Flux of an Electric Field — Question 4

Question 4

An angularly varying normal field separates hemisphere fluxes

On a sphere of radius R, the outward normal field is $E_n(\theta)=E_0(1+\eta\cos\theta)$, with polar angle $0\le\theta\le\pi$, $E_0>0$ and $|\eta|<1$.

1. Find the total outward flux by direct surface integration.

2. Find the northern and southern hemisphere fluxes and recover $\eta$ from their difference divided by their sum.

Question 4 - Solution

An angularly varying normal field separates hemisphere fluxes

(a) Method and calculation

$$\Phi=2\pi R^2E_0\int_0^\pi(1+\eta\cos\theta)\sin\theta\,d\theta.$$$$\Phi=4\pi R^2E_0.$$

(b) Interpretation and further calculation

$$\Phi_N=2\pi R^2E_0(1+\eta/2),\quad\Phi_S=2\pi R^2E_0(1-\eta/2).$$$$\frac{\Phi_N-\Phi_S}{\Phi_N+\Phi_S}=\frac\eta2.$$Total flux alone cannot reveal this north-south asymmetry.

Physical check

The term odd in $\cos\theta$ cancels over the whole sphere but not over a single hemisphere.