Gauss' Law — Question 9

Question 9

An unchanged Gaussian average can conceal a field reversal at one pole

A positive charge q is at the center of a sphere of radius R. Add a uniform field $E_0\hat{\mathbf z}$. Write $E_c=q/(4\pi\epsilon_0R^2)$ for the radial field magnitude of q at the sphere.

1. Find the total flux and average outward normal field over the sphere.

2. Find the normal fields at the north and south poles. For $E_0=2E_c$, determine their signs and explain why one cannot infer a constant surface field from Gauss law alone.

Question 9 - Solution

An unchanged Gaussian average can conceal a field reversal at one pole

(a) Method and calculation

$$E_n(\theta)=E_c+E_0\cos\theta,\quad\int\cos\theta\,dA=0.$$$$\Phi=4\pi R^2E_c=q/\epsilon_0,\quad\langle E_n\rangle=E_c.$$

(b) Interpretation and further calculation

$$E_{n,N}=E_c+E_0=3E_c,\quad E_{n,S}=E_c-E_0=-E_c.$$$$\Phi\ne E_n(P)\,4\pi R^2\quad\text{for a general surface point }P.$$The added field breaks spherical symmetry without changing enclosed charge or total flux.

Physical check

The south-pole outward normal points downward; a sufficiently strong upward field therefore gives negative normal flux there.