14.2 Problems
649
14.2.3 Phase Velocity and Group Velocity
14.85 Using the results of prob. (14.93) and the following table, estimate the fractional difference between the phase and group velocity in air at a wavelength
of 5000 Å
Free space wavelength (Å)
(n − 1) for air
4800
2.786 × 10 −4
5000
2.781 × 10 −4
5200
2.777 × 10 −4
[University of Manchester 1972]
14.86 Given the dispersion relation ω = ak 2 , calculate (a) phase velocity and (b)
group velocity.
14.87 (a) Write down an expression for the phase velocity v p of an electromagnetic wave in a medium with permittivity ε and permeability μ.
(b) The relative permittivity, ε r , in an ionized gas is given by
ε r = 1 −
D 2
ω 2
where D is a constant and ω is the angular frequency.
Find an expression for the refractive index n and thus show that
ω
2
= D
2
+ c
2 k
2
where k is the wavenumber and c is the speed of light in vacuum.
(c) Hence show that v p v g = c 2 .
(d) In a particular gas, D has the value 1.2 × 10 11 /s. Determine the phase
and group velocities at 20 GHz.
Comment on the result.
14.88 Show that the group velocity v g can be expressed as
v g = v p + k
dv p
dk
where v p is the phase velocity and k = 2π/λ.
14.89 Show that the group velocity can be expressed in the form
v g =
c
n
+
λc
n 2
dn
dλ
where n is the refractive index.
14.90 Show that if the phase velocity varies inversely with the wavelength then the
group velocity is twice the phase velocity.