Gauss' Law and Coulomb's Law — Question 2

Question 2

A cone captures a solid-angle fraction of a point-source flux

A point charge q is at the center of a sphere. Consider the outward spherical cap $0\le\theta\le\alpha$, where $0\le\alpha\le\pi$. Use $k=1/(4\pi\epsilon_0)$.

1. Derive its solid angle and electric flux from Coulomb field.

2. Find the cone half-angle that captures one quarter of the total signed flux. State whether increasing the sphere radius changes this fraction.

Question 2 - Solution

A cone captures a solid-angle fraction of a point-source flux

(a) Method and calculation

$$d\Omega=\sin\theta\,d\theta\,d\phi,\quad\Omega=2\pi\int_0^\alpha\sin\theta\,d\theta=2\pi(1-\cos\alpha).$$$$\Phi_{\mathrm{cap}}=kq\Omega=\frac{q}{2\epsilon_0}(1-\cos\alpha).$$

(b) Interpretation and further calculation

$$\frac{\Phi_{\mathrm{cap}}}{q/\epsilon_0}=\frac{1-\cos\alpha}{2}=\tfrac14\Rightarrow\alpha=60^\circ.$$$$E_r\propto R^{-2},\quad A_{\mathrm{cap}}\propto R^2\Rightarrow\Phi_{\mathrm{cap}}\text{ is independent of }R.$$

Physical check

The radius cancellation is the geometric signature of the inverse-square field.