Lecture 8

Lossy Media, Lorentz Force Law,

and Drude-Lorentz-Sommerfeld
Model

8.1 Plane Waves in Lossy Conductive Media

Previously, we have derived the plane wave solution for a lossless homogeneous medium.
The derivation can be generalized to a lossy conductive medium by invoking mathematical
homomorphism. When conductive loss is present, σ 6= 0, and J = σE. Then generalized
Ampere’s law becomes
 
σ
∇ × H = jωεE + σE = jω ε + E (8.1.1)
jω

A complex permittivity can be defined as ε = ε − j ωσ . Eq. (8.1.1) can be rewritten as

e
∇ × H = jωεE (8.1.2)
e
This equation is of the same form as source-free Ampere’s law in the frequency domain for a
lossless medium where ε is completely real. Using the same method as before, a wave solution

E = E0 e−jk·r (8.1.3)

will have the dispersion relation which is now given by

ky2 + ky2 + kz2 = ω 2 µε (8.1.4)

e
Since ε is complex now, kx , ky , and kz cannot be all real. Equation (8.1.4) has been derived
previously
e by assuming that k is a real vector. When k = k0 − jk00 is a complex vector, some
of the derivations may not be correct previously. It is also difficult to visualize a complex k

73
74 Electromagnetic Field Theory

vector that is suppose to indicate the direciton with which the wave is propagating. Here,
the wave can decay and oscillate in different directions.
So again, we look at the simplified case where

E = x̂Ex (z) (8.1.5)

√
so that ∇ · E = ∂x Ex (z) = 0, and let k = ẑk = ẑω µε. This wave is constant in the xy
plane, and hence, is a plane wave. Furthermore, in thisemanner, we are requiring that the
wave decays and propagates (or oscillates) only in the z direction. For such a simple plane
wave,

E = x̂Ex (z) = x̂E0 e−jkz (8.1.6)

√
where k = ω µε, since k · k = k 2 = ω 2 µε is still true.
Faraday’s law
e gives rise to e
s
k×E kEx (z) ε
H= = ŷ = ŷ e Ex (z) (8.1.7)
ωµ ωµ µ
√
or by letting k = ω µε, then
e
s
µ
Ex /Hy = (8.1.8)
ε
e
When the medium is highly conductive, σ → ∞, and ε ≈ −j ωσ . Thus, the following
approximation can be made, namely, e
r
q jσ p
k = ω µε ' ω −µ = −jωµσ (8.1.9)
e ω
√
Taking −j = √12 (1 − j), we have
r
ωµσ
k = (1 − j) = k 0 − jk 00 (8.1.10)
2

For a plane wave, e−jkz , and then it becomes

0 00
e−jkz = e−jk z−k z
(8.1.11)

This plane wave decays exponentially in the z direction. The penetration depth of this wave
is then
r
1 2
δ = 00 = (8.1.12)
k ωµσ
This distance δ, the penetration depth, is called the skin depth of a plane wave propagating
in a highly lossy conductive medium where conduction current dominates over displacement
Lossy Media, Lorentz Force Law, and Drude-Lorentz-Sommerfeld Model 75

current, or that σ  ωε. This happens for radio wave propagating in the saline solution of
the ocean, the Earth, or wave propagating in highly conductive metal, like your induction
cooker.
When the conductivity is low, namely, when the displacement current is larger than the
σ
conduction current, then ωε  1, we have
r  s  
σ jσ
k =ω µ ε−j = ω µε 1 −
ω ωε
 
√ 1 σ
≈ ω µε 1 − j = k 0 − jk 00 (8.1.13)
2 ωε
σ
The term ωε is called the loss tangent of a lossy medium.
In general, in a lossy medium ε = ε0 − jε00 , ε00 /ε0 is called the loss tangent of the medium.
It is to be noted that in the optics and physics community, e−iωt time convention is preferred.
In that case, we need to do the switch j → −i, and a loss medium is denoted by ε = ε0 + iε00 .

8.2 Lorentz Force Law

The Lorentz force law is the generalization of the Coulomb’s law for forces between two
charges. Lorentz force law includes the presence of a magnetic field. The Lorentz force law
is given by

F = qE + qv × B (8.2.1)

The first term electric force similar to the statement of Coulomb’s law, while the second term
is the magnetic force called the v × B force. This law can be also written in terms of the
force density f which is the force on the charge density, instead of force on a single charge.
By so doing, we arrive at

f = %E + %v × B = %E + J × B (8.2.2)

where % is the charge density, and one can identified the current J = %v.
Lorentz force law can also be derived from the integral form of Faraday’s law, if one
assumes that the law is applied to a moving loop intercepting a magnetic flux [60]. In other
words, Lorentz force law and Faraday’s law are commensurate with each other.

8.3 Drude-Lorentz-Sommerfeld Model

In the previous lecture, we have seen how loss can be introduced by having a conduction
current flowing in a medium. Now that we have learnt the versatility of the frequency domain
method, other loss mechanism can be easily introduced with the frequency-domain method.
First, let us look at the simple constitutive relation where

D = ε0 E + P (8.3.1)
76 Electromagnetic Field Theory

We have a simple model where

P = ε0 χ0 E (8.3.2)
where χ0 is the electric susceptibility. To see how χ0 (ω) can be derived, we will study the
Drude-Lorentz-Sommerfeld model. This is usually just known as the Drude model or the
Lorentz model in many textbooks although Sommerfeld also contributed to it. This model
can be unified in one equation as shall be shown.
We can first start with a simple electron driven by an electric field E in the absence of a
magnetic field B. If the electron is free to move, then the force acting on it, from the Lorentz
force law, is −eE where e is the charge of the electron. Then from Newton’s law, assuming a
one dimensional case, it follows that
d2 x
me = −eE (8.3.3)
dt2
where the left-hand side is due to the inertial force of the mass of the electron, and the right-
hand side is the electric force acting on a charge of −e coulomb. Here, we assume that E
points in the x-direction, and we neglect the vector nature of the electric field. Writing the
above in the frequency domain for time-harmonic fields, and using phasor technique, one gets
−ω 2 me x = −eE (8.3.4)
From this, we have
e
x= E (8.3.5)
ω 2 me
This for instance, can happen in a plasma medium where the atoms are ionized, and the
electrons are free to roam [61]. Hence, we assume that the positive ions are more massive,
and move very little compared to the electrons when an electric field is applied.

Figure 8.1: Polarization of an atom in the presence of an electric field. Here, it is assumed
that the electron is weakly bound or unbound to the nucleus of the atom.

The dipole moment formed by the displaced electron away from the ion due to the electric
field is
e2
p = −ex = − E (8.3.6)
ω 2 me
Lossy Media, Lorentz Force Law, and Drude-Lorentz-Sommerfeld Model 77

for one electron. When there are N electrons per unit volume, the dipole density is then
given by
N e2
P = Np = − E (8.3.7)
ω 2 me
In general, P and E point in the same direction, and we can write
N e2 ωp 2
P=− E = − ε0 E (8.3.8)
ω 2 me ω2
where we have defined ωp 2 = N e2 /(me ε0 ). Then,
ωp 2
 
D = ε0 E + P = ε0 1 − 2 E (8.3.9)
ω
In this manner, we see that the effective permittivity is
ωp 2
 
ε(ω) = ε0 1 − 2 (8.3.10)
ω
This gives the interesting result that in the frequency domain, ε < 0 if
p
ω < ωp = N/(me ε0 )e
√
Here, ωp is the plasma frequency. Since k = ω µε, if ε is negative, k = −jα becomes pure
imaginary, and a wave such as e−jkz decays exponentially as e−αz . This is also known as
an evanescent wave. In other words, the wave cannot propagate through such a medium:
Our ionosphere is such a medium. So it was extremely fortuitous that Marconi, in 1901, was
able to send a radio signal from Cornwall, England, to Newfoundland, Canada. Nay sayers
thought his experiment would never succeed as the radio signal would propagate to outer
space and never return. It is the presence of the ionosphere that bounces the radio wave
back to Earth, making his experiment a resounding success and a very historic one! The
experiment also heralds in the age of wireless communications.

Figure 8.2:

The above model can be generalized to the case where the electron is bound to the ion,
but the ion now provides a restoring force similar to that of a spring, namely,
d2 x
me + κx = −eE (8.3.11)
dt2
78 Electromagnetic Field Theory

We assume that the ion provides a restoring force just like Hooke’s law. Again, for a time-
harmonic field, (8.3.11) can be solved easily in the frequency domain to yield
e e
x= E == E (8.3.12)
(ω 2 m e − κ) (ω 2 − ω0 2 )me

where we have defined ω0 2 me = κ. The above is the typical solution of a lossless harmonic
oscillator (pendulum) driven by an external force, in this case the electric field.
Equation (8.3.11) can be generalized to the case when frictional or damping forces are
involved, or that

d2 x dx
me + me Γ + κx = −eE (8.3.13)
dt2 dt
The second term on the left-hand side is a force that is proportional to the velocity dx/dt of
the electron. This is the hall-mark of a frictional force. Frictional force is due to the collision
of the electrons with the background ions or lattice. It is proportional to the destruction (or
change) of momentum of an electron. The momentum of the electron is given by me dx dt . In
the average sense, the destruction of the momentum is given by the product of the collision
frequency and the momentum. In the above, Γ has the unit of frequency, and for plasma,
and conductor, it can be regarded as a collision frequency.
Solving the above in the frequency domain, one gets
e
x= E (8.3.14)
(ω 2 − jωΓ − ω0 2 )me
Following the same procedure in arriving at (8.3.7), we get

−N e2
P = E (8.3.15)
(ω 2 − jωΓ − ω0 2 )me
In this, one can identify that

−N e2
χ0 (ω) =
(ω 2 − jωΓ − ω0 2 )me ε0
ωp 2
=− 2 (8.3.16)
ω − jωΓ − ω0 2
where ωp is as defined before. A function with the above frequency dependence is also called
a Lorentzian function. It is the hallmark of a damped harmonic oscillator.
If Γ = 0, then when ω = ω0 , one sees an infinite resonance peak exhibited by the DLS
model. But in the real world, Γ 6= 0, and when Γ is small, but ω ≈ ω0 , then the peak value
of χ0 is

ωp 2 ωp 2
χ0 ≈ + = −j (8.3.17)
jωΓ ωΓ
χ0 exhibits a large negative imaginary part, the hallmark of a dissipative medium, as in the
conducting medium we have previously studied.
Lossy Media, Lorentz Force Law, and Drude-Lorentz-Sommerfeld Model 79

The DLS model is a wonderful model because it can capture phenomenologically the
essence of the physics of many electromagnetic media, even though it is a purely classical
model.1 It captures the resonance behavior of an atom absorbing energy from light excitation.
When the light wave comes in at the correct frequency, it will excite electronic transition
within an atom which can be approximately modeled as a resonator with behavior similar to
that of a pendulum oscillator. This electronic resonances will be radiationally damped [33],
and the damped oscillation can be modeled by Γ 6= 0.
Moreover, the above model can also be used to model molecular vibrations. In this case,
the mass of the electron will be replaced by the mass of the atom involved. The damping of
the molecular vibration is caused by the hindered vibration of the molecule due to interaction
with other molecules [62]. The hindered rotation or vibration of water molecules when excited
by microwave is the source of heat in microwave heating.
In the case of plasma, Γ 6= 0 represents the collision frequency between the free electrons
and the ions, giving rise to loss. In the case of a conductor, Γ represents the collision frequency
between the conduction electrons in the conduction band with the lattice of the material.2
Also, if there is no restoring force, then ω0 = 0. This is true for sea of electron moving in the
conduction band of a medium. Also, for sufficiently low frequency, the inertial force can be
ignored. Thus, from (8.3.16)
ωp 2
χ0 ≈ −j (8.3.18)
ωΓ
and
ωp 2
 
ε = ε0 (1 + χ0 ) = ε0 1 − j (8.3.19)
ωΓ
We recall that for a conductive medium, we define a complex permittivity to be
 
σ
ε = ε0 1 − j (8.3.20)
ωε0
Comparing (8.3.19) and (8.3.20), we see that
ωp 2
σ = ε0 (8.3.21)
Γ
The above formula for conductivity can be arrived at using collision frequency argument as
is done in some textbooks [65].
Because the DLS is so powerful, it can be used to explain a wide range of phenomena
from very low frequency to optical frequency.
The fact that ε < 0 can be used to explain many phenomena. The ionosphere is essentially
a plasma medium described by
ωp 2
 
ε = ε0 1 − 2 (8.3.22)
ω
1 What we mean here is that only Newton’s law has been used, and no quantum theory as yet.
2 Itis to be noted that electron has a different effective mass in a crystal lattice [63, 64], and hence, the
electron mass has to be changed accordingly in the DLS model.
80 Electromagnetic Field Theory

Radio wave or microwave can only penetrate through this ionosphere when ω > ωp , so that
ε > 0.

8.3.1 Frequency Dispersive Media

The DLS model shows that, except for vacuum, all media are frequency dispersive. It is
prudent to digress to discuss more on the physical meaning of a frequency dispersive medium.
The relationship between electric flux and electric field, in the frequency domain, still follows
the formula
D(ω) = ε(ω)E(ω) (8.3.23)
When the effective permittivity, ε(ω), is a function of frequency, it implies that the above
relationship in the time domain is via convolution, viz.,
D(t) = ε(t) ~ E(t) (8.3.24)
Since the above represents a linear time-invariant system, it implies that an input is not
followed by an instantaneous output. In other words, there is a delay between the input and
the output. The reason is because an electron has a mass, and it cannot respond immediately
to an applied force: or it has inertial. In other words, the system has memory of what it was
before when you try to move it.
When the effective permittivity is a function of frequency, it also implies that different
frequency components will propagate with different velocities through such a medium. Hence,
a narrow pulse will spread in its width because different frequency components are not in phase
after a short distance of travel.
Also, the Lorentz function is great for data fitting, as many experimentally observed
resonances have finite Q and a line width. The Lorentz function models that well. If multiple
resonances occur in a medium or an atom, then multi-species DLS model can be used. It
is now clear that all media have to be frequency dispersive because of the finite mass of
the electron and the inertial it has. In other words, there is no instantaneous response in a
dielectric medium due to the finiteness of the electron mass.
Even at optical frequency, many metals, which has a sea of freely moving electrons in the
conduction band, can be modeled approximately as a plasma. A metal consists of a sea of
electrons in the conduction band which are not tightly bound to the ions or the lattice. Also,
in optics, the inertial force due to the finiteness of the electron mass (in this case effective
mass) can be sizeable compared to other forces. Then, ω0  ω or that the restoring force is
much smaller than the inertial force, in (8.3.16), and if Γ is small, χ0 (ω) resembles that of a
plasma, and ε of a metal can be negative.

8.3.2 Plasmonic Nanoparticles

When a plasmonic nanoparticle made of gold is excited by light, its response is given by (see
homework assignment)
a3 cos θ εs − ε0
ΦR = E 0 (8.3.25)
r2 εs + 2ε0
Lossy Media, Lorentz Force Law, and Drude-Lorentz-Sommerfeld Model 81

In a plasma, εs can be negative, and thus, at certain frequency, if εs = −2ε0 , then ΦR → ∞.

Therefore, when light interacts with such a particle, it can sparkle brighter than normal. This
reminds us of the saying “All that glitters is not gold!” even though this saying has a different
intended meaning.
Ancient Romans apparently knew about the potent effect of using gold and silver nanopar-
ticles to enhance the reflection of light. These nanoparticles were impregnated in the glass
or lacquer ware. By impregnating these particles in different media, the color of light will
sparkle at different frequencies, and hence, the color of the glass emulsion can be changed
(see website [66]).

Figure 8.3: Ancient Roman goblets whose laquer coating glisten better due to the presence
of gold nanoparticles (courtesy of Smithsonian.com).
82 Electromagnetic Field Theory

x
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