Gauss' Law and Coulomb's Law — Question 4

Question 4

Displacing a charge inside a sphere preserves flux but destroys constant field

A positive charge q lies a distance b from the center of a spherical Gaussian surface of radius R, with $0<b<R$. The near and far poles lie on the displacement axis. Here $k=1/(4\pi\epsilon_0)$.

1. Find the outward normal fields at the two poles and their ratio.

2. Evaluate that ratio for $b=R/2$, then find the total flux. Explain why either pole field times $4\pi R^2$ gives the wrong total.

Question 4 - Solution

Displacing a charge inside a sphere preserves flux but destroys constant field

(a) Method and calculation

$$E_{n,\mathrm{near}}=\frac{kq}{(R-b)^2},\quad E_{n,\mathrm{far}}=\frac{kq}{(R+b)^2}.$$$$\frac{E_{n,\mathrm{near}}}{E_{n,\mathrm{far}}}=\left(\frac{R+b}{R-b}\right)^2.$$

(b) Interpretation and further calculation

$$b=R/2:\quad E_{n,\mathrm{near}}/E_{n,\mathrm{far}}=9.$$$$\Phi=q/\epsilon_0.$$The charge is still enclosed, but the sphere is not centered on the source. Neither constant magnitude nor a radial field relative to the sphere center can be assumed.

Physical check

Both pole contributions are outward for an interior positive charge, although their magnitudes differ.