A point charge sits at the origin

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Conditions of Validity, Ch Gauss, Stokes and Green proved, and the first inversion of a differential operator

Problem

A point charge q=2.0 nCq = 2.0\,\mathrm{nC} sits at the origin, producing E=q4πϵ0r2r^\mathbf{E} = \dfrac{q}{4\pi\epsilon_0 r^2}\hat{\mathbf{r}}, with ϵ0=8.854×10−12 F m−1\epsilon_0 = 8.854\times10^{-12}\,\mathrm{F\,m^{-1}}. Someone argues: "∇⋅E=0\nabla\cdot\mathbf{E} = 0 everywhere except at the origin, which is a single point of zero volume. A single point contributes nothing to a volume integral, so ∭V∇⋅E dV=0\iiint_V\nabla\cdot\mathbf{E}\,\mathrm{d}V = 0 for a sphere VV of radius R=0.10 mR = 0.10\,\mathrm{m} centred on the charge. By the divergence theorem this volume integral equals the flux ∮∂VE⋅dS\oint_{\partial V}\mathbf{E}\cdot\mathrm{d}\mathbf{S} through the sphere, so that flux must be zero." ((a)) Find the error in this reasoning, naming the hypothesis of the divergence theorem it violates. ((b)) Compute the actual flux ∮∂VE⋅dS\oint_{\partial V}\mathbf{E}\cdot\mathrm{d}\mathbf{S} and confirm it disagrees with the conclusion above.

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