Page 175 - Electromagnetics

Physical interpretation of M in a magnetic material. In (3.137)we presented
an expression for the vector potential produced by a magnetized material in terms of
equivalent magnetization surface and volume currents. Suppose a magnetized medium
is separated into volume regions with bounding surfaces across which the permeability
is discontinuous. With J M =∇ × M and J Ms =−ˆ n × M we obtain

µ 0 J(r ) µ 0 J s (r )

A(r) = dV + dS +
4π V |r − r | 4π S |r − r |

µ 0

∇ × M(r ) −ˆ n × M(r )

+ dV + dS . (3.148)
4π |r − r | |r − r |

i V i S i
Here ˆ n points outward from region V i . Using the curl theorem on the fourth term and
employing the vector identity (B.43), we have

µ 0 J(r ) µ 0 J s (r )

A(r) = dV + dS +
4π V |r − r | 4π S |r − r |

µ 0

1

+ M(r ) ×∇ dV . (3.149)
4π |r − r |

i V i
ˆ
2
But ∇ (1/R) = R/R , hence the third term is a sum of integrals of the form

ˆ
µ 0 R

M(r ) × dV .
4π R 2
V i
Comparison with (3.146)shows that this integral represents a volume superposition of
dipole moments where M is a volume density of magnetic dipole moments. Hence a
magnetic material with permeability µ is equivalent to a volume distribution of magnetic
dipoles in free space. As with our interpretation of the polarization vector in a dielectric,
we base this conclusion only on Maxwell’s equations and the assumption of a linear,
isotropic relationship between B and H.
3.3.3 Boundary conditions for the magnetostatic field
The boundary conditions found for the dynamic magnetic field remain valid in the
magnetostatic case. Hence
(3.150)
ˆ n 12 × (H 1 − H 2 ) = J s
and
ˆ n 12 · (B 1 − B 2 ) = 0, (3.151)
where ˆ n 12 points into region 1 from region 2. Since the magnetostatic curl and divergence
equations are independent, so are the boundary conditions (3.150)and (3.151). We can
also write (3.151)in terms of equivalent sources by (3.118):
ˆ n 12 · (H 1 − H 2 ) = ρ Ms1 + ρ Ms2 , (3.152)
where ρ Ms = ˆ n · M is called the equivalent magnetization surface charge density. Here ˆ n
points outward from the material body.
For a linear, isotropic material described by B = µH, equation (3.150)becomes

B 1 B 2
ˆ n 12 × − = J s .
µ 1 µ 2
© 2001 by CRC Press LLC