A charge is spread smoothly through a small ball centred on the origin

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Tensors and differential geometry

Invariance, Ch Integrating forms: Stokes's theorem, holes and Maxwell's equations

Problem

A charge Q=3.0 nCQ = 3.0\,\mathrm{nC} is spread smoothly through a small ball centred on the origin, so that ρ\rho is a smooth function. Let SS be any closed surface enclosing the ball, oriented outward, and let ρ dV/ε0\rho\,\mathrm{d}V/\varepsilon_0 be the source three-form of Gauss's law, with ε0=8.854×10−12 F m−1\varepsilon_0 = 8.854\times10^{-12}\,\mathrm{F\,m^{-1}}. Writing the electric flux two-form as α\alpha with dα=ρ dV/ε0\mathrm{d}\alpha = \rho\,\mathrm{d}V/\varepsilon_0, use the generalised Stokes theorem to find ∫Sα=∮SE⋅dS\int_S\alpha = \oint_S\mathbf{E}\cdot\mathrm{d}\mathbf{S}.

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