Potential of a Charged Isolated Conductor — Question 9

Question 9

An off-center cavity charge leaves a spherical outer surface at a uniform potential

An isolated conductor has a spherical outer surface of radius b and a closed cavity containing a fixed point charge q. Its net conductor charge is Q. The cavity may be irregular and q need not be centered. There are no external sources. Use uniqueness of the exterior electrostatic boundary-value problem.

1. Find the total inner and outer surface charges and the conductor potential relative to infinity.

2. Find the outer surface charge density and explain which quantities cannot be inferred from these data alone.

Question 9 - Solution

An off-center cavity charge leaves a spherical outer surface at a uniform potential

(a) Method and calculation

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

(b) Interpretation and further calculation

$$\sigma_{\mathrm{outer}}=\frac{Q+q}{4\pi b^2}\quad\text{uniform}.$$$$\int_{\mathrm{inner}}\sigma_{\mathrm{inner}}\,dA=-q$$fixes only the total; local inner density and force on q require the cavity shape and source position.

Physical check

The spherical exterior and constant conductor boundary determine the outside solution even when the cavity field has no spherical symmetry.