Mathematics of Physics and Engineering

Equations in Material Media 177

so that Ampere's Law in material media becomes

curlH = Je. (3.3.43)

EXERCISE 3.3.20. B Verify that, in material media, Maxwell's equation

(3.3.5) becomes

cm\H = Je + ^-. (3.3.44)
at
Hint: use the same arguments as in the derivation of (3.3.5).
In paramagnetic materials, the magnetic dipoles are aligned so that the
induced magnetic field B is stronger than in vacuum; the vector M has, on
average, the same direction as H: Jff(M • H) dV > 0. In diamagnetic
G
materials, the magnetic dipoles are aligned so that the induced magnetic
field B is weaker than in vacuum; the directions of M and H are, on the
average, opposite: JJf(M • H) dV < 0.
G
A material is called linear if magnetization M depends linearly on H,
that is, M = XmH, where Xm is called the magnetic susceptibility of
the material. This linear relation holds for most materials when \H\ is
sufficiently small. By (3.3.42),

B = ii H, where /x = fi 0 (l + Xm)

is called the permeability of the material. In vacuum, Xm — 0 and fi =

I/O- Similar to (electric) permittivity, permeability is, in general, a tensor

field and the value of /i depends on both the location and the direction.

For time-varying magnetic fields, the value of \i can also depend on the

frequency of the field. The value of it usually depends also on temperature

and density. In linear homogeneous isotropic materials, /x is a number and

(3.3.43) becomes

VxB = iiJe, (3.3.45)

which is the same as (3.3.11), but with \i instead of \i§.

EXERCISE 3.3.21. c Assuming the frequencies of B and E are constant,

verify that, in a linear homogeneous isotropic material, equation (3.3-44)

becomes

cmlB = nJ dE

e + /j,e~. (3.3.46)

Hint: use (3.3.36).