Applying Gauss' Law: Spherical Symmetry — Question 5

Question 5

A surface-field measurement can determine the cavity radius of a charged shell

A thick spherical insulating shell has known outer radius b and uniform density $\rho>0$ in $a<r<b$, but unknown inner radius a. Its cavity is empty and concentric. There are no other sources.

1. Derive the field in the cavity, material and exterior.

2. Given the outward field $E_b$ at b, solve for a/b and state the range of $E_b$ compatible with $0\le a<b$.

Question 5 - Solution

A surface-field measurement can determine the cavity radius of a charged shell

(a) Method and calculation

$$Q_{\mathrm{enc}}=\frac{4\pi\rho}{3}(r^3-a^3)\quad(a<r<b),\qquad E_r=0\quad(r<a).$$$$E_r=\frac\rho{3\epsilon_0}\left(r-\frac{a^3}{r^2}\right)\quad(a<r<b),\quad E_r=\frac{\rho(b^3-a^3)}{3\epsilon_0r^2}\quad(r>b).$$

(b) Interpretation and further calculation

$$E_b=\frac{\rho b}{3\epsilon_0}\left[1-(a/b)^3\right]\Rightarrow\frac ab=\left(1-\frac{3\epsilon_0E_b}{\rho b}\right)^{1/3}.$$$$0\le a<b\Rightarrow0<E_b\le\frac{\rho b}{3\epsilon_0}.$$The upper endpoint is a solid sphere; vanishing field is approached only as shell volume tends to zero.

Physical check

An inferred negative radicand would rule out this uniform-density shell model for the measured field.