导电媒质中的平面波1(双语)

In this way, a sinusoidal electromagnetic field then satisfies the following homogeneous vector Helmholtz equation:
2 2
E H
2e E 0 2e H 0
Let
kc
e
If , the first Maxwell’s equation becomes
H E jE j( j )E
If let
e
j
Then the above equation can be rewritten as
H je E
where e is called the equivalent permittivity.
H (z)
1 Z0
ez
E
ey
1 6π
e j2πz
A/m
(c) The energy flow density vector is
Sc
E
H*
ez
10 3π
W/m2
(d) The phase and energy velocities are
vp
ve
k
3108
m/s
3. Plane Waves in Conducting Media
We construct a cylinder of long l and cross-section A along the direction of energy flow, as shown in the figure.
Suppose the distribution of ຫໍສະໝຸດ Baiduhe energy is
Solution: (a) The frequency is The wavelength is
f
2π
6π 108 2π
3108
Hz
2π k
2π 2π
1m
(b) The electric field intensity is
E (z) ex 20e j2πz V/m
The magnetic field intensity is
By spatial Fourier transform, a non-plane wave can be expressed in terms of the sum of many plane waves, which proves to be useful sometimes
Example. A uniform plane wave is propagating along with the positive direction of the z-axis in vacuum, and the instantaneous value of the electric field intensity is
If the observer is very far away from the source, the wave front is very large while the observer is limited to the local area, the wave can be approximately considered as a uniform plane wave.
Considering
Sav
Ex20 Z
and wav 2weav Ex20
, we find
ve
1
vp
The wave front of a uniform plane wave is an infinite plane and the amplitude of the field intensity is uniform on the wave front, and the energy flow density is constant on the wave front. Thus this uniform plane wave carries infinite energy. Apparently, an ideal uniform plane wave does not exist in nature.
( j )
We obtain
2 2
E H
kc2 E 0 kc2 H 0
If we let
E Exeax s before, and
E x x
Sav A
wavlA t
wav
A
l t
Obviously, the ratiotl stands for the displacement of the energy in
time
t,
and
it
is
called
the
energy velocity,
ve
Sav wav
denoted
as
ve.
We
obtain
S A uniform in the cylinder. The average value
of the energy density is wav , and that of the
l
energy flow density is Sav.
Then the total energy in the cylinder is wav Al , and the total energy
flowing across the cross-sectional area A per unit time is Sav A.
If all energy in the cylinder flows across the area A in the time
interval t, then
Sav At wavlA
Find:
E(z, t) ex 20 2 sin(6π 108t 2πz) V/m
(a) The frequency and the wavelength. (b) The complex vectors of the electric and the magnetic field
intensities. (c) The complex energy flow density vector. (d) The phase velocity and the energy velocity.