Potential of a Charged Isolated Conductor — Question 6

Question 6

A central cavity charge shifts an isolated shell potential through its outer charge

A conducting spherical shell has inner radius a, outer radius b>a and net charge Q. A point charge q is held at its center. There are no external sources; take V(infinity)=0.

1. Find the charges on both conductor surfaces and the conductor potential.

2. Find the potential in the cavity for $0<r<a$, checking its boundary value.

Question 6 - Solution

A central cavity charge shifts an isolated shell potential through its outer charge

(a) Method and calculation

$$Q_{\mathrm{inner}}=-q,\quad Q_{\mathrm{outer}}=Q+q.$$$$V_{\mathrm{metal}}=\frac{k(Q+q)}b.$$

(b) Interpretation and further calculation

$$V(r)=\frac{kq}{r}-\frac{kq}{a}+\frac{k(Q+q)}b\quad(0<r<a).$$$$V(a^-)=\frac{k(Q+q)}b=V_{\mathrm{metal}},\quad-\frac{dV}{dr}=\frac{kq}{r^2}.$$

Physical check

The cavity field depends on q, while the constant offset also depends on the outer surface charge and its radius.