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Electric Potential & Capacitance
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 equilateral triangle of
side r. What is the potential at point C?
𝑘𝑄
2𝑘𝑄
𝑘𝑄
2𝑘𝑄
−𝑘𝑄
(A) 2
(B) 2
(C)
(D)
(E)
𝑟
𝑟
𝑟
𝑟
𝑟
3. A positive charge +2Q is located at point -3 m from the origin. What is the sing
and size of the second charge Qx located at point 2 m from the origin, if the
electric potential at the origin is zero?
4𝑄
−4𝑄
5𝑄
−5𝑄
𝑄
(A)
(B)
(C)
(D)
(E)
3
3
3
3
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)
(D)
(B)
(C)
(E)
5. A positive charged particle q is accelerated through a uniform electric field E. 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
2
(D) kQ/r
0
2
(E) kQ/R
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
2
(A) kQ/R
kQ/R
(B) kQ/R
kQ/R
(C) kQ/R
kQ/r
2
(D) kQ/r
kQ/r
2
(E) kQ/r
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?
kQ
V (x )  2
x  a2
(A)
kQ
V (x ) 
ax
V (x ) 
(B)
kQ
a x
2
2
(C)
V (x ) 
kQ
x2
(D)
V (x ) 
kQ
x
(E)
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? Assuming that electric
potential is zero at infinite distance from the spheres.
𝑄
𝑄
𝑄
𝑄
𝑄
A)
B)
C)
D)
E)
2
2
4𝜋Ɛ0 𝑟
2𝜋Ɛ0 𝑟
2𝜋Ɛ0 𝑟
2𝜋Ɛ0 𝑏
4𝜋Ɛ0 𝑟
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? Assuming that electric
potential is zero at infinite distance from the spheres.
A)
𝑄
4𝜋Ɛ0 𝑎
B)
𝑄
2𝜋Ɛ0
𝑏2
C)
𝑄
2𝜋Ɛ0 𝑏
D)
𝑄
1
1
2𝜋Ɛ0 𝑎
𝑏
( + )
E)
𝑄
1
1
4𝜋Ɛ0 𝑟
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?
V0
V0
A) 2V0
B) V0
C) zero
D)
E)
2
4
14. Two charges +Q and +2Q 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 the electric field is 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)
1
e
4 0 R
(B)
e2
8 0 R
1
(C) 
e2
8 0 R
1
(D)
e2
4 0 R 2
1
(E) 
e2
4 0 R 2
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 the point of distance 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. The
capacitance of the second capacitor is most nearly
(A) ¼Co
(B) ½Co
(C) Co
(D) 2Co
(E) 4Co
1
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. 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)
4𝜋Ɛ0 𝑎𝑏
𝑏+𝑎
(B)
4𝜋Ɛ0 𝑎𝑏
𝑏−𝑎
(C)
4𝜋Ɛ0
𝑏+𝑎
(D)
2𝜋Ɛ0 𝑎𝑏
𝑏−𝑎
(E)
√2𝜋Ɛ0 𝑎𝑏
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) 2F
(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 1/3d of separation.
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 second capacitor in
terms of C0?
(𝜅+1)𝐶0
𝜅𝐶
𝐶
𝜅𝐶
(𝜅+1)𝐶0
(A)
(B) 0
(C) 0
(D) 0
(E)
3
3
3
2
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 is
equal to
𝑒𝑄𝑑
Ɛ 𝑄𝑑
Ɛ 𝑄𝐴
Ɛ 𝑄
Ɛ 𝑄
(A)
(B) 0
(C) 0
(D) 0
(E) 0
Ɛ0 𝐴
𝑒𝐴
𝑑
𝑑𝐴
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?
𝜅𝑈
𝑈
(𝜅+1)𝑈0
(A) 0
(B) 0
(C)
(D) κ𝑈0
(E) (κ+1)𝑈0
2
𝜅
2
34. What is the capacitance of a 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 any 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) Vo2/d
36. A sheet of insulating plastic material is inserted between the plates of 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 require to separate the plates is
(A) Q2/4C
(B) Q2/2C
(C) Q2/C
Question 38-40
(D) 2Q2/C
(E) 4Q2/C
A
C
B
5V
E
D
10V
15V
38. What is the direction of the magnetic field at point C?
(A)
(B)
(C)
(D)
(E)
39. How much 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
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.C
21.E
22.B
23.D
24.D
25.B
26.B
27.A
28.C
29.A
30.D
31.A
32.A
33.B
34.A
35.B
36.D
37.B
38.A
39.C
40.C