Chapter 23 Electric Potential (Voltage)

Transcription

1 Chapter 23 Electric Potential (Voltage)

2 Electric potential energy Recall how a conservative force is related to the potential energy associated with that force: The electric potential energy: Change in potential energy is independent of path since the electric field is a conservative field.

3 Consider a uniform electric field (say inside a parallel capacitor) If a proton is taken from location B to location C, how does its potential energy change? 1. it decreases 2. it increases 3. it doesn t change

4 Suppose a proton is released from rest just below the top (positive) plate of an parallel plate capacitor with an electric field strength E = 100 N/C. If the distance between the plates is d = 3 mm, how fast is it moving when it hits the bottom (negative) plate?

5 Electric Potential Energy of Two Point Charges What is the change in potential energy of the charge q 0 as it goes from position a to position b? U AB = Z B A q 0 ~ E ~ dr U AB = Z rb r a q 0 kq r 2 dr U AB = kq 0 q r r b r a = k q 0q r b k q 0q r a If we let r a be infinity (the zero point), and r b an arbitrary distance r, then U(r) =k q 0q r

6 Example Rutherford scattering. A helium nucleus of mass 4 m p is emitted with an initial speed of v 0 = 4.9 x 10 5 m/s towards a gold nucleus of charge q 2 = 79 e. What is the minimum distance between the two particles (assume the gold nucleus doesn t move)?

7 Electric Potential Energy with Several Point Charges Consider the potential energy of a charge q o due to several point charges: U 0 = X i kq 0 q i r i U 0 = kq 0 (q 1 /r 1 + q 2 /r 2 + q 3 /r 3 )

8 Electric Potential (Voltage) Electric potential, or voltage, at a point in space is defined as the electric potential energy per unit charge associated with a test charge at that point. V (~r )= U(~r ) q 0 Potential energy deals with the energy of a particle. Voltage deals with all locations in space (no particle needs to be there). Unit of electric potential is the volt (V). 1 V = 1 J/C.

9 Potential Difference (Voltage Difference) Voltage difference is given by: V = U q 0 In a uniform field, the potential difference becomes V AB = ~ E ~r

10 Clicker Question In a parallel plate capacitor, the electric field is uniform and is directed from the positive plate to the negative plate. An electron goes from location A to location C. Which statement is true? A) The electron goes from a high voltage to a lower voltage. B) The electron goes from a low voltage to a higher voltage. C) The voltage is the same at both locations.

11 Clicker question The figure shows three straight paths AB of the same length, each in a different electric field. Which one of the three has the largest magnitude of a voltage difference between the two points? A. (a) B. (b) C. (c)

12 Voltage of a point charge Recall the potential energy of two point charges: U(r) =k q 0q r Thus the voltage a distance r from the charge q is given by V (r) =U(r)/q 0 = kq r Alternatively, V (r) = Z r 1 ~E ~dr = Z r 1 kq r 2 dr = kq r

13 Clicker Question A conducting sphere of radius R has a net charge Q (on its surface). What is the voltage at the center of the sphere? 1. V = kq/r 2. V > kq/r 3. 0 < V < kq/r 4. V=0

14 V (r = R) = Z R 1 kq r 2 dr = kq R V = V (r = 0) V (r = R) =0 ) V (r = 0) = kq R

15 Clicker Question A solid sphere of radius R has a UNIFORM charge density per unit volume ρ and net charge Q. The voltage at the center of the sphere is 1. V = k Q/R 2. 0 < V < kq/r 3. V > kq/r 4. V=0

16 Clicker Question A solid sphere of radius R has a UNIFORM charge density per unit volume ρ and net charge Q. The voltage at the center of the sphere is 1. V = k Q/R 2. 0 < V < kq/r 3. V > kq/r 4. V=0 V (r = 0) = Z R 1 kq r 2 dr Z 0 R kqr R 3 dr V (r = 0) = kq R + kq 2R = 3kQ 2R

17 Voltage due to a charge distribution If the electric field of the charge distribution is known, the voltage can be found by integration. Alternatively, the voltage can be found by summing point-charge potentials: For discrete point charges, V (P )= V ( P)= Z i ~E d~r = kq i r i Z X For a continuous charge distribution, V (P )= Z kdq r i ~E i! d~r = X i Z ~E i d~r = X i V i (P )

18 Question Hollywood Square Four point charges are arranged at the corners of a square. Find the electric field E and the potential V at the center of the square. 1) E = 0 V = 0 2) E = 0 V 0 3) E 0 V 0 4) E 0 V = 0 5) E = V regardless of the value -Q +Q -Q +Q

19 Clicker Question Two identical positive charges of charge Q are a distance d apart. What is the voltage at the midway point between the charges? a) k Q/d b) 2 k Q/d c) 4 k Q/d d) 8 k Q/d e) 0

20 Voltage due to a charged ring For a uniformly charged ring of total charge Q, integration gives the potential Z on the ring axis: kdq V = dq = ad r V (x, y, z) = Z 2 Very hard integral in general! If P is on x axis, then r only depends on x and a. 0 k a d r(,x,y,z) On x axis: kdq k kq r r x + a ( ) = = dq= 2 2 V x

21 Maximum voltage of a Van de Graaff generator. Molecules in air get ionized for electric fields greater than roughly E max = 3 x 10 6 V/m. What is the maximum voltage of a charged sphere of radius R=0.2 m?

22 Millikan s Oil Drop Experiment Charged oil droplets made to levitate inside capacitor Measure voltage difference across plates Release and measure terminal velocity (which gives droplet radius and thus mass) Determine net charge on droplet. qe = q V d = mg charge always a multiple of -1.6 x C

23 Equipotential Diagrams An equipotential is a surface on which the potential (voltage) is constant. In two-dimensional drawings, equipotential curves are similar to contours of an elevation map.

24 Clicker Question

25 Clicker question The figure shows cross sections through two equipotential surfaces. In both diagrams the potential difference between adjacent equipotentials is the same. Which of these two could represent the field of a point charge? A. (a) B. (b) C. neither (a) nor (b)

26 Conductors There s no electric field inside a conductor in electrostatic equilibrium. At the surface there s no parallel component of the electric field. Therefore in electrostatic equilibrium, the entire conductor is at the same potential

27 Determining E from V Voltage can be determined if electric field is known Can electric field be determined if voltage Z is known? For a very small displacement, V = ~E d~r = E ~ ~r Suppose Then E x = E r = E x x V x = V x Can do the same thing in other direction: E = rv = r = x î V x î + V y ĵ + V z ˆk The derivatives here are partial derivatives, expressing the variation with respect to one variable alone. (gradient of V)

28 Clicker Question For which region is the magnitude of the electric field the highest?

29 Clicker Question If a proton were released from rest at location B, in which direction would it start moving? 1. To the right 2. To the left 3. Up 4. Down 5. It would remain at rest

30 Example: Getting E from V Determine the electric field along axis of charged ring of charge Q and radius a: E = rv = V x î + V y ĵ + V z ˆk Recall that the voltage due to a charged ring is: kdq k kq r r x + a ( ) = = dq= 2 2 V x Use this to determine E(x): E(x) = dv/dx = kqx (x 2 + a 2 ) 3/2

31 Which group of charges took more work to bring together from a very large initial distance apart? d d d Both took the same amount of work. +1 d +1

32 End of Chapter Problems What is V at location A?

33 A very long wire carries a uniform linear charge density λ. Using a voltmeter to measure potential difference, you find that when one probe of the meter is placed 2.70 cm from the wire and the other probe is 1.50 cm farther from the wire, the meter reads 600 V. What is λ?

34 A disk with radius a has uniform surface charge density σ. What is the voltage a distance y above the center?