Page 45 - Electromagnetics Handbook

Generally a material will have either its electric or magnetic properties dominant. If
µ r = 1 and
r
= 1 then the material is generally called a perfect dielectric or a perfect
insulator, and is said to be an electric material. If
r = 1 and µ r
= 1, the material is
said to be a magnetic material.
A linear isotropic material may also have conduction properties. In a conducting
material, a constitutive relation is generally used to describe the mechanical interaction
of field and charge by relating the electric field to a secondary electric current. For
a nondispersive isotropic material, the current is aligned with, and proportional to, the
electric field; there are no temporal operators in the constitutive relation, which is simply

J = σE. (2.28)

This is known as Ohm’s law. Here σ is the conductivity of the material.
If µ r ≈ 1 and σ is very small, the material is generally called a good dielectric.If
σ is very large, the material is generally called a good conductor. The conditions by
which we say the conductivity is “small” or “large” are usually established using the
frequency response of the material. Materials that are good dielectrics over broad ranges
of frequency include various glasses and plastics such as fused quartz, polyethylene,
and teflon. Materials that are good conductors over broad ranges of frequency include
common metals such as gold, silver, and copper.
For dispersive linear isotropic materials, the constitutive parameters become nonsta-
tionary (time dependent), and the constitutive relations involve time operators. (Note
that the name dispersive describes the tendency for pulsed electromagnetic waves to
spread out, or disperse, in materials of this type.) If we assume that the relationships
given by (2.26), (2.27), and (2.28) retain their product form in the frequency domain,
then by the convolution theorem we have in the time domain the constitutive relations

t

D(r, t) =
0 E(r, t) + χ e (r, t − t )E(r, t ) dt , (2.29)
−∞
t

B(r, t) = µ 0 H(r, t) + χ m (r, t − t )H(r, t ) dt , (2.30)
−∞
t

J(r, t) = σ(r, t − t )E(r, t ) dt . (2.31)
−∞
These expressions were first introduced by Volterra in 1912[199]. We see that for a linear
dispersive material of this type the constitutive operators are time integrals, and that
the behavior of D(t) depends not only on the value of E at time t, but on its values at
all past times. Thus, in dispersive materials there is a “time lag” between the effect of
the applied field and the polarization or magnetization that results. In the frequency
domain, temporal dispersion is associated with complex values of the constitutive pa-
rameters, which, to describe a causal relationship, cannot be constant with frequency.
The nonzero imaginary component is identified with the dissipation of electromagnetic
energy as heat. Causality is implied by the upper limit being t in the convolution inte-
grals, which indicates that D(t) cannot depend on future values of E(t). This assumption
leads to a relationship between the real and imaginary parts of the frequency domain
constitutive parameters as described through the Kronig–Kramers equations.

Constitutive relations for fields in perfect conductors. In a perfect electric con-
ductor (PEC) or a perfect magnetic conductor (PMC) the fields are exactly specified as

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