A system consists of two thin concentric conducting spherical shells. The inner shell has a radius $

Question

A system consists of two thin concentric conducting spherical shells. The inner shell has a radius $R_1 = 10 	ext{ cm}$ and carries a charge $q_1 = 4 \mu	ext{C}$. The outer shell has a radius $R_2 = 20 	ext{ cm}$ and carries a charge $q_2 = -2 \mu	ext{C}$. Find the total electrostatic energy of this system in milli-Joules (mJ). (Take $\frac{1}{4\pi\epsilon_0} = 9 	imes 10^9 	ext{ N m}^2	ext{/C}^2$).

Professor PC Thomas Classes · Answer

The total electrostatic energy $U$ of the system can be calculated using $U = \frac{1}{2} q_1 V_1 + \frac{1}{2} q_2 V_2$.
The potential of the inner shell is $V_1 = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1}{R_1} + \frac{q_2}{R_2} ight)$. 
The potential of the outer shell is $V_2 = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1 + q_2}{R_2} ight)$. 
Substituting these into the energy equation: 
$U = \frac{1}{8\pi\epsilon_0} \left( \frac{q_1^2}{R_1} + \frac{q_1 q_2}{R_2} + \frac{q_1 q_2}{R_2} + \frac{q_2^2}{R_2} ight) =\ \frac{1}{8\pi\epsilon_0} \left( \frac{q_1^2}{R_1} + \frac{q_2^2 + 2q_1 q_2}{R_2} ight)$.
 Given $\frac{1}{8\pi\epsilon_0} = 4.5 	imes 10^9$.