636
14 Electromagnetism II
δ =
2
μσ ω
(14.44)
where μ is the permeability and σ is the conductivity.
Phase Velocity (v ph ) and Group Velocity (v g )
Phase velocity is just due to the nodes of the wave that are moving and not energy
or information. In fact the phase velocity can be greater than c, the velocity of light
v ph = ω/k
(14.45)
In order to know how fast signals will travel one must calculate the speed of
pulses or modulation caused by the interference of a wave of one frequency with
one or more waves of slightly different frequencies. The speed of the envelope of
such a group of waves is called the group velocity and is given by
v g = dω/dk
(14.46)
Waveguides are hollow metallic structures in which the electromagnetic waves
are guided to travel from one place to another without much attenuation. Here, we
shall be concerned only with rectangular guide of cross-section of x = a and y = b,
with the wave propagated in the z-direction.
k
2
=
ω 2
c 2 −
mπ
a
2 −
nπ
b
2
(14.47)
where c is the velocity of light in free space, m and n are integers. Equation (14.47)
is valid both for TE mn and TH mn waves.
For the simplest case m = 1 and n = 0. For TE 10 wave.
v ph =
c
1 −
λ 0
2a
2
(14.48)
where λ 0 is the free space wavelength.
v g = c
1 −
λ
2a
2
(14.49)
v ph .v g = c
2
(14.50)
λ g =
λ 0
1 −
λ 0
2a
2
(14.51)
14 Electromagnetism II
δ =
2
μσ ω
(14.44)
where μ is the permeability and σ is the conductivity.
Phase Velocity (v ph ) and Group Velocity (v g )
Phase velocity is just due to the nodes of the wave that are moving and not energy
or information. In fact the phase velocity can be greater than c, the velocity of light
v ph = ω/k
(14.45)
In order to know how fast signals will travel one must calculate the speed of
pulses or modulation caused by the interference of a wave of one frequency with
one or more waves of slightly different frequencies. The speed of the envelope of
such a group of waves is called the group velocity and is given by
v g = dω/dk
(14.46)
Waveguides are hollow metallic structures in which the electromagnetic waves
are guided to travel from one place to another without much attenuation. Here, we
shall be concerned only with rectangular guide of cross-section of x = a and y = b,
with the wave propagated in the z-direction.
k
2
=
ω 2
c 2 −
mπ
a
2 −
nπ
b
2
(14.47)
where c is the velocity of light in free space, m and n are integers. Equation (14.47)
is valid both for TE mn and TH mn waves.
For the simplest case m = 1 and n = 0. For TE 10 wave.
v ph =
c
1 −
λ 0
2a
2
(14.48)
where λ 0 is the free space wavelength.
v g = c
1 −
λ
2a
2
(14.49)
v ph .v g = c
2
(14.50)
λ g =
λ 0
1 −
λ 0
2a
2
(14.51)