A point charge sits at the origin

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

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

Problem

A point charge q=5.0×10−9 Cq = 5.0\times10^{-9}\,\mathrm{C} sits at the origin. Its charge density, written in spherical coordinates (r,θ,ϕr,\theta,\phi as usual, with θ\theta from the +z+z axis), is ρq(r,θ,ϕ)=q δ(r)4πr2.\rho_q(r,\theta,\phi) = q\,\frac{\delta(r)}{4\pi r^2}. ((a)) Evaluate ∫0∞ ⁣ ⁣∫0π ⁣ ⁣∫02πρq r2sin⁡θ dϕ dθ dr\displaystyle\int_0^{\infty}\!\!\int_0^{\pi}\!\!\int_0^{2\pi} \rho_q\,r^2\sin\theta\,\mathrm{d}\phi\,\mathrm{d}\theta\,\mathrm{d}r, the total charge in all space. ((b)) Evaluate the same integral but with rr running only from 1.0 cm1.0\,\mathrm{cm} to 2.0 cm2.0\,\mathrm{cm} (a shell that misses the origin).

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