Mathematics of Physics and Engineering

176 Maxwell's Equations

and (3.3.37) becomes

A(P) = g fjj M(Q) x grad (j^j dV. (3.3.39)

G

EXERCISE 3.3.18.^5 Verify (3.3.38). Hint: use (3.1.33) on page UO, taking P

as the origin of the frame.

By (3.1.30) on page 139,

M(Q) x grad (^) = ^ curl A*(Q) - curl (^M) , (3.3.40)

and then we use relation (3.2.6) on page 153 to write (3.3.39) as

A{P) =

£ \IIIwQ\

cmlMiQ)dV +

IIwQ\

M{Q) X HGda
) '
\ G dG J
(3.3.41)
where dG is the boundary of G, which we assume to be piece-wise smooth,
closed, and orientable, and ng is the outside unit normal to dG. We
call Jb = curliW the bound current, and Kb = M x no, the surface
density of the bound current. Recall that a magnetic dipole is mod-
elled by a circular movement of electric charges, and the vector J& can be
interpreted as the total current produced by the charge movement in all
the point magnetic dipoles in G.
EXERCISE 3.3.19. (a)c Verify that (3.3.41) indeed follows from (3.3.39),
(3.3.40), and (3.2.6). (b)B Find the magnetic field corresponding to the
potential (3.3.41). Hint: recall that the magnetic field is curl A; in the resulting
computations, you now treat P as a variable and Q as fixed.
Suppose that the material has non-zero conductivity and there is a
steady current Je flowing in the material due to an external source of
electric field, for example, two electrodes embedded in the material and
connected to an external battery. The total current density in the material
is then J = J\, + Je = curliVT + Je. We assume the the system is in
equilibrium so that both J\, and Je do not depend on time. By the original
Ampere's Law (3.3.11) on page 166, the resulting magnetic field B satisfies
curlB = /x 0 (curlM + Je) or curl (B//xo — M) = Je. We therefore define
the net magnetic field H, also called the magnetic field strength, as

H = B/fio - M, (3.3.42)