Physics For Scientists And Engineers 6E - part 269
Consider a rectangle of width dx and height ! lying in the xy plane, as shown in
Figure 34.5. To apply Equation 34.3, we must first evaluate the line integral of
E ! ds
around this rectangle. The contributions from the top and bottom of the rectangle are
zero because
E is perpendicular to ds for these paths. We can express the electric field
on the right side of the rectangle as
while the field on the left side
2
is simply E(x, t). Therefore, the line integral over this
rectangle is approximately
(34.15)
Because the magnetic field is in the z direction, the magnetic flux through the rectangle
of area ! dx is approximately (
B
"
B ! dx. (This assumes that dx is very small compared
with the wavelength of the wave.) Taking the time derivative of the magnetic flux gives
(34.16)
Substituting Equations 34.15 and 34.16 into Equation 34.3 gives
This expression is Equation 34.6.
In a similar manner, we can verify Equation 34.7 by starting with Maxwell’s fourth
equation in empty space (Eq. 34.5). In this case, the line integral of
B!ds is evaluated
around a rectangle lying in the xz plane and having width dx and length !, as in Figure
34.6. Noting that the magnitude of the magnetic field changes from B(x, t) to B(x ' dx, t)
over the width dx and that the direction in which we take the line integral is as shown in
Figure 34.6, the line integral over this rectangle is found to be approximately
(34.17)
The electric flux through the rectangle is (
E
"
E! dx, which, when differentiated with
respect to time, gives
(34.18)
Substituting Equations 34.17 and 34.18 into Equation 34.5 gives
which is Equation 34.7.
+
B
+
x
" )
&
0
#
0
+
E
+
t
)
!
"
+
B
+
x
#
dx " &
0
#
0
!
dx
"
+
E
+
t
#
+
(
E
+
t
"
!
dx
+
E
+
t
!
B!d
s " [B(x, t)]! ) [B(x ' dx, t)]! $ )!
"
+
B
+
x
#
dx
+
E
+
x
" )
+
B
+
t
!
"
+
E
+
x
#
dx " )!
dx
+
B
+
t
d
(
B
dt
"
!
dx
dB
dt
%
x constant
"
!
dx
+
B
+
t
!
E!d
s " [E(x ' dx, t)]! ) [E(x, t)]! $ !
"
+
E
+
x
#
dx
E(x ' dx, t)
$ E(x, t) '
d
E
dx
%
t constant
dx " E(x, t) '
+
E
+
x
dx
S E C T I O N 3 4 . 2 • Plane Electromagnetic Waves
1073
E + d E
E
dx
!
y
x
z
B
Figure 34.5 At an instant when a
plane wave moving in the ' x
direction passes through a
rectangular path of width dx lying in
the xy plane, the electric field in the
y direction varies from E to E ' d E.
This spatial variation in E gives rise
to a time-varying magnetic field
along the z direction, according to
Equation 34.6.
2
Because dE/dx in this equation is expressed as the change in E with x at a given instant t, dE/dx is
equivalent to the partial derivative +E/+x. Likewise, dB/dt means the change in B with time at a particu-
lar position x, so in Equation 34.16 we can replace dB/dt with +B/+t.
B
E
B + d B
dx
z
y
x
!
Figure 34.6 At an instant when a
plane wave passes through a
rectangular path of width dx lying
in the xz plane, the magnetic field
in the z direction varies from B to
B ' d B. This spatial variation in B
gives rise to a time-varying electric
field along the y direction,
according to Equation 34.7.