Potential of a Charged Isolated Conductor — Question 8

Question 8

Nested isolated conductors can have zero exterior potential and a nonzero internal voltage

A conducting sphere of radius a carries +Q. A concentric thin conducting shell of radius b>a carries -Q. There are no other sources; set V(infinity)=0.

1. Find V everywhere, including within the inner sphere and outside the shell.

2. Find the voltage between the conductors and the quasistatic work moving a small positive probe q from just inside b to just outside a.

Question 8 - Solution

Nested isolated conductors can have zero exterior potential and a nonzero internal voltage

(a) Method and calculation

$$V(r)=\begin{cases}kQ(1/a-1/b)&r\le a,\\kQ(1/r-1/b)&a<r<b,\\0&r\ge b.\end{cases}$$$$V_a-V_b=kQ(1/a-1/b)>0.$$

(b) Interpretation and further calculation

$$W_{\mathrm{ext}}=q(V_a-V_b)=kQq(1/a-1/b).$$$$\mathbf E=\frac{kQ}{r^2}\hat{\mathbf r}\quad(a<r<b),\quad\mathbf E=0\quad(r<a\text{ or }r>b).$$

Physical check

The total charge vanishes, so the external field and infinity-referenced potential vanish; charge separation still produces an internal voltage.