Conductors and Insulators — Question 5

Question 5

Charge inside a cavity fixes the induced inner-surface total

A conducting shell has net charge $+7.0$ nC. An insulated object carrying $-3.0$ nC is held inside a closed cavity without touching the shell. Use the electrostatic equilibrium result that the total charge induced on a cavity wall equals the negative of the enclosed object charge.

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

2. The object is moved off center without touching. Identify which charge totals stay fixed and which spatial distributions may change.

Question 5 - Solution

Charge inside a cavity fixes the induced inner-surface total

(a) Method and calculation

$$Q_{\mathrm{inner}}=-q=+3.0\,\mathrm{nC}.$$$$Q_{\mathrm{shell}}=Q_{\mathrm{inner}}+Q_{\mathrm{outer}}=7.0\,\mathrm{nC}\Rightarrow Q_{\mathrm{outer}}=+4.0\,\mathrm{nC}.$$

(b) Interpretation and further calculation

$$q\prime=q,\quad Q_{\mathrm{inner}}\prime=+3.0\,\mathrm{nC},\quad Q_{\mathrm{outer}}\prime=+4.0\,\mathrm{nC}.$$$$q+Q_{\mathrm{shell}}=+4.0\,\mathrm{nC}.$$The inner-wall density redistributes when the object moves. For a fixed isolated shell with unchanged external surroundings, the external electrostatic configuration is unchanged.

Physical check

The given induced-total rule does not imply a uniform inner-wall density when the object is off center.