Electric Potential & Capacitance Multiple Choice

1. Two charges +Q and -3Q are placed in opposite corners of a square. The

work required to move a test charge q from point A to point B is:

(A) dependent on the path taken from A to B

(B) directly proportional to the distance between A and B

(C) positive (D) zero (E) negative

2. Two charges +2Q and –Q are placed at two corners of an equilateral triangle.

What is the potential at point C?

(A) 𝑘𝑄

𝑟2 (B)

2𝑘𝑄

𝑟2 (C)

𝑘𝑄

𝑟 (D)

2𝑘𝑄

𝑟 (E)

−𝑘𝑄

𝑟

3. A positive charge +2Q is located at point -3 m from the origin. What is the sign

and size of the second charge Qx located at point 2 m from the origin, if the

electric potential at the origin is zero?

(A) 4𝑄

3 (B)

−4𝑄

3 (C)

5𝑄

3 (D)

−5𝑄

3 (E)

𝑄

3

4. A conducting sphere of radius a is charged positively with a charge +Q. A

conducting spherical shell with radius b and charge –Q surrounds the sphere.

Which of the following graphs best represents the electric potential V as a

function of r?

(A) (B) (C)

(D) (E)

5. A positive charged particle q is accelerated through a uniform electric field E

starting from rest. If v0 is the velocity of the particle at the end of distance d,

what is the velocity of the particle at the end of 2d?

(A) √2v0 (B) √3v0 (C) 2v0 (D) 4v0 (E) 8v0

6. A conducting sphere of radius R is charged with a positive charge +Q. Which of

the following represents the magnitude of the electric field E and the potential

V as functions of r, the distance from the center of the sphere, when r < R?

E V (A) 0 kQ/R

(B) 0 kQ/r

(C) 0 0

(D) kQ/r2 0

(E) kQ/R2 0

7. A conducting sphere of radius R is charged with a positive charge +Q. Which of

the following represents the magnitude, of the electric field E and the potential

V as functions of r, the distance from the center of sphere, when r > R?

E V

(A) kQ/R2 kQ/R

(B) kQ/R kQ/R

(C) kQ/R kQ/r

(D) kQ/r2 kQ/r

(E) kQ/r2 kQ/r2

8. Positive charge Q is uniformly distributed over a thin ring of radius a, that lies in

a plane perpendicular to the x-axis with its center at the origin 0, as shown above.

The potential V at points on the x-axis is represented by which of the following

functions?

(A) V x

kQ

x a( )

2 2 (B) V x

kQ

a x( )

2 2 (C)

V xkQ

x( ) 2 (D)

V xkQ

x( )

(E)

V xkQ

a x( )

9. Two positive and two negative charges are arranged at the corners of a square,

as shown above. At the center C of the square, the potential due to one charge

alone is Vo and the electric field due to one charge alone has magnitude Eo.

Which of the following correctly gives the electric potential and the magnitude

of the electric field at the center of the square due to all four charges?

Electric Potential Electric Field

A) Zero Zero

B) Zero √2 (2Eo)

C) 2 Vo 4Eo

D) 4 Vo Zero

E) 4 Vo 2Eo

10. What is the radial component of the electric field associated with the

potential V = a𝑟−3 where a is a constant?

A) -3a𝑟−4 B) -2a𝑟−3 C) -4a𝑟−2 D) 3a𝑟−4 E) 2ar

11. Two concentric, spherical conducting shells have radii a and b and equal

charges +Q, as shown above. Which of the following represents the electric

potential as a function of distance r in the region r > b? Assume that electric

potential is zero at an infinite distance from the spheres.

A) 𝑄

4𝜋Ɛ0𝑟2 B)

𝑄

2𝜋Ɛ0𝑟2 C)

𝑄

2𝜋Ɛ0𝑟 D)

𝑄

2𝜋Ɛ0𝑏 E)

𝑄

4𝜋Ɛ0𝑟

12. Two concentric, spherical conducting shells have radii a and b and equal

charges +Q, as shown above. Which of the following represents the electric

potential as a function of distance r in the region a < r < b? Assume that electric

potential is zero at an infinite distance from the spheres.

A) 𝑄

4𝜋Ɛ0𝑎 B)

𝑄

2𝜋Ɛ0𝑏2 C)

𝑄

2𝜋Ɛ0𝑏 D)

𝑄

2𝜋Ɛ0(

1

𝑎 +

1

𝑏) E)

𝑄

4𝜋Ɛ0(

1

𝑟 +

1

𝑏)

13. The electric potential of an isolated conducting sphere of radius R is V0. What

is the electric potential at the center of the sphere?

A) 2V0 B) V0 C) zero D) V0

2 E)

V0

4

14. Two charges +Q and +2Q are located on the x-axis. The electric potential is

zero at which of the following points?

(A) A (B) B (C) C (D) D (E) None of these points the potential is zero.

15. The graph shows the electric potential V as a function of distance r. At which

point is the electric field positive with respect to +r?

(A) A (B) B (C) C (D) D (E) E

16. A positive charge of 6µC moves between two points through an electric field.

Work done by the electric field on the charge is 30 mJ. What is the potential

difference between the two points?

(A) 3kV (B) 4kV (C) 5kV (D) 6kV (E) 9kV

17. An electron -e moves in a circular orbit around a proton +e of a constant radius

R. Assuming that the proton is stationary, which of the following represents the

kinetic energy of the two-particle system?

(A) R

e

04

1

(B)

R

e 2

08

1

(C)

R

e 2

08

1

(D)

2

2

04

1

R

e

(E)

2

2

04

1

R

e

18. In a region of spherical symmetry, the electric potential as a function of

distance x is presented by the following formula V(x) = bx3.What is the

magnitude of the electric field at a point of distance x = a from the origin?

(A) 2ba2 (B) ba (C) 3ba2 (D) -3ba2 (E) -2ba3

19. A parallel plate capacitor has a capacitance Co. A second parallel plate

capacitor has plates with twice the area and twice the separation of the first

capacitor. The capacitance of the second capacitor is most nearly:

(A) ¼Co (B) ½Co (C) Co (D) 2Co (E) 4Co

20. Three 12-microfarad capacitors are connected in series. The equivalent

capacitance of the set of capacitors is:

(A) 4 F (B) 2 F (C) 3 F (D) 9 F (E) 18 F

21. Two 4-microfarad capacitors are connected in series with a 12-volt battery. The

energy stored in each capacitor is:

(A) 144 J (B) 4 J (C) 12 J (D) 36 J (E) 72 J

22. An isolated capacitor with air between its plates has a potential difference Vo

and a charge Qo. After the space between the plates is filled with oil, the

difference in potential is V and the charge is Q. Which of the following pairs of

relationships is correct?

(A) Q=Qo and V>Vo (B) Q=Qo and V<Vo (C) Q>Qo and V=Vo

(D) Q< Qo and V<Vo

(E) Q>Qo and V>Vo

23. Which of the following can be used along with fundamental constants, but no

other quantities, to calculate the magnitude of the electric field between the

plates of a parallel plate capacitor whose plate dimensions and spacing are not

known?

(A) The flux between the plates

(B) The total charge on either plate

(C) The potential difference between the plates

(D) The surface charge density on either plate

(E) The total energy stored in the capacitor

24. When two identical parallel plate capacitors are connected in series, which of

the following is true of the equivalent capacitance?

(A) It depends on the charge on each capacitor.

(B) It depends on the potential difference across both capacitors.

(C) It is larger than the capacitance of each capacitor.

(D) It is smaller than the capacitance of each capacitor.

(E) It is the same as the capacitance of each capacitor.

25. Two conducting spheres have charges +Q and –Q respectively. The inner shell

has a radius a and outer shell has a radius b. What is the capacitance of the

system of two shells?

(A) 4𝜋Ɛ0𝑎𝑏

𝑏+𝑎 (B)

4𝜋Ɛ0𝑎𝑏

𝑏−𝑎 (C)

4𝜋Ɛ0

𝑏+𝑎 (D)

2𝜋Ɛ0𝑎𝑏

𝑏−𝑎 (E)

√2𝜋Ɛ0𝑎𝑏

𝑏−𝑎

26. A sheet of mica is inserted between the plates of an isolated charged parallel

plate capacitor. Which of the following statements is true?

(A) The capacitance decreases

(B) The potential difference across the capacitor decreases

(C) The energy of the capacitor does not change

(D) The charge on the capacitor plates decreases

(E) The electric field between the capacitor plates increases

27. Five 4-µF capacitors are connected in one network. What is the capacitance of

the circuit?

(A) 2F (B) 4 F (C) 6 F (D) 12 F (E) 24 F

28. Five 4-µF capacitors are connected in one network. What potential difference

must be applied between points Y and Z if the circuit is connected to a 12 V

battery?

A) 4 V B) 3V C) 6 V D) 9 V E) 18 V

29. A capacitor C0 has two parallel plates each of area A separated by a distance d.

Another capacitor with identical dimensions is filled with a dielectric of

dielectric constant κ that covers the area of one plate and 1/3 of the separation

d. What is the capacitance of second capacitor in terms of C0?

(A) 3𝜅𝐶0

2𝜅+1 (B)

3𝐶0

2𝜅 (C)

3𝐶0

𝜅 2 (D)

2𝐶0

2𝜅+1 (E)

2𝐶0

𝜅+1

30. What is the potential difference between two points in an electric field if 1 J of

work is required to move 1 C of charge between the points?

(A) 2 V (B) 3 V C) 5 V D) 1 V E) 0 V

31. A capacitor C0 has two parallel plates each of area A separated by a distance d.

Another capacitor with identical dimensions is filled with a dielectric of

dielectric constant κ that fills in the separation between the plates and covers a

half of the area of each plate. What is the capacitance of the second capacitor in

terms of C0?

(A) (𝜅+1)𝐶0

3 (B)

𝜅𝐶0

3 (C)

𝐶0

3 (D)

𝜅𝐶0

2 (E)

(𝜅+1)𝐶0

2

32. A parallel plate capacitor has charge +Q on one plate and charge -Q on the

other. The plates, each of area A, are a distance d apart and are separated by a

vacuum. A single proton of charge +e, released from rest at the surface of the

positively charged plate, will arrive at the other plate with kinetic energy equal

to:

(A)𝑒𝑄𝑑

Ɛ0𝐴 (B)

Ɛ0𝑄𝑑

𝑒𝐴 (C)

Ɛ0𝑄𝐴

𝑑 (D)

Ɛ0𝑄𝑒

𝑑𝐴 (E)

Ɛ0𝑄

𝑑𝐴

33. An isolated parallel plate capacitor is charged to a maximum energy U0. A

dielectric plate with a dielectric constant κ is inserted in the space between the

plates. What is the energy of the capacitor with the dielectric plate?

(A) 𝜅𝑈0

2 (B)

𝑈0

𝜅 (C)

(𝜅+1)𝑈0

2 (D) κ𝑈0 (E) (κ+1)𝑈0

34. What is the capacitance of this given circuit?

(A) 4.4 µF (B) 3 µF (C) 4 µF (D) 8 µF (E) 4.6 µF

35. A capacitor is made of two identical conducting plates parallel to each other

and separated by a distance d. The capacitor is charged to a potential difference of

Vo by a battery, which is then disconnected. If edge effects are negligible, what is

the magnitude of the electric field between the plates?

(A) Vod (B) Vo/d (C) d/Vo (D) Vo/d2 (E) Vo

2/d

36. A sheet of insulating plastic material is inserted between the plates of a fully

charged capacitor, which is disconnected from a battery. What effect does this

have on the charge and potential difference between the plates?

(A) Charge increases and voltage decreases

(B) Charge decreases and voltage increases

(C) Charge stays unchanged and voltage increases

(D) Charge stays unchanged and voltage decreases

(E) Voltage stays unchanged and charge increases

37. An isolated parallel plate capacitor has capacitance C and charge Q. The

separation between the plates is doubled. The work required to separate the plates

is:

(A) Q2/4C (B) Q2/2C (C) Q2/C (D) 2Q2/C (E) 4Q2/C

Question 38-40

38. What is the direction of the electric field at point C?

(A) (B) (C) (D) (E)

39. How much work is required by an external force to move a particle of charge

6x10-6 C from point A to point E?

(A) -6x10-7 J (B) -3x10-5 J (C) 6x10-5 J (D) 3x10-7 J (E) 6x10-4 J

40. If the distance between B and D is 10-3 meters, what is the strength of the

electric field at B?

(A) 5x10-2 N/C (B) 10x103 N/C (C) 5x103 N/C

(D) 10x10-2 N/C (E) 5x10-1 N/C

A

B

C

D

E

5V

10V

15V

Answers

1. D

2. C

3. B

4. B

5. A

6. A

7. D

8. B

9. B

10. D

11. C

12. E

13. B

14. E

15. D

16. C

17. B

18. D

19. C

20. A

21. E

22. B

23. D

24. D

25. B

26. B

27. A

28. C

29. A

30. D

31. E

32. A

33. B

34. A

35. B

36. D

37. B

38. A

39. C

40. C