A Charged Isolated Conductor — Question 1

Question 1

A surface-field limit constrains the charge on an isolated sphere

An isolated conducting sphere of radius R carries charge Q and is in electrostatic equilibrium with no external sources. A specified model permits a maximum surface field magnitude $E_{\mathrm{lim}}$.

1. Derive its uniform surface density and the field immediately outside using Gauss law.

2. Find the largest allowed $|Q|$. Compare the changes in surface field and charge capacity when R doubles.

Question 1 - Solution

A surface-field limit constrains the charge on an isolated sphere

(a) Method and calculation

$$\sigma=\frac Q{4\pi R^2},\quad E_r(R^+)4\pi R^2=Q/\epsilon_0.$$$$E_r(R^+)=\frac Q{4\pi\epsilon_0R^2}=\frac\sigma{\epsilon_0},\quad\mathbf E(R^-)=\mathbf0.$$

(b) Interpretation and further calculation

$$|Q|_{\max}=4\pi\epsilon_0R^2E_{\mathrm{lim}}.$$$$E_s(2R;Q)=E_s(R;Q)/4,\quad |Q|_{\max}(2R)=4|Q|_{\max}(R).$$The threshold is a supplied field limit, not a universal breakdown value.

Physical check

The exterior field has a discontinuity at the charged surface; the zero interior value must not be used as the exterior field.