A long cylindrical rod of radius carries a magnetisation

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Vector calculus and field geometry

The Boundary-Value Problem, Ch Magnetisation, bound currents and H, completed

Problem

A long cylindrical rod of radius RR carries a magnetisation M(s)=M0(s/R)2 z^\mathbf{M}(s) = M_0(s/R)^2\,\hat{\mathbf{z}}, where ss is the distance from the axis and M0M_0 is a constant. There is no free current anywhere. ((a)) Starting from the vector potential of a magnetised body, A(r)=μ04π∫VM(r′)×∇′1∣r−r′∣ dτ′\mathbf{A}(\mathbf{r}) = \dfrac{\mu_0}{4\pi}\displaystyle\int_{\mathcal V}\mathbf{M}(\mathbf{r}')\times\nabla'\dfrac{1}{|\mathbf{r}-\mathbf{r}'|}\,d\tau', carry out one integration by parts to identify the volume and surface bound current densities in general. ((b)) Evaluate Jb\mathbf{J}_b and Kb\mathbf{K}_b for this rod. ((c)) Show that the total bound current crossing a half-plane ϕ=const\phi = \text{const}, per unit length along z^\hat{\mathbf{z}}, is zero.

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