Gauss' Law — Question 3

Question 3

Overlapping Gaussian volumes require inclusion and exclusion

Two closed surfaces enclose overlapping volumes A and B. Their outward fluxes are $7\Phi_0$ and $5\Phi_0$. The closed boundary of their intersection has flux $2\Phi_0$, with $\Phi_0>0$. No charge lies on any boundary.

1. Find the charge in A only, B only and the overlap.

2. Find the outward flux through the boundary of the union and explain why simply adding the first two fluxes fails.

Question 3 - Solution

Overlapping Gaussian volumes require inclusion and exclusion

(a) Method and calculation

$$Q_{A\cap B}=2\epsilon_0\Phi_0,\quad Q_{A\setminus B}=(7-2)\epsilon_0\Phi_0=5\epsilon_0\Phi_0.$$$$Q_{B\setminus A}=(5-2)\epsilon_0\Phi_0=3\epsilon_0\Phi_0.$$

(b) Interpretation and further calculation

$$Q_{A\cup B}=Q_A+Q_B-Q_{A\cap B}=10\epsilon_0\Phi_0.$$$$\Phi_{A\cup B}=10\Phi_0.$$Adding the separate enclosed charges counts the intersection twice; the union contains it only once.

Physical check

The cancellation of internal boundary pieces is consistent with their opposite outward normals.