Classical Electrodynamics · Problem 5.8

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · azimuthal-current-vector-potential-multipoles

Problem

A localized cylindrically symmetric current distribution is such that the current flows only in the azimuthal direction; the current density is a function only of rr and θ\theta (or ρ\rho and zz): J=ϕ^J(r,θ)\mathbf{J} = \hat{\boldsymbol{\phi}} J(r, \theta). The distribution is ``hollow'' in the sense that there is a current-free region near the origin, as well as outside. (a) Show that the magnetic field can be derived from the azimuthal component of the vector potential, with a multipole expansion Aϕ(r,θ)=−μ04π∑LmLrLPL1(cos⁡θ)A_\phi(r,\theta) = -\frac{\mu_0}{4\pi}\sum_L m_L r^L P_L^1(\cos\theta) in the interior and Aϕ(r,θ)=−μ04π∑LμLr−L−1PL1(cos⁡θ)A_\phi(r,\theta) = -\frac{\mu_0}{4\pi}\sum_L \mu_L r^{-L-1} P_L^1(\cos\theta) outside the current distribution. (b) Show that the internal and external multipole moments are mL=−1L(L+1)∫d3x r−L−1PL1(cos⁡θ) J(r,θ)m_L = -\frac{1}{L(L+1)}\int d^3x\, r^{-L-1} P_L^1(\cos\theta)\, J(r,\theta) and μL=−1L(L+1)∫d3x rLPL1(cos⁡θ) J(r,θ)\mu_L = -\frac{1}{L(L+1)}\int d^3x\, r^L P_L^1(\cos\theta)\, J(r,\theta)

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