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:

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1 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. Two charges +Q and Q are placed at two corners of an equilateral triangle. What is the potential at point C? (A) kq r kq (B) r (C)kQ r (D) kq r (E) kq r 3. A positive charge +Q is located at point -3 m from the origin. What is the sign and size of the second charge Q x located at point 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

2 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)

3 5. A positive charged particle q is accelerated through a uniform electric field E starting from rest. If v 0 is the velocity of the particle at the end of distance d, what is the velocity of the particle at the end of d? (A) v 0 (B) 3v 0 (C) v 0 (D) 4v 0 (E) 8v 0 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/r 0 (E) kq/r 0

4 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/r kq/r (B) kq/r kq/r (C) kq/r kq/r (D) kq/r kq/r (E) kq/r kq/r 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 kq V ( x) a x (B) V ( x ) a kq x kq x (C) V ( x ) V x (D) ( ) kq x (E)

5 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 V o and the electric field due to one charge alone has magnitude E o. 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 (E o ) C) V o 4E o D) 4 V o Zero E) 4 V o E o 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) -ar 3 C) -4ar D) 3ar 4 E) ar 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. Q A) B) Q C) Q Q Q D) E) 4πƐ 0 r πɛ 0 r πɛ 0 r πɛ 0 b 4πƐ 0 r

6 1. 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 4πƐ 0 a B) Q C) Q πɛ 0 b πɛ 0 b D) Q πɛ 0 ( 1 a + 1 b ) E) Q 4πƐ 0 ( 1 r + 1 b ) 13. The electric potential of an isolated conducting sphere of radius R is V 0. What is the electric potential at the center of the sphere? A) V 0 B) V 0 C) zero D) V0 E) V Two charges +Q and +Q 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.

7 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) e R (B) e R (C) e (D) R e R 1 (E) In a region of spherical symmetry, the electric potential as a function of distance x is presented by the following formula V(x) = bx 3.What is the magnitude of the electric field at a point of distance x = a from the origin? (A) ba (B) ba (C) 3ba (D) -3ba (E) -ba 3 0 e R 19. A parallel plate capacitor has a capacitance C o. 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) ¼C o (B) ½C o (C) C o (D) C o (E) 4C o

8 0. Three 1-microfarad capacitors are connected in series. The equivalent capacitance of the set of capacitors is: (A) 4 F (B) F (C) 3 F (D) 9 F (E) 18 F 1. Two 4-microfarad capacitors are connected in series with a 1-volt battery. The energy stored in each capacitor is: (A) 144 J (B) 4 J (C) 1 J (D) 36 J (E) 7 J. An isolated capacitor with air between its plates has a potential difference V o and a charge Q o. 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=Q o and V>V o (B) Q=Q o and V<V o (C) Q>Q o and V=V o (D) Q< Q o and V<V o (E) Q>Q o and V>V o 3. 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 4. 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.

9 5. 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) 4πƐ 0ab b+a (B) 4πƐ 0ab b a (C) 4πƐ 0 b+a (D) πɛ 0ab b a (E) πɛ 0ab b a 6. 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 7. Five 4-µF capacitors are connected in one network. What is the capacitance of the circuit? (A) F (B) 4 F (C) 6 F (D) 1 F (E) 4 F 8. 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 1 V battery? A) 4 V B) 3V C) 6 V D) 9 V E) 18 V

10 9. A capacitor C 0 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 C 0? (A) 3κC 0 κ+1 (B) 3C 0 κ (C) 3C 0 κ (D) C 0 κ+1 (E) C 0 κ 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) V (B) 3 V C) 5 V D) 1 V E) 0 V 31. A capacitor C 0 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 C 0? (A) (κ+1)c 0 3 (B) κc 0 3 (C) C 0 3 (D) κc 0 (E) (κ+1)c 0

11 3. 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 Ɛ 0 A (B) Ɛ 0Qd ea (C) Ɛ 0QA d (D) Ɛ 0Qe da (E) Ɛ 0Q da 33. An isolated parallel plate capacitor is charged to a maximum energy U 0. 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) κu 0 (B) U 0 κ (C) (κ+1)u 0 (D) κu 0 (E) (κ+1)u 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 V o by a battery, which is then disconnected. If edge effects are negligible, what is the magnitude of the electric field between the plates? (A) V o d (B) V o /d (C) d/v o (D) V o /d (E) V o /d

12 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) Q /4C (B) Q /C (C) Q /C (D) Q /C (E) 4Q /C Question A C E B D 5V 10V 15V 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 - N/C (B) 10x10 3 N/C (C) 5x10 3 N/C (D) 10x10 - N/C (E) 5x10-1 N/C

13 Answers 1. D. C 3. B 4. B 5. A 6. A 7. D 8. B 9. B 10. D 11. C 1. E 13. B 14. E 15. D 16. C 17. B 18. D 19. C 0. A 1. E. B 3. D 4. D 5. B 6. B 7. A 8. C 9. A 30. D 31. E 3. A 33. B 34. A 35. B 36. D 37. B 38. A

14 39. C 40. C

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