A point charge sits at the origin

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential

Mathematical Methods for Physics, Ch Delta functions, distributions and generalised functions

Problem

A point charge q=3.2 nCq = 3.2\,\mathrm{nC} sits at the origin. Its field is E(r)=q4πϵ0r^r2\mathbf{E}(\mathbf{r}) = \dfrac{q}{4\pi\epsilon_0}\dfrac{\hat{\mathbf{r}}}{r^2}, and the checked identity ∇⋅(r^/r2)=4πδ3(r)\nabla\cdot(\hat{\mathbf{r}}/r^2) = 4\pi\delta^3(\mathbf{r}) gives the local form of Gauss's law, ∇⋅E=q δ3(r)/ϵ0\nabla\cdot\mathbf{E} = q\,\delta^3(\mathbf{r})/\epsilon_0. Take ϵ0=8.854×10−12 F m−1\epsilon_0=8.854\times10^{-12}\,\mathrm{F\,m^{-1}}. ((a)) Integrate the local Gauss's law over the ball of radius 2.0 cm2.0\,\mathrm{cm} centred on the origin and over the ball of radius 8.0 cm8.0\,\mathrm{cm}, and compute the flux Φ=∮E⋅dS\Phi = \oint \mathbf{E}\cdot\mathrm{d}\mathbf{S} directly from E\mathbf{E} through each bounding sphere. Give both numerically and confirm that the two sides of the divergence theorem agree at both radii. ((b)) A third sphere of radius 1.0 cm1.0\,\mathrm{cm} is centred at the point (10.0 cm,0,0)(10.0\,\mathrm{cm}, 0, 0), so that it does not enclose the origin. State, with a reason drawn from the same identity, the value of the flux of E\mathbf{E} through this sphere.

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