A Charged Isolated Conductor — Question 6

Question 6

A neutral collection inside a cavity can induce zero net wall charge and nonzero fields

A closed cavity inside an isolated neutral conductor contains two fixed, separated charges $+q$ and $-q$, neither touching the wall. The whole system is in electrostatic equilibrium.

1. Find the total induced charge on the cavity wall and the total outer-surface charge.

2. Explain why the zero cavity-wall charge total does not imply zero surface density everywhere or zero cavity field. Use a small Gaussian sphere enclosing only $+q$.

Question 6 - Solution

A neutral collection inside a cavity can induce zero net wall charge and nonzero fields

(a) Method and calculation

$$Q_{\mathrm{contents}}=q-q=0,\quad Q_{\mathrm{wall}}=-Q_{\mathrm{contents}}=0.$$$$Q_{\mathrm{outer}}=Q_c-Q_{\mathrm{wall}}=0.$$

(b) Interpretation and further calculation

$$\int_{\mathrm{wall}}\sigma\,dA=0\quad\text{does not require}\quad\sigma(P)=0.$$$$\oint_{S_+}\mathbf E\cdot d\mathbf A=q/\epsilon_0\ne0.$$A field that vanished everywhere in the cavity could not satisfy the last equation. Local induced charges enforce zero field in the surrounding metal.

Physical check

A cavity containing separated charges is not an empty cavity, even when its net charge is zero.