Electromagnetism / Electrodynamics I · Electrostatics and distributions of charge
A charge q is at the origin
Problem
A charge $q$ is at the origin. A sphere of radius $R$ is centered at $d\hat{\mathbf z}$ and oriented outward. Find the flux for $d<R$ and $d>R$. At $d=R$, the field is undefined at the touching point. Parameterize the sphere by $R\hat{\mathbf z}+R\hat{\mathbf n}$, let $\theta$ be the angle between $\hat{\mathbf n}$ and $+\hat{\mathbf z}$, and evaluate the natural improper surface integral obtained by deleting a shrinking cap around the touching point. Compare that direct value with the two one-sided limits obtained as the center approaches tangency from $d<R$ and $d>R$, and explain why Gauss law alone does not assign the boundary case.
Hint
The origin lies inside exactly when $d<R$.
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