Page 44 - Electromagnetics

Historically it has been the practice to define µ 0 , measure c, and compute
0 . In SI units
µ 0 = 4π × 10 −7 H/m,
8
c = 2.998 × 10 m/s,

0 = 8.854 × 10 −12 F/m.

With the two constitutive equations we have enough information to put Maxwell’s
equations into definite form. Traditionally (2.22) and (2.23) are substituted into (2.1)–
(2.2) to give

∂B
∇× E =− , (2.24)
∂t
∂E
∇× B = µ 0 J + µ 0
0 . (2.25)
∂t
These are two vector equations in two vector unknowns (equivalently, six scalar equations
in six scalar unknowns).
In terms of the general constitutive relation (2.18), we find that free space is isotropic
with
1
¯ ¯ ¯ ¯ ¯
P = Q = I, L = M = 0,
η 0
where η 0 = (µ 0 /
0 ) 1/2 is called the intrinsic impedance of free space. This emphasizes
the fact that free space has, along with c, only a single empirical constant associated
with it (i.e.,
0 or η 0 ). Since no derivative or integral operators appear in the constitutive
relations, free space is nondispersive.

Constitutive relations in a linear isotropic material. In a linear isotropic mate-
rial there is proportionality between D and E and between B and H. The constants of
proportionality are the permittivity
and the permeability µ. If the material is nondis-
persive, the constitutive relations take the form
D =
E, B = µH,

where
and µ may depend on position for inhomogeneous materials. Often the permit-
tivity and permeability are referenced to the permittivity and permeability of free space
according to

=
r
0 , µ = µ r µ 0 .
Here the dimensionless quantities
r and µ r are called, respectively, the relative permit-
tivity and relative permeability.
When dealing with the Maxwell–Boffi equations (§ 2.4) the difference between the
material and free space values of D and H becomes important. Thus for linear isotropic
materials we often write the constitutive relations as

D =
0 E +
0 χ e E, (2.26)
B = µ 0 H + µ 0 χ m H, (2.27)
where the dimensionless quantities χ e =
r − 1 and χ m = µ r − 1 are called, respectively,
the electric and magnetic susceptibilities of the material. In terms of (2.18) we have
¯
r ¯ ¯ 1 ¯ ¯ ¯
P = I, Q = I, L = M = 0.
η 0 η 0 µ r

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