Gauss' Law and Coulomb's Law — Question 3

Question 3

Equal cube faces share flux by symmetry but equal patches need not

A point charge q is at the center of a cube of side 2a. There are no other sources. On the face $x=a$, use coordinates y,z and the outward positive-x normal.

1. Use symmetry and Gauss law to find the flux through one whole face.

2. Derive the local normal field on that face and compare its value at the face center and at a corner. Explain why equal-area patches are not guaranteed equal flux.

Question 3 - Solution

Equal cube faces share flux by symmetry but equal patches need not

(a) Method and calculation

$$\Phi_{\mathrm{cube}}=q/\epsilon_0,\quad\Phi_1=\cdots=\Phi_6.$$$$\Phi_{\mathrm{face}}=q/(6\epsilon_0).$$

(b) Interpretation and further calculation

$$E_n(y,z)=\frac{qa}{4\pi\epsilon_0(a^2+y^2+z^2)^{3/2}}.$$$$\frac{E_n(0,0)}{E_n(a,a)}=3\sqrt3.$$Cube symmetry maps whole faces to one another, but it does not map arbitrary equal-area patches on the same face to equivalent positions.

Physical check

The face center and corner differ both in source distance and in field projection onto the face normal.