Page 386 - Engineering Electromagnetics, 8th Edition

368 ENGINEERING ELECTROMAGNETICS

written in terms of E and H only as

∂E
∇× H = 0 (1)
∂t
∂H
∇× E =−µ 0 (2)
∂t
∇ · E = 0 (3)
∇ · H = 0 (4)

Now let us see whether wave motion can be inferred from these four equations
without actually solving them. Equation (1) states that if electric field E is changing
with time at some point, then magnetic field H has curl at that point; therefore H varies
spatially in a direction normal to its orientation direction. Also, if E is changing with
time, then H will in general also change with time, although not necessarily in the
same way. Next, we see from Eq. (2) that a time-varying H generates E, which,
having curl, varies spatially in the direction normal to its orientation. We now have
once more a changing electric field, our original hypothesis, but this field is present
a small distance away from the point of the original disturbance. We might guess
(correctly) that the velocity with which the effect moves away from the original point
is the velocity of light, but this must be checked by a more detailed examination of
Maxwell’s equations.
We postulate the existence of a uniform plane wave, in which both fields, E and
H, lie in the transverse plane—that is, the plane whose normal is the direction of
propagation. Furthermore, and by definition, both fields are of constant magnitude in
the transverse plane. For this reason, such a wave is sometimes called a transverse
electromagnetic (TEM) wave. The required spatial variation of both fields in the
direction normal to their orientations will therefore occur only in the direction of
travel—or normal to the transverse plane. Assume, for example, that E = E x a x ,or
that the electric field is polarized in the x direction. If we further assume that wave
travel is in the z direction, we allow spatial variation of E only with z. Using Eq. (2),
we note that with these restrictions, the curl of E reduces to a single term:

∂E x ∂H ∂ H y
∇× E = a y =−µ 0 =−µ 0 a y (5)
∂z ∂t ∂t

The direction of the curl of E in (5) determines the direction of H, which we observe
to be along the y direction. Therefore, in a uniform plane wave, the directions of E and
H and the direction of travel are mutually orthogonal. Using the y-directed magnetic
field, and the fact that it varies only in z, simplifies Eq. (1) to read

∂ H y ∂E ∂E x
∇× H =− a x = 0 = 0 a x (6)
∂z ∂t ∂t