A point charge sits not at the origin but at the spherical-polar location , ,

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=3.0 nCq = 3.0\,\mathrm{nC} sits not at the origin but at the spherical-polar location r0=2.0 cmr_0 = 2.0\,\mathrm{cm}, θ0=π/3\theta_0 = \pi/3, ϕ0=π/2\phi_0 = \pi/2. The scale factors of spherical coordinates are hr=1h_r=1, hθ=rh_\theta=r, hϕ=rsin⁡θh_\phi=r\sin\theta, so the volume element is dV=r2sin⁡θ dr dθ dϕ\mathrm{d}V = r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi. ((a)) Write the charge density ρq(r,θ,ϕ)\rho_q(r,\theta,\phi) as qq times a product of one-dimensional delta functions in rr, θ\theta and ϕ\phi, with whatever coefficient makes ∫ρq dV=q\int\rho_q\,\mathrm{d}V = q over all space. ((b)) Evaluate that coefficient numerically, in m−2\mathrm{m^{-2}}. ((c)) State, without further calculation, the value of ∫ρq dV\int\rho_q\,\mathrm{d}V over a small ball centred away from (r0,θ0,ϕ0)(r_0,\theta_0,\phi_0) that does not contain that point.

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