Page 392 - Engineering Electromagnetics, 8th Edition

374 ENGINEERING ELECTROMAGNETICS

In general, we find from (32) that the electric and magnetic field amplitudes of
the forward-propagating wave in free space are related through

µ 0
E x0 = H y0 = η 0 H y0 (34a)
0
We also find the backward-propagating wave amplitudes are related through

µ 0
E x0 =− H y0 =−η 0 H y0 (34b)

0
where the intrinsic impedance of free space is defined as

µ 0 .
η 0 = = 377 = 120π
(35)
0

The dimension of η 0 in ohms is immediately evident from its definition as the ratio of
E (in units of V/m) to H (in units of A/m). It is in direct analogy to the characteristic
impedance, Z 0 ,ofa transmission line, where we defined the latter as the ratio of
voltage to current in a traveling wave. We note that the difference between (34a) and
(34b)isa minus sign. This is consistent with the transmission line analogy that led to
Eqs. (25a) and (25b)in Chapter 10. Those equations accounted for the definitions of
positive and negative current associated with forward and backward voltage waves. In
a similar way, Eq. (34a) specifies that in a forward-z propagating uniform plane wave
whose electric field vector lies in the positive x direction at a given point in time and
space, the magnetic field vector lies in the positive y direction at the same space and
time coordinates. In the case of a backward-z propagating wave having a positive
x-directed electric field, the magnetic field vector lies in the negative y direction. The
physical significance of this has to do with the definition of power flow in the wave,
2
as specified through the Poynting vector, S = E×H (in watts/m ). The cross product
of E with H must give the correct wave propagation direction, and so the need for
the minus sign in (34b)is apparent. Issues relating to power transmission will be
addressed in Section 11.3.
Some feeling for the way in which the fields vary in space may be obtained from
Figures 11.1a and 11.1b. The electric field intensity in Figure 11.1a is shown at t = 0,
and the instantaneous value of the field is depicted along three lines, the z axis and
arbitrary lines parallel to the z axis in the x = 0 and y = 0 planes. Since the field
is uniform in planes perpendicular to the z axis, the variation along all three of the
lines is the same. One complete cycle of the variation occurs in a wavelength, λ. The
values of H y at the same time and positions are shown in Figure 11.1b.
A uniform plane wave cannot exist physically, for it extends to infinity in two
dimensions at least and represents an infinite amount of energy. The distant field of
a transmitting antenna, however, is essentially a uniform plane wave in some limited
region; for example, a radar signal impinging on a distant target is closely a uniform
plane wave.