Page 399 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 381

Consider the Maxwell curl equation (23) which, using (42), becomes:
(53)
∇× H s = jω( − j )E s = ω E s + jω E s
This equation can be expressed in a more familiar way, in which conduction current
is included:
∇× H s = J s + jω E s (54)
We next use J s = σE s , and interpret in (54) as . The latter equation becomes:

(55)
∇× H s = (σ + jω )E s = J σs + J ds
which we have expressed in terms of conduction current density, J σs = σE s , and
displacement current density, J ds = jω E s . Comparing Eqs. (53) and (55), we find

that in a conductive medium:
σ
(56)
=
ω
Let us now turn our attention to the case of a dielectric material in which the loss
is very small. The criterion by which we should judge whether or not the loss is small
is the magnitude of the loss tangent, / . This parameter will have a direct influence

on the attenuation coefficient, α,as seen from Eq. (44). In the case of conducting
media, to which (56) applies, the loss tangent becomes σ/ω .By inspecting (55),

we see that the ratio of conduction current density to displacement current density
magnitudes is
J σs σ
= = (57)
J ds j jω
That is, these two vectors point in the same direction in space, but they are 90 out of
◦
phase in time. Displacement current density leads conduction current density by 90 ,
◦
just as the current through a capacitor leads the current through a resistor in parallel
with it by 90 in an ordinary electric circuit. This phase relationship is shown in
◦
Figure 11.2. The angle θ (not to be confused with the polar angle in spherical
coordinates) may therefore be identified as the angle by which the displacement
current density leads the total current density, and

σ
tan θ = = (58)
ω
The reasoning behind the term loss tangent is thus evident. Problem 11.16 at the end
of the chapter indicates that the Q of a capacitor (its quality factor, not its charge)
that incorporates a lossy dielectric is the reciprocal of the loss tangent.
If the loss tangent is small, then we may obtain useful approximations for the
attenuation and phase constants, and the intrinsic impedance. The criterion for a small
loss tangent is / 1, which we say identifies the medium as a good dielectric.