A point charge is fixed on the axis at height above the origin

PHYSICS · Subjective · 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=3.0×10−9 Cq = 3.0\times10^{-9}\,\mathrm{C} is fixed on the zz axis at height d=0.040 md = 0.040\,\mathrm{m} above the origin. In cylindrical coordinates (ρ,ϕ,z\rho,\phi,z, ρ\rho the distance from the zz axis) its density is ρq(ρ,ϕ,z)=q δ(ρ)2πρ δ(z−d).\rho_q(\rho,\phi,z) = q\,\frac{\delta(\rho)}{2\pi\rho}\,\delta(z-d). Verify, by evaluating ∫02π ⁣ ⁣∫0R ⁣ ⁣∫−LLρq ρ dz dρ dϕ\displaystyle\int_0^{2\pi}\!\!\int_0^{R}\!\!\int_{-L}^{L} \rho_q\,\rho\,\mathrm{d}z\,\mathrm{d}\rho\,\mathrm{d}\phi with R=0.10 mR = 0.10\,\mathrm{m} and L=0.10 mL = 0.10\,\mathrm{m} (a cylinder that encloses the charge), that the enclosed charge equals qq.

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