A Charged Isolated Conductor — Question 4

Question 4

A centered cavity source produces three distinct radial field regions

A spherical conducting shell has inner radius a and outer radius b. A point charge q is fixed at the common center, and the shell has net charge $Q_c$. There are no external sources; the system is in electrostatic equilibrium.

1. Find the field for $0<r<a$, $a<r<b$ and $r>b$.

2. Find both surface densities and verify the field jump at each interface using normals directed from metal into vacuum.

Question 4 - Solution

A centered cavity source produces three distinct radial field regions

(a) Method and calculation

$$E_r(r)=\frac{q}{4\pi\epsilon_0r^2}\quad(0<r<a),\qquad E_r=0\quad(a<r<b).$$$$E_r(r)=\frac{q+Q_c}{4\pi\epsilon_0r^2}\quad(r>b).$$

(b) Interpretation and further calculation

$$\sigma_a=-\frac{q}{4\pi a^2},\quad\sigma_b=\frac{q+Q_c}{4\pi b^2}.$$$$E_{n,a}=\mathbf E(a^-)\cdot(-\hat{\mathbf r})=\sigma_a/\epsilon_0,\quad E_{n,b}=E_r(b^+)=\sigma_b/\epsilon_0.$$At the cavity wall, the outward normal from the metal points inward toward the center.

Physical check

The cavity contains a source, so zero field in the metal does not imply zero field in the cavity.