A point charge sits at the centre of a closed rectangular box measuring

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

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

Problem

A point charge q=3.5 nCq = 3.5\,\mathrm{nC} sits at the centre of a closed rectangular box measuring 4.0 cm×6.0 cm×10.0 cm4.0\,\mathrm{cm}\times 6.0\,\mathrm{cm}\times 10.0\,\mathrm{cm}, with no other charge nearby. Taking the origin at the charge, its field is E(r)=q4πϵ0r^r2\mathbf E(\mathbf r) = \dfrac{q}{4\pi\epsilon_0}\dfrac{\hat{\mathbf r}}{r^2}, with ϵ0=8.854×10−12 F m−1\epsilon_0 = 8.854\times10^{-12}\,\mathrm{F\,m^{-1}}. ((a)) Using excision of a small ball round the charge and the divergence theorem, rather than evaluating a face integral, find the outward flux ∮E⋅dS\oint\mathbf E\cdot\mathrm d\mathbf S through the box's surface. ((b)) State in one sentence why the box's three side lengths never entered the calculation.

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