts, it would look like this:

61 Dear Cecil: A friend of mine is selling these "corona discharge" air filters--she has severe asthma and swears this little gizmo is better than pooping gold bricks. Is this truly the miracle hand.we asthma We even sufferers tested laundryare looking balls, for? remember? This unit So claims I sent toJane be to a rummage"miniature through miracle" the made archives.possible byThe "a assembled revolutionary research discovery presents in the electrodynamic usual inconclusive negative muddle. corona Some discharge observations: purification technology." Supposedly it's "more advanced than any filter/fan air purifier or HEPA filtration device" and cleans air "naturally. . . . Similar to a thunderstorm, a  Air ionization is to today's scientists what magnetism was to medical types in the 19th corona discharge of negative high energy is passed through the pollutants." In layman's century. In other words, a possible cure for just about anything. A computer search terms it's supposed to zap any particle large enough to be an allergen that passes turned up nearly 400 scientific articles on air ionization, claiming that negative ions do through it. Or at least that's how it was marketed to me. How about it, Cecil? Is it the everything from reduce asthma symptoms to improve your mood and enhance answer to my prednisone prayers or a crock of shit? --Breathlessly awaiting your children's learning ability. Yet the methodology of these studies is sometimes suspect. answer, L. Hayes, Oregon For example, a couple papers claim that daily sessions with a high-output negative ionizer will alleviate seasonal affective disorder (winter blues). But one study Cecil replies: measured the efficacy of "bright-light" therapy (which is pretty much what it sounds Maybe it's just me, but when I hear somebody go on about revolutionary discoveries, negative like) in alleviating SAD while using negative ion therapy as a control--that is, as a high energy, or anything to do with ions (which are what a corona discharge device produces), presumably ineffective therapy intended to provide a baseline comparison. Bright-light I make sure I've got a firm grip on my wallet. Still, one never likes to dismiss things out of therapy alleviated SAD symptoms, but no more than negative ion therapy did. Normally you'd consider this evidence of the placebo effect -- the participants thought the therapy would do some good, so they persuaded themselves it did. In this case, though, the investigators stood this conclusion on its head: bright lights and air ionization aren't equally bad, they're equally good! Whatever you say, guys.  Air ionizers may improve some things, but they don't improve asthma. I'm not saying ionizers are useless. Studies found that they reduced transmission of airborne disease in poultry, cut down on microbial pollution in a dentist's office, etc. But they didn't help asthma and arguably made things worse (Warner et al, 1993).  HEPA filters don't do much for asthmatics either. HEPA filters are often recommended to screen allergens out of the air and alleviate asthma, but Antonicelli et al (1991) concluded the filters didn't do squat. Reisman et al (1990) say "the overall impression" is that HEPA filters reduce asthma, but once you look closely at the study, your overall impression is that Reisman and company are kidding themselves. One possible explanation, which may also account for the ineffectiveness of ionizers, is that asthma is caused less by airborne particles than by contact with objects, e.g., your allergen-infested mattress.  Nobody knows what the "HEPA" in "HEPA filter" stands for. Writers in the scientific journals almost universally believe HEPA stands for "high efficiency particulate air," whatever that is. Consumer Reports, however, says HEPA stands for "high efficiency particulate absorbing." I vote for CR.  Negative ions do something to the learning ability of kids, but God knows what. Morton and Kershner (1990) report, "On a dichotic listening task using consonant- vowel combinations, both [learning disabled and normal-achieving children] showed an ion-induced increase in the normal right ear advantage." The normal right ear advantage?  To cure asthma, we must turn to the mysterious East. Studies claim that asthma symptoms can be reduced by yoga (Nagendra and Nagarathna, 1986) and by transcendental meditation (Wilson et al, 1975). Sure. My advice: don't get rid of the prednisone yet. 62 --CECIL ADAMS

Breathes There the Man (from The Lay of the Last Minstrel)

BREATHES there the man with soul so dead, Who never to himself hath said, This is my own, my native land! Whose heart hath ne'er within him burn'd, As home his footsteps he hath turn'd From wandering on a foreign strand! If such there breathe, go, mark him well; For him no Minstrel raptures swell; High though his titles, proud his name, Boundless his wealth as wish can claim; Despite those titles, power, and pelf, The wretch, concentred all in self, Living, shall forfeit fair renown, And, doubly dying, shall go down To the vile dust, from whence he sprung, Unwept, unhonored, and unsung.

--- Sir Walter Scott