1. $E_y$ is the negative of the slope of the V-versus-y graph. $E_y$ is positive because $\vec{E}$ points up, so the graph has a negative slope. $E_y$ has constant magnitude, so the slope has a constant value.

2. b. $\vec{E}$ points “downhill,” so the potential must decrease from right to left. The field is everywhere perpendicular to equipotential surfaces, which rules out d. The field is stronger at the ends, where the field lines are closer together, so the contour lines at the ends must be closer together than in the middle.

3. b. Because of the connecting wire, the three spheres form a single conductor in electrostatic equilibrium. Thus all points are at the same potential. The electric field of a sphere is related to the sphere’s potential by $E=V/R$, so a smaller-radius sphere has a larger E.

4. 5.0 V. The potentials add, but $\text{Δ}V=-1.0\text{V}$ because the charge escalator goes down by 1.0 V.

5. ${(C_{eq})}_B>(C_{eq}{)}_A=(C_{eq}{)}_D>(C_{eq}{)}_C.$ ${(C_{\text{eq}})}_{\text{B}}=3\mu \text{F + }3\mu \text{F = }6\mu \text{F}$. The equivalent capacitance of series capacitors is less than any capacitor in the group, so ${(C_{\text{eq}})}_{\text{C}}<3\mu \text{F}$. Only d requires any real calculation. The two $4\mu \text{F}$ capacitors are in series and are equivalent to a single $2\mu \text{F}$ capacitor. The $2\mu \text{F}$ equivalent capacitor is in parallel with $3\mu \text{F}$, so ${(C_{\text{eq}})}_{\text{D}}=5\mu \text{F}$.

1. $+5.0\text{kV}$

3. $-400\text{V}$

5. $E_1>E_2$

7. $2.0\times {10}^4\text{V}\text{/}\text{m}$ $45\text{°}$ ccw from the $-x$-axis

9. $(3750\text{V}\text{/}\text{m}\text{,}\text{down})$, $(7500\text{V}\text{/}\text{m}\text{,}\text{up})$

11.

13. $-1.0\text{kV}\text{/}\text{m}$

15.

1. 0 V/m

2. $-0.2\text{kV}\text{/}\text{m}$

17. 3.0 C

19. 1.5 V

21.

1. 13 pF

2. 1.3 nC

23. 4.8 cm

25. $3.0\mu \text{F}$

27. $1.5\mu \text{F}$

29. $150\mu \text{F}$, in series

31. 1.4 kV

33. 1.2 kV/m

35.

1. 0.15 nF

2. 12 kV

37. 89 pF

39.

• a.

  

  Full Alternative Text

• b. 12V

41.

1. $-(1.4\times {10}^7î)\text{V/m, }7\times {10}^4\text{V}$

2. $\text{0}\text{.0 V/m,}1.4\times {10}^5\text{ V}$

3. $1.4\times {10}^7î\text{V/m, }7\times {10}^4\text{V}$

43. ${\displaystyle \frac{Q}{4\pi {ϵ}_0{z}^2}}$

45. $\text{1000 V/m, }127\text{°}$ ccw from $\text{+ }x\text{-axis}$

47.

49.

1. $V={\displaystyle \frac{2kq}{\sqrt{a^2+x^2}}}$

2. ${\displaystyle \frac{2kq}{a}}(1-{\displaystyle \frac{x^2}{2a^2}})$

3. ${\displaystyle \frac{2kqx}{a^3}}$

4. $F_x=-{\displaystyle \frac{2kqex}{a^3}}$

5. 8.9 GHz

51. $Q_{1\text{f}}=2\text{nC, }Q{}_{2\text{f}}=4\text{nC}$

53. 5.0 cm

55.

1. $\pm 32\text{pC}$

2. $\pm 16.0\text{pC}$

57. $V_1=12\text{ V, }{V}_2=7.2\text{ V}\text{,}{V}_3=4.8\text{}\text{ V; }{Q}_1=190\mu \text{C, }{Q}_2=290\mu \text{C, }{Q}_3=290\mu \text{C}$

59.

1. $\frac{3}{2}\text{pC}$

2. 0

61. 12 V

63. $Q_1^{\prime }=33\mu \text{C}\text{,}{Q}_2^{\prime }=67\mu \text{C},\text{ Δ}{V}_1^{\prime }=\text{Δ}{V}_2^{\prime }=3.3\text{ V }$

65. $11\mu \text{F}$

67. $5.0\mu \text{F}$

69.

1. 8.0 kV

2. $4.3\times {10}^7\text{V}\text{/}\text{m}$

71. $C_0{\displaystyle \frac{2\kappa }{1+\kappa }}$

73. $6.1\times {10}^7\text{ions}$

75.

2. $(10-a{z}^2)\text{V, with }z\text{in m}$

77.

2. $2\mu \text{F}$

79. ${\displaystyle \frac{\rho R^2}{2{ϵ}_0}}$

81.

1. $C={\displaystyle \frac{Q}{\text{Δ}{V}_{\text{C}}}}4\pi {ϵ}_0{({\displaystyle \frac{1}{R_1}}-{\displaystyle \frac{1}{R_2}})}^{-1}$

2. 5.9 cm, 6.1 cm

83. 2 C