Page 25 - Electrical Properties of Materials

8 The electron as a particle

Then,

∂ ∂
≡ ik, ≡ –iω, (1.32)
∂z ∂t
which reduces our differential equations to the algebraic equations

ikE x =iωB y (1.33)

and

–ikB y =(μσ –iωμ )E x . (1.34)

This is a homogeneous equation system. By the rules of algebra, there is a
solution, apart from the trivial E x = B y = 0, only if the determinant of the
coefficients vanishes, that is

–ik
iω
= 0. (1.35)

μσ –iωμ ik
Expanding the determinant we get
2
k –iω(μσ –iωμ ) = 0. (1.36)
Different people call this equa-
tion by different names. Char- Essentially, the equation gives a relationship between the frequency, ω, and
acteristic, determinantal, and dis- the wavenumber, k, which is related to phase velocity by v P = ω/k. Thus,
persion equation are among the
unless ω and k are linearly related, the various frequencies propagate with dif-
names more frequently used. We
ferent velocities and at the boundary of two media are refracted at different
shall call it the dispersion equation
angles. Hence the name dispersion.
because that name describes best
A medium for which σ = 0 and μ and are independent of frequency is
what is happening physically.
nondispersive. The relationship between k and ω is simply
c m c is the velocity of the elec- √ ω
k = ω μ = . (1.37)
tromagnetic wave in the medium.
c m
Solving eqn (1.36) formally, we get
2
∗ k =(ω μ +iωμσ) 1/2 . (1.38)
The negative sign is also permissible
though it does not give rise to an expon-
entially increasing wave as would follow Thus, whenever σ = 0, the wavenumber is complex. What is meant by a com-
from eqn (1.39). It would be very nice to
make an amplifier by putting a piece of plex wavenumber? We can find this out easily by looking at the exponent of
lossy material in the way of the electro- eqn (1.30). The spatially varying part is
magnetic wave. Unfortunately, it violates
the principle of conservation of energy.
Without some source of energy at its dis- exp(ikz)=exp i(k real +ik imag )z
posal no wave can grow. So the wave =exp(ik real z)exp(–k imag z). (1.39)
which seems to be exponentially grow-
ing is in effect a decaying wave which
travels in the direction of the negative Hence, if the imaginary part of k is positive, the amplitude of the electromag-
z-axis. netic wave declines exponentially. ∗