Page 430 - Engineering Electromagnetics, 8th Edition

412 ENGINEERING ELECTROMAGNETICS

The magnetic field intensities are
100
H y10 = 100 = 1.00 A/m
+
50
H y10 =− 100 =−0.50 A/m
−
Using Eq. (77) from Chapter 11, we find that the magnitude of the average incident
power density is
1 1

S 1i = Re{E s × H } = E + H + = 50 W/m 2
∗
2 2 x10 y10
s
The average reflected power density is
1
S 1r =− E − H − = 12.5 W/m 2
2 x10 y10
In region 2, using (10),
E x20 = τ E x10 = 150 V/m
+
+
and
150
H y20 = 300 = 0.500 A/m
+
Therefore, the average power density that is transmitted through the boundary into
region 2 is
1 2
+
+
S 2 = E x20 H y20 = 37.5 W/m
2
We may check and confirm the power conservation requirement:
S 1i = S 1r + S 2

A general rule on the transfer of power through reflection and transmission can
be formulated. We consider the same field vector and interface orientations as before,
but allow for the case of complex impedances. For the incident power density, we have
1 1 1 1 1 2
+
+∗
+∗
S 1i = Re E xs1 H ys1 = Re E + E x10 = Re E x10
x10
+
2 2 η ∗ 1 2 η ∗ 1
The reflected power density is then
1 1 1 1 1
S 1r =− Re E − H −∗ = Re E + E +∗ = Re E + 2 | | 2

∗
2 xs1 ys1 2 x10 η 1 ∗ x10 2 η ∗ 1 x10
We thus find the general relation between the reflected and incident power:
2
S 1r =| | S 1i (15)
In a similar way, we find the transmitted power density:
1 1 1 1 1 2
∗
S 2 = Re E + H +∗ = Re τ E + τ E +∗ = Re E + |τ| 2
2 xs2 ys2 2 x10 η ∗ 2 x10 2 η ∗ 2 x10