A cloud of charge fills the whole of space with volume charge density

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

Mathematical Methods for Physics, Ch Multiple integrals and change of variables

Problem

A cloud of charge fills the whole of space with volume charge density ρq(r)=ρq0 e−r2/b2\rho_q(r) = \rho_{q0}\,e^{-r^2/b^2}, where rr is the distance from the origin, ρq0=2.0×10−6 C m−3\rho_{q0} = 2.0\times10^{-6}\,\mathrm{C\,m^{-3}} and b=0.050 mb = 0.050\,\mathrm{m}. ((a)) Write the total charge as a triple integral in spherical coordinates, giving the volume element and the limits. ((b)) Evaluate it, using ∫0∞r2e−r2/b2 dr=b3π4\int_0^{\infty} r^2 e^{-r^2/b^2}\,\mathrm{d}r = \dfrac{b^3\sqrt\pi}{4} (obtained by parts from the Gaussian integral) and ∫0πsin⁡θ dθ=2\int_0^\pi \sin\theta\,\mathrm{d}\theta = 2. ((c)) Give the total charge to two significant figures.

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