Applying Gauss' Law: Spherical Symmetry — Question 8

Question 8

An eccentric spherical cavity has a uniform field with a three-dimensional factor

A uniformly charged insulating sphere of density $\rho$ and radius R has a spherical cavity of radius a whose center is displaced by vector $\mathbf d$, with $d+a<R$. Let $\mathbf r$ be measured from the original center and $\mathbf r\prime=\mathbf r-\mathbf d$ from the cavity center.

1. Use superposition of two full uniform spheres to derive the cavity field.

2. Compare its magnitude with that for an analogous infinite cylindrical cavity, whose field is $\rho\mathbf d/(2\epsilon_0)$. Bound the spherical cavity field using the containment condition.

Question 8 - Solution

An eccentric spherical cavity has a uniform field with a three-dimensional factor

(a) Method and calculation

$$\mathbf E_{\mathrm{full}}=\frac\rho{3\epsilon_0}\mathbf r,\quad\mathbf E_{\mathrm{removed}}=-\frac\rho{3\epsilon_0}\mathbf r\prime.$$$$\mathbf E_{\mathrm{cavity}}=\frac\rho{3\epsilon_0}\mathbf d.$$

(b) Interpretation and further calculation

$$\frac{|E_{\mathrm{spherical}}|}{|E_{\mathrm{cylindrical}}|}=\frac23\quad(\rho d\ne0).$$$$d<R-a\Rightarrow |E_{\mathrm{cavity}}|<\frac{|\rho|(R-a)}{3\epsilon_0}.$$The different coefficient comes from spherical versus cylindrical enclosed-volume growth, not from a conducting shielding effect.

Physical check

Superposition is needed because the cavity breaks the symmetry of the full distribution, although each auxiliary sphere remains symmetric.