Page 169 - Electromagnetics

Figure 3.17: Magnetic potential.

Because V m21 is not unique, the field H is nonconservative. In point form this is
shown by the fact that ∇× H is not identically zero. We are not too concerned about
energy-related implications of the nonconservative nature of H; the electric point charge
has no magnetic analogue that might fail to conserve potential energy if moved around
in a magnetic field.
Assuming a linear, isotropic region where B(r) = µ(r)H(r), we can substitute (3.124)
into (3.123)and expand to obtain
2
∇µ(r) ·∇ m (r) + µ(r)∇ m (r) = 0.
For a homogeneous medium this reduces to Laplace’s equation
2
∇ m = 0.
We can also obtain an analogue to Poisson’s equation of electrostatics if we use
B = µ 0 (H + M) =−µ 0 ∇ m + µ 0 M

in (3.123); we have
2
∇ m =−ρ M (3.125)
where
ρ M =−∇ · M
is called the equivalent magnetization charge density. This form can be used to describe
fields of permanent magnets in the absence of J. Comparison with (3.98)shows that ρ M
is analogous to the polarization charge ρ P .
Since m obeys Poisson’s equation, the details regarding uniqueness and the construc-
tion of solutions follow from those of the electrostatic case. If we include the possibility of
a surface density of magnetization charge, then the integral solution for m in unbounded
space is

1 ρ M (r ) 1 ρ Ms (r )

m (r) = dV + dS . (3.126)

4π V |r − r | 4π S |r − r |
Here ρ Ms , the surface density of magnetization charge, is identified as ˆ n · M in the
boundary condition (3.152).
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