HW 11

10-10

(a) Since the two spheres have the same potential, $q_1/r_1 = q_2/r_2$ where $q_1+q_2 = Q = 20.0~\rm\mu C$, $r_1 = 0.0400~\rm m$, and $r_2 = 0.0600~\rm m$. So we get $q_1 = 8.00~\rm\mu C$ and $q_2=12 ~\rm\mu C$. Then the electric field near the surface of the first sphere is $E_1 = {q_1\over 4\pi \epsilon_0 r_1^2}=4.50 \times 10^{7} ~\rm N/C$, and that of the second sphere is $E_2 = {q_2\over 4\pi \epsilon_0 r_2^2}=3.00 \times 10^{7} ~\rm N/C$. (b) The electric potential is the same for each sphere: $V = {q_1\over 4\pi \epsilon_0 r_1}= 1.80\times 10^6 ~\rm V$.

11-1

(a) From $CV=Q$, $C= 3.5\times 10^{-12}~\rm F$. (b) The capacitance is not dependent on the charges. (c) The new potential is $V=Q/C = 57~\rm V$.

11-2

From ${Q\over 4\pi\epsilon_0 r}= 15~\rm kV$, $Q=4.2\times 10^{-7} ~\rm C$.

11-3

From $C= A\epsilon_0/d$, we get $d= A\epsilon_0/C = 3.10 ~\rm nm$.

11-4

(a) The equivalent capacitance is $C = C_1C_2/(C_1+C_2) = 1.6 ~\rm\mu F$. So the total charge is $Q = CV = 4.8\times 10^{-4} ~\rm C$. All capacitors have the same amount of charge. Therefore the potential difference for each capacitor is $V_1=Q/C_1 = 240~\rm V$ and $V_2=V-V_1 = 60 ~\rm V$. (b) In this situation, the overall voltage is zero, which means that both capacitors have the same magnitude of the potential difference. The sum of the charges needs to be the same as twice the accumulated charge in (a), which means that $V(C_1+C_2)= 9.6\times 10^{-4}~\rm C$. So we get $V=96~\rm V$. For the first capacitor, $Q_1 = C_1V =1.92\times 10^{-4}~\rm C$ and $Q_2=7.68\times 10^{-4}~\rm C$. (c) In this case, the steady state maintains no net charge or potential differences.

11-5

Repeatedly using the equivalence formulae part by part, we get ${11\over 6}C$.

11-6

The potential of the metal sphere is determined from $V = {Q\over 4\pi\epsilon_0 R}=8000~\rm V$ with $R= 0.1~\rm m$. Then the electric field on the surface of the sphere is $E=V/R=8\times 10^4 ~\rm N/C$. So the energy density at the surface is $u = {1\over 2}\epsilon_0E^2 = 2.83\times 10^{-2} ~\rm J/m^3$.