Question

Three conducting spherical shells have radii a, b, and c such that a < b< c (Fig. 3.113). Initially, the inner shell is uncharged,the middle shell has a positive charge Q, and the outer shell has a negative charge -Q.

a. Find the electric potential of the three shells.

b. If the inner and outer shells are now connected by a wire that is insulated as it passes through the middle shell, what is the electric potential of each of the three shells? Also, what is the final charge on each shell?

Answer 1

Potential of shell A
$V_A=V_{DuetochangeonA}+V_{DuetochangeonB}+V_{DuetochangeonC}$
$=0+\frac{Q}{4\pi {\epsilon }_{0}b}-\frac{Q}{4\pi {\epsilon }_{0}c}$
Potential of shell B
$V_B=V_{DuetochangeonA}+V_{DuetochangeonB}+V_{DuetochangeonC}$
$=0+\frac{Q}{4\pi {\epsilon }_{0}b}-\frac{Q}{4\pi {\epsilon }_{0}c}$
Potential of shell C
$V_C=V_{DuetochangeonA}+V_{DuetochangeonB}+V_{DuetochangeonC}$
$=0+\frac{Q}{4\pi {\epsilon }_{0}b}-\frac{Q}{4\pi {\epsilon }_{0}c}=0$
b. After connecting inner and outer shells, we get
$V_A=V_C$
or $\frac{q}{a}-\frac{q}{b}+\frac{Q+q}{b}-\frac{Q+q}{c}=\frac{q-q+Q+q-(Q+q)}{c}$
or $q=\frac{Qa}{b}\frac{(b-c)}{(c-a)}$
Final potential of $A$ and $C$ is zero
$V_B=k[\frac{q}{a}-\frac{q}{b}+\frac{Q+q}{b}-\frac{(Q+q)}{c}]=k(Q+q)\left[\frac{c-b}{bc}\right]$
$=k[Q+\frac{Qa(b-c)}{b(c-a)}]\left[\frac{c-b}{bc}\right]=kQ\left[\frac{(b-a)(c-b)}{b^2(c-a)}\right]$