Single-Particle Electrodynamics - Assassination Science

sheet wrapping the sphere, with the charge density being positive in the
direction of d, negative antiparallel to d, and zero perpendicular to d (more
specifically, varying as sin θ, where θ = π/2 in the direction of d, and −π/2
antiparallel to d).
But this cannot, of course, be generalised to the magnetic case, since
we would then have a magnetic charge sheet wrapping the spherical volume,
which is, as noted, fundamentally disallowed by the Maxwell equations. However,
we do know that the field external to the spherical source volume will
be the dual of the electric case. Let us then write
⎧
− µ
⎪⎨ 4πε + 3 Bµ M(r), r < ε,
B µ (r) =
3(n·µ)n − µ
⎪⎩
, r > ε,
4πr 3
where the extra magnetic Maxwell field, B µ M(r), will be constructed so as to
repair the breach of Maxwell’s equations that would otherwise occur. Now,
since there is no net electric current in the interior of the sphere (since the
effects of any current loop at one position would be cancelled out by that
of its neighbour—except on the surface of the sphere—due to the fact that
the polarisation is uniform), and since everything is electromagnetostatic,
Maxwell’s equations require
∇·B µ M = 0
and
∇×B µ M = 0;
hence, B µ M must be constant throughout the sphere. We must therefore only
determine its magnitude and direction. Clearly, the direction of B µ M singles
out one particular direction in space; since the only special direction we have
available to us is µ, we must have
B M(r) µ = B µ M ˆµ.
205