Ch23

© 2000 by Harcourt College Publishers. All rights reserved.

Chapter 23

Electric Fields

Multiple Choice

1. Each of two small non-conducting spheres is charged positively, the combined chargebeing 40 µC. If each sphere is repelled from the other by a force having a magnitude of2.0 N when the two spheres are 50 cm apart, determine the charge on the sphere havingthe smaller charge.

a . 1.4 µCb. 1.1 µCc. 2.0 µCd. 3.3 µCe. 17 µC

2. A particle (charge = +40 µC) is located on the x axis at the point x = –20 cm, and a secondparticle (charge = –50 µC) is placed on the x axis at x = +30 cm. What is the magnitude ofthe total electrostatic force on a third particle (charge = –4.0 µC) placed at the origin(x = 0)?

a . 41 Nb. 16 Nc. 56 Nd. 35 Ne. 72 N

3. In the figure, if Q = 30 µC, q = 5.0 µC, and d = 30 cm, what is the magnitude of theelectrostatic force on q?

Q

d 2d

q 2Q

a . 15 Nb. 23 Nc. zerod. 7.5 Ne. 38 N

2 Chapter 23


4. A charge of +80 µC is placed on the x axis at x = 0. A second charge of –50 µC is placed onthe x axis at x = 50 cm. What is the magnitude of the electrostatic force on a third chargeof 4.0 µC placed on the x axis at x = 30 cm?

a . 13 Nb. 77 Nc. 39 Nd. 25 Ne. 45 N

5. Three point charges are positioned on the x axis. If the charges and correspondingpositions are +32 µC at x = 0, +20 µC at x = 40 cm, and –60 µC at x = 60 cm, what is themagnitude of the electrostatic force on the +32-µC charge?

a . 84 Nb. 12 Nc. 36 Nd. 50 Ne. 48 N

6. A particle (m = 50 g, q = 5.0 µC) is released from rest when it is 50 cm from a secondparticle (Q = –20 µC). Determine the magnitude of the initial acceleration of the 50-gparticle.

a . 54 m/s2

b. 90 m/s2

c. 72 m/s2

d. 65 m/s2

e. 36 m/s2

7. A point charge Q is placed on the x axis at x = 2.0 m. A second point charge, –Q, is placedat x = 3.0 m. If Q = 40 µC, what is the magnitude of the electrostatic force on a 30-µCcharge placed at the origin?

a . 7.2 Nb. 3.9 Nc. 1.5 Nd. 14 Ne. 8.1 N

8. A point charge Q is placed on the x axis at x = –2.0 m. A second point charge, –Q, is placedat x = 1.0 m. If Q = 60 µC, what is the magnitude of the electrostatic force on a 40-µCcharge placed at the origin?

a . 16 Nb. 27 Nc. 32 Nd. 11 Ne. 3.0 N

Chapter 23 3


9. A point charge Q is placed on the x axis at the origin. An identical point charge is placedon the x axis at x = –1.0 m and another at x = +1.0 m. If Q = 40 µC, what is the magnitudeof the electrostatic force on the charge at x = +1.0 m?

a . 29 Nb. 14 Nc. 11 Nd. 18 Ne. 7.0 N

10. If a = 3.0 mm, b = 4.0 mm, Q1 = 60 nC, Q2 = –80 nC, and q = 36 nC in the figure, what is themagnitude of the total electric force on q?

Q1 q Q2

a b

a . 5.0 Nb. 4.4 Nc. 3.8 Nd. 5.7 Ne. 0.60 N

11. If a = 3.0 mm, b = 4.0 mm, Q1 = –60 nC, Q2 = 80 nC, and q = 30 nC in the figure, what is themagnitude of the total electric force on q?

Q1 Q2 q

a b

a . 1.4 Nb. 1.0 Nc. 1.7 Nd. 2.0 Ne. 0.50 N

12. If a = 3.0 mm, b = 4.0 mm, Q1 = –60 nC, Q2 = 80 nC, and q = 40 nC in the figure, what is themagnitude of the total electric force on q?

q Q1 Q2

a b

a . 1.8 Nb. 2.3 Nc. 2.7 Nd. 3.0 Ne. 4.2 N

4 Chapter 23


13. If a = 3.0 mm, b = 4.0 mm, Q1 = 60 nC, Q2 = 80 nC, and q = 24 nC in the figure, what is themagnitude of the total electric force on q?

q

Q2

Q1

b

a

90°90°

a . 2.7 Nb. 1.9 Nc. 2.3 Nd. 1.5 Ne. 0.52 N


q

Q2

Q1

b

a

a

a . 1.6 Nb. 1.3 Nc. 1.9 Nd. 2.2 Ne. 0.04 N


q

Q2

Q1

b

a

90°90°

a . 0.78 Nb. 0.68 Nc. 0.58 Nd. 0.88 Ne. 0.62 N

Chapter 23 5


16. A particle (charge = +50 µC) is placed on the y axis at the point y = +40 cm, and a secondparticle (charge = –20 µC) is placed at the origin (x = y = 0). What is the direction of thetotal electrostatic force on a third particle with respect to the +x-axis (charge = –5.0 µC)placed on the x axis at the point x = +30 cm?

a . 33°b. 57°c. –53°d. 127°e. 63°

17. Three point charges, two positive and one negative, each having a magnitude of 20 µC areplaced at the vertices of an equilateral triangle (30 cm on a side). What is the magnitudeof the electrostatic force on the negative charge?

a . 80 Nb. 40 Nc. 69 Nd. 57 Ne. 75 N

18. Three point charges, two positive and one negative, each having a magnitude of 20 µC areplaced at the vertices of an equilateral triangle (30 cm on a side). What is the magnitudeof the electrostatic force on one of the positive charges?

a . 69 Nb. 40 Nc. 80 Nd. 57 Ne. 20 N

19. Identical point charges Q are placed at two of the vertices of an equilateral triangle(length of each side = 50 cm). A third charge, –2Q, is placed at the third vertex. IfQ = 20 µC, what is the magnitude of the electrostatic force on either of the positivecharges?

a . 76 Nb. 56 Nc. 39 Nd. 25 Ne. 29 N

20. A point charge Q is placed at the origin. A second charge, 2Q, is placed on the x axis atx = –3.0 m. If Q = 50 µC, what is the magnitude of the electrostatic force on a third pointcharge, –Q, placed on the y axis at y = +4.0 m?

a . 2.5 Nb. 3.0 Nc. 3.7 Nd. 4.4 Ne. 1.8 N

6 Chapter 23


21. Three identical point charges Q are placed at the vertices of an equilateral triangle(length of each side = 2.0 m). If Q = 60 µC, what is the magnitude of the electrostaticforce on any one of the charges?

a . 25 Nb. 19 Nc. 14 Nd. 22 Ne. 16 N

22. Identical point charges Q are placed at each of the four corners of a 3.0 m × 4.0 mrectangle. If Q = 40 µC, what is the magnitude of the electrostatic force on any one of thecharges?

a . 3.0 Nb. 2.4 Nc. 1.8 Nd. 3.7 Ne. 2.0 N

23. A point charge (–5.0 µC) is placed on the x axis at x = 4.0 cm, and a second charge (+5.0 µC)is placed on the x axis at x = –4.0 cm. What is the magnitude of the electric force on athird charge (+2.5 µC) placed on the y axis at y = 3.0 cm?

a . 90 Nb. 45 Nc. 54 Nd. 72 Ne. 36 N

24. If Q = 25 µC, q = 10 µC, and L = 40 cm in the figure, what is the magnitude of theelectrostatic force on q?

+Q −Q

LL

q

90°90°

a . 28 Nb. 22 Nc. 20 Nd. 14 Ne. 10 N

Chapter 23 7


25. If Q = 20 µC and L = 60 cm, what is the magnitude of the electrostatic force on any one ofthe charges shown?

L

L L

−Q +Q

L+Q −Q

a . 25 Nb. 19 Nc. 15 Nd. 9.1 Ne. 14 N

26. If a = 60 cm, b = 80 cm, Q = –4.0 nC, and q = 1.5 nC, what is the magnitude of the electricfield at point P shown?

b

a a

q

bQ P

a . 68 N/Cb. 72 N/Cc. 77 N/Cd. 82 N/Ce. 120 N/C

27. If a = 60 cm, b = 80 cm, Q = –6.0 nC, and q = 4.0 nC, what is the magnitude of the electricfield at point P shown?

b

a

q

Q

P90°90°


8 Chapter 23


28. If a = 60 cm, b = 80 cm, Q = –6.0 nC, and q = 6.0 nC, what is the magnitude of the electricfield at point P in the figure?

a

b

Q q90°90°

P


29. If a = 60 cm, b = 80 cm, Q = –6.0 nC, and q = 3.0 nC in the figure, what is the magnitude ofthe electric field at point P?

a

a

b

Q

q

P90°90°


30. Two particles, having oppositely signed charges of +12 nC, are placed at two of thevertices of an equilateral triangle (length of each side = 2.0 m). What is the magnitudeof the electric field at the third vertex of the triangle?


Chapter 23 9


31. If Q = 16 nC, a = 3.0 m, and b = 4.0 m, what is the magnitude of the electric field at point Pshown?

Q Q

Q

b

a

P

90°90°


32. If Q = 80 nC, a = 3.0 m, and b = 4.0 m in the figure, what is the magnitude of the electricfield at point P?

2Q

2Q

−Q Pb

a

a

90°90°


33. A +2.0-nC point charge is placed at one corner of a square (1.5 m on a side), and a –3.0-nCcharge is placed on a corner diagonally away from the first charge. What is themagnitude of the electric field at either of the two unoccupied corners?

a . 20 N/Cb. 14 N/Cc. 4.0 N/Cd. 12 N/Ce. 8.0 N/C

34. A +15-nC point charge is placed on the x axis at x = 1.5 m, and a –20-nC charge is placed onthe y axis at y = –2.0m. What is the magnitude of the electric field at the origin?


10 Chapter 23


35. A +20-nC point charge is placed on the x axis at x = 2.0 m, and a –25-nC point charge isplaced on the y axis at y = –3.0 m. What is the direction of the electric field at theorigin?

a . 209°b. 61°c. 29°d. 241°e. 151°

36. A charge Q is placed on the x axis at x = +4.0 m. A second charge q is located at the origin.If Q = +75 nC and q = –8.0 nC, what is the magnitude of the electric field on the y axis aty = +3.0 m?


37. A 40-µC charge is positioned on the x axis at x = 4.0 cm. To produce a net electric field ofzero at the origin where should a –60-µC charge be placed?

a . –5.3 cmb. 5.7 cmc. 4.9 cmd. –6.0 cme. +6.0 cm

38. A charge of 80 nC is uniformly distributed along the x axis from x = 0 to x = 2.0 m.Determine the magnitude of the electric field at a point on the x axis with x = 8.0 m.


39. A charge (uniform linear density = 9.0 nC/m) is distributed along the x axis from x = 0 tox = 3.0 m. Determine the magnitude of the electric field at a point on the x axis withx = 4.0 m.


Chapter 23 11


40. A charge of 25 nC is uniformly distributed along a circular arc (radius = 2.0 m) that issubtended by a 90-degree angle. What is the magnitude of the electric field at the centerof the circle along which the arc lies?


41. Charge of uniform density 4.0 nC/m is distributed along the x axis from x = –2.0 m tox = +3.0 m. What is the magnitude of the electric field at the point x = +5.0 m on thex axis?

a . 16 N/Cb. 13 N/Cc. 19 N/Cd. 26 N/Ce. 5.0 N/C

42. A uniformly charged rod (length = 2.0 m, charge per unit length = 5.0 nC/m) is bent to formone quadrant of a circle. What is the magnitude of the electric field at the center of thecircle?


43. A uniformly charged rod (length = 2.0 m, charge per unit length = 3.0 nC/m) is bent to forma semicircle. What is the magnitude of the electric field at the center of the circle?


44. A 16-nC charge is distributed uniformly along the x axis from x = 0 to x = 4 m. Which ofthe following integrals is correct for the magnitude (in N/C) of the electric field atx = +10 m on the x axis?

a . ⌡⌠

0

4

36 dx

(10 – x)2

b. ⌡⌠

0

4

154 dx

(10 – x)2

c. ⌡⌠

0

4

36 dx

x2

12 Chapter 23


d. ⌡⌠

6

10

154 dx

x2

e. none of these

Chapter 23 13


45. A uniform linear charge of 2.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.Which of the following integrals is correct for the y component of the electric field aty = 4 m on the y axis?

a . ⌡⌠

0

3

72 dx

(16 + x2)3/2

b. ⌡⌠

0

3

18 dx

(16 + x2)3/2

c. ⌡⌠

0

3

72 dx

16 + x2

d. ⌡⌠

3

0

18 dx

16 + x2

e. none of these

46. A 12-nC charge is distributed uniformly along the y axis from y = 0 to y = 4 m. Which ofthe following integrals is correct for the x component of the electric field at x = 2 m on thex axis?

a . ⌡⌠

0

4

216 dy

(4 + y2)3/2

b. ⌡⌠

0

4

54 dy

(4 + y2)3/2

c. ⌡⌠

0

4

108 dy4 + y2

d. ⌡⌠

0

4

27 dy4 + y2

e. none of these

47. A uniform linear charge of 3.0 nC/m is distributed along the y axis from y = –3 to y = 2m.Which of the following integrals is correct for the magnitude of the electric field aty = 4 m on the y axis?

a . ⌡⌠

–3

2

27 dy

(4 – y)2

b. ⌡⌠

–3

2

27 dy

16 + y2

c. ⌡⌠

–3

2

135 dy(4 – y)2

d. ⌡⌠

–3

2

135 dy16 + y2

e. none of these

14 Chapter 23


48. A uniform linear charge of 2.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.Which of the following integrals is correct for the x component of the electric field aty = 2 m on the y axis?

a . ⌡⌠

0

3

–18x dx

(4 + x2)3/2

b. ⌡⌠

0

3

–18 dx(2 + x)2

c. ⌡⌠

0

3

–18 dx4 + x2

d. ⌡⌠

0

3

–18x dx(2 + x)3

e. none of these

49. A rod (length = 2.0 m) is uniformly charged and has a total charge of 40 nC. What is themagnitude of the electric field at a point which lies along the axis of the rod and is 3.0 mfrom the center of the rod?


50. A charge of 50 nC is uniformly distributed along the y axis from y = 3.0 m to y = 5.0 m.What is the magnitude of the electric field at the origin?


51. A linear charge of uniform density equal to 8.0 nC/m is distributed along the x axis fromx = –2.0 m to x = 3.0 m. What is the magnitude of the electric field at the point x = 6.0 m onthe x axis?


Chapter 23 15


52. A 24-nC charge is distributed uniformly along the x axis from x = 2 m to x = 6 m. Whichof the following integrals is correct for the magnitude (in N/C) of the electric field atx = +8 m on the x axis?

a . ⌡⌠

2

6

54 dx

(8 – x)2

b. ⌡⌠

2

6

216 dx(8 – x)2

c. ⌡⌠

2

6

54 dx

x2

d. ⌡⌠

2

6

216 dx

x2

e. none of these

53. A uniform linear charge density of 7.0 nC/m is distributed along the y axis from y = 2 m toy = 5 m. Which of the following integrals is correct for the magnitude (in N/C) of theelectric field at y = 0 on the y axis?

a . ⌡⌠

2

5

63 dy

y2

b. ⌡⌠

2

5

21 dy

y2

c. ⌡⌠

2

5

63 dy

(5 – y)2

d. ⌡⌠

2

5

21 dy

(5 – y)2

e. none of these

54. A uniform linear charge of 2.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.What is the x component of the electric field at y = 2 m on the y axis?

a . –5.0 N/Cb. –4.0 N/Cc. –5.7 N/Cd. –6.2 N/Ce. –9.0 N/C

55. A particle (mass = 4.0 g, charge = 80 mC) moves in a region of space where the electricfield is uniform and is given by Ex = –2.5 N/C, Ey = Ez = 0. If the velocity of the particle att = 0 is given by vx = 80 m/s, vy = vz = 0, what is the speed of the particle at t = 2.0 s?

a . 40 m/sb. 20 m/sc. 60 m/sd. 80 m/se. 180 m/s

16 Chapter 23


56. A particle (mass = 5.0 g, charge = 40 mC) moves in a region of space where the electricfield is uniform and is given by Ex = 2.5 N/C, Ey = Ez = 0. If the velocity of the particle att = 0 is given by vy = 50 m/s, vx = vz = 0, what is the speed of the particle at t = 2.0 s?


57. A particle (mass = 5.0 g, charge = 40 mC) moves in a region of space where the electricfield is uniform and is given by Ex = –5.5 N/C, Ey = Ez = 0. If the position and velocity ofthe particle at t = 0 are given by x = y = 0 and vx = 50 m/s, vy = vz = 0, what is the distancefrom the origin to the particle at t = 2.0 s?

a . 60 mb. 28 mc. 44 md. 12 me. 88 m

58. A particle (mass = 5.0 g, charge = 40 mC) moves in a region of space where the electricfield is uniform and is given by Ex = –2.3 N/C, Ey = Ez = 0. If the position and velocity ofthe particle at t = 0 are given by x = y = 0 and vz = 20 m/s, vx = vy = 0, what is the distancefrom the origin to the particle at t = 2.0 s?

a . 60 mb. 54 mc. 69 md. 78 me. 3.2 m

59. An electron enters a region of uniform electric field (E = 50 N/C) with an initial velocityof 40 km/s directed the same as the electric field. What is the speed of the electron 1.5 nsafter entering this region?

a . 53 km/sb. 27 km/sc. 18 km/sd. 62 km/se. 42 km/s

60. A particle (q = 3.0 mC, m = 20 g) has a speed of 20 m/s when it enters a region where theelectric field has a constant magnitude of 80 N/C and a direction which is the same asthe velocity of the particle. What is the speed of the particle 3.0 s after it enters thisregion?


Chapter 23 17


61. A particle (q = 4.0 mC, m = 50 g) has a velocity of 25 m/s in the positive x direction whenit first enters a region where the electric field is uniform (60 N/C in the positive ydirection). What is the speed of the particle 5.0 s after it enters this region?


62. A charge of 50 µC is placed on the y axis at y = 3.0 cm and a 77-µC charge is placed on thex axis at x = 4.0 cm. If both charges are held fixed, what is the magnitude of the initialacceleration of an electron released from rest at the origin?

a . 1.2 × 1020 m/s2

b. 1.5 × 1020 m/s2

c. 1.0 × 1020 m/s2

d. 1.8 × 1020 m/s2

e. 2.0 × 1020 m/s2

63. The velocity of a particle (m = 10 mg, q = –4.0 C) at t = 0 is 20 m/s in the positive xdirection. If the particle moves in a uniform electric field of 20 N/C in the positive xdirection, what is the particle's speed at t = 5.0 s?


64. A particle (m = 20 mg, q = –5.0 C) moves in a uniform electric field of 60 N/C in thepositive x direction. At t = 0, the particle is moving 25 m/s in the positive x direction andis passing through the origin. How far is the particle from the origin at t = 2.0 s?

a . 80 mb. 20 mc. 58 md. 10 me. 30 m

65. A particle (m = 20 mg, q = –5.0 µC) moves in a uniform electric field of 60 N/C in thepositive x direction. At t = 0, the particle is moving 30 m/s in the positive x direction andis passing through the origin. Determine the maximum distance beyond x = 0 the particletravels in the positive x direction.

a . 25 mb. 20 mc. 15 md. 30 me. 60 m

18 Chapter 23


Conceptual Problems

66. The electron gun in a television tube is to accelerate electrons (mass 9.1 × 10–31 kg, charge1.6 × 10–19 C) from rest to 3.0 × 107 m/s within a distance of 2.0 cm. What electric field isrequired?

67. An alpha particle (charge = +2e) is sent at high speed toward a gold nucleus (charge+79e). What is the electrical force acting on the alpha particle when it is at a distance of2 × 10–14 m away from the gold nucleus? (e = 1.6 × 10–19 C)

68. A proton moving at 3 × 104 m/s is projected at an angle of 30° above a horizontal plane. Ifan electric field of 400 N/C is acting down, how long does it take the proton to return tothe horizontal plane? (HINT: Ignore gravity) [mPROTON = 1.67 × 10–27 kg,qPROTON = +1.6 × 10–19 C]

69. Imagine for a minute that the moon is held in its orbit about the Earth by electricalforces rather than by gravitation. What is the necessary electrical charge –Q that wouldbe on the earth and +Q that would be on the moon to hold the moon in a circular orbit of27.3 days? The Earth-moon distance is 384,000 km and the mass of the moon is7.35 × 1022 kg.

70. Electric field lines in the space surrounding a charge distribution show:

a . the directions of the forces that exist in space at all times.b. only the directions in which static charges would accelerate when at points on those

linesc. only the directions in which moving charges would accelerate when at points on those

lines.d. the directions in which either static or moving charges would accelerate when

passing through points on those lines.e. the paths static or moving charges would take.

71. When a positive charge q is placed in the field created by two other charges Q1 and Q2,each a distance a away from q, the acceleration of q is:

a . in the direction of the charge Q1 or Q2 of smaller magnitude.b. in the direction of the charge Q1 or Q2 of greater magnitude.c. in the direction of the negative charge if Q1 and Q2 are of opposite sign.d. in the direction of the positive charge if Q1 and Q2 are of opposite sign.e. in a direction determined by the vector sum of the electric fields of Q1 and Q2.

72. Two charged particles, Q1 and Q2, are a distance r apart. Q2 = 5Q1. Compare the forcesthey exert on each other. F1 is the force Q2 exerts on Q1. F2 is the force Q1 exerts on Q2.

a . F2 = 5F1.b. F2 = –5F1.c. F2 = F1.d. F2 = –F1.e. 5F2 = F1.

Chapter 23 19


73. Rubber rods charged by rubbing with cat fur repel each other. Glass rods charged byrubbing with silk repel each other. A rubber rod and a glass rod charged respectively asabove attract each other. A possible explanation is that:

a . Any two rubber rods charged this way have opposite charges on them.b. Any two glass rods charged this way have opposite charges on them.c. A rubber rod and a glass rod charged this way have opposite charges on them.d. All rubber rods always have an excess of positive charge on them.e. All glass rods always have an excess of negative charge on them.

74. Which one of the diagrams below is not a possible electric field configuration for a regionof space which does not contain any charges?

a

d e

b c

75. A positively charged particle is moving in the +y-direction when it enters a region witha uniform electric field pointing in the +x-direction. Which of the diagrams below showsits path while it is in the region where the electric field exists. The region with the fieldis the region between the plates bounding each figure. The field lines always point to theright. The x-direction is to the right; the y-direction is up.

d

b c

q

e

q

q

a

q q

20 Chapter 23


76. A negatively charged particle is moving in the +x-direction when it enters a region witha uniform electric field pointing in the +x-direction. Which graph gives its position as afunction of time correctly? (Its initial position is x = 0 at t = 0.)

x

a b c

d e

t

x

t

x

t

x

t

x

t

Chapter 23 21


Chapter 23 1


Chapter 23

Electric Fields

1. Answer: a

2. Answer: c

3. Answer: d

4. Answer: b

5. Answer: b

6. Answer: c

7. Answer: c

8. Answer: b

9. Answer: d

10. Answer: c

11. Answer: b

12. Answer: a

13. Answer: d

14. Answer: b

15. Answer: c

16. Answer: b

17. Answer: c

18. Answer: b

19. Answer: d

20. Answer: b

21. Answer: c

22. Answer: b

23. Answer: d

2 Chapter 23


24. Answer: c

25. Answer: d

26. Answer: a

27. Answer: a

28. Answer: d

29. Answer: d

30. Answer: a

31. Answer: c

32. Answer: d

33. Answer: b

34. Answer: c

35. Answer: a

36. Answer: b

37. Answer: c

38. Answer: b

39. Answer: c

40. Answer: d

41. Answer: b

42. Answer: c

43. Answer: d

44. Answer: a

45. Answer: a

46. Answer: b

47. Answer: a

48. Answer: a

49. Answer: b

50. Answer: c

Chapter 23 3


51. Answer: d

52. Answer: a

53. Answer: a

54. Answer: b

55. Answer: b

56. Answer: c

57. Answer: d

58. Answer: b

59. Answer: b

60. Answer: c

61. Answer: b

62. Answer: a

63. Answer: b

64. Answer: b

65. Answer: d

66. Answer: 128,000 N/C

67. Answer: 91 N

68. Answer: 7.8 × 10–7 s

69. Answer: Q = 5.73 × 1013 C

70. Answer: d

71. Answer: e

72. Answer: d

73. Answer: c

74. Answer: d

75. Answer: d

76. Answer: c

4 Chapter 23


Ch23

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