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Chapter 24
Electromagnetic Waves
X-ray
Radio waves
Unpolarized visible light
Polarized visible light
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24.1 Electromagnetic
Waves, Introduction
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Electromagnetic (EM) waves permeate
our environment
EM waves can propagate through a
vacuum
Much of the behavior of mechanical
wave models is similar for EM waves
Maxwell’s equations form the basis of
all electromagnetic phenomena
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Conduction Current
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A conduction current is carried by charged
particles in a wire
The magnetic field associated with this
current can be calculated by using Ampère’s
Law:
 B  ds   I
o
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The line integral is over any closed path through
which the conduction current passes
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Conduction Current, cont.
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Ampère’s Law in this form is
valid only if the conduction
current is continuous in
space
In the example, the
conduction current passes
through only S1 but not S2
This leads to a contradiction
in Ampère’s Law which
needs to be resolved
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James Clerk Maxwell
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1831 – 1879
Developed the
electromagnetic theory
of light
Developed the kinetic
theory of gases
Explained the nature of
color vision
Explained the nature of
Saturn’s rings
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Displacement Current
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Maxwell proposed the resolution to the
previous problem by introducing an
additional term called the displacement
current
The displacement current is defined as
d E
Id   o
dt
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Displacement current
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The electric flux through S2
is EA
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S2 is the gray circle
A is the area of the capacitor
plates
E is the electric field between
the plates
If q is the charge on the
plates, then Id = dq/dt
This is equal to the
conduction current through
S1
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Displacement Current
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The changing electric field may be
considered as equivalent to a current
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For example, between the plates of a capacitor
This current can be considered as the
continuation of the conduction current in a
wire
This term is added to the current term in
Ampère’s Law
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Ampère-Maxwell Law

The general form of Ampère’s Law is
also called the Ampère-Maxwell Law
and states:
d E
 B  ds  o (I  Id )  oI  oo dt

Magnetic fields are produced by both
conduction currents and changing
electric fields
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24.2 Maxwell’s Equations,
Introduction
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In 1865, James Clerk Maxwell provided a
mathematical theory that showed a close
relationship between all electric and magnetic
phenomena
Maxwell’s equations also predicted the
existence of electromagnetic waves that
propagate through space
Einstein showed these equations are in
agreement with the special theory of relativity
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Maxwell’s Equations
Gauss’ Law (electric flux)
Gauss’ Law for magnetism
Faraday’s Law of induction
Ampère-Maxwell Law
q
 E  dA  
o
d B
 E  ds   dt
 B  dA  0
d E
 B  ds  oI  o o dt
The equations are for free space
No dielectric or magnetic material is present
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Lorentz Force

Once the electric and magnetic fields
are known at some point in space, the
force of those fields on a particle of
charge q can be calculated:
F  qE  qv  B
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The force is called the Lorentz force
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24.3 Electromagnetic Waves
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In empty space, q = 0 and I = 0
Maxwell predicted the existence of
electromagnetic waves
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The electromagnetic waves consist of oscillating
electric and magnetic fields
The changing fields induce each other which
maintains the propagation of the wave
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A changing electric field induces a magnetic field
A changing magnetic field induces an electric field
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Plane EM Waves
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We assume that the vectors
for the electric and magnetic
fields in an EM wave have a
specific space-time behavior
that is consistent with
Maxwell’s equations
Assume an EM wave that
travels in the x direction with
the electric field in the y
direction and the magnetic
field in the z direction
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Plane EM Waves, cont
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The x-direction is the direction of propagation
Waves in which the electric and magnetic
fields are restricted to being parallel to a pair
of perpendicular axes are said to be linearly
polarized waves
We assume that at any point in space, the
magnitudes E and B of the fields depend
upon x and t only
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Equations of the Linear EM
Wave
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From Maxwell’s equations applied to empty
space, E and B are satisfied by the following
equations
 2E
 2E
 o  o 2
2
x
t
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and
 2B
 2B
 o  o 2
2
x
t
These are in the form of a general wave
equation, with v  c  1 o o
Substituting the values for o and o gives
c = 2.99792 x 108 m/s
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Solutions of the EM wave
equations
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The simplest solution to the partial differential
equations is a sinusoidal wave:
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The angular wave number is k = 2 p / l
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E = Emax cos (kx – wt)
B = Bmax cos (kx – wt)
l is the wavelength
The angular frequency is w = 2 p ƒ
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ƒ is the wave frequency
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Ratio of E to B
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The speed of the electromagnetic
wave is
w 2p ƒ

 lƒ  c
k 2p l
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Taking partial derivations also gives
Emax w E
Bmax

k

B
c
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Properties of EM Waves
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The solutions of Maxwell’s are wave-like, with
both E and B satisfying a wave equation
Electromagnetic waves travel at the speed of
light
1
c
oo
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This comes from the solution of Maxwell’s
equations
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Properties of EM Waves, 2
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The components of the electric and
magnetic fields of plane electromagnetic
waves are perpendicular to each other
and perpendicular to the direction of
propagation
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The electromagnetic waves are transverse
waves
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Properties of EM Waves, 3
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The magnitudes of the fields in empty
space are related by the expression
c E
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B
This also comes from the solution of the
partial differentials obtained from Maxwell’s
Equations
Electromagnetic waves obey the
superposition principle
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EM Wave Representation
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This is a pictorial
representation, at
one instant, of a
sinusoidal, linearly
polarized plane
wave moving in the
x direction
E and B vary
sinusoidally with x
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Rays
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A ray is a line along which the wave travels
All the rays for the type of linearly polarized
waves that have been discussed are parallel
The collection of waves is called a plane
wave
A surface connecting points of equal phase
on all waves, called the wave front, is a
geometric plane
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Doppler Effect for Light
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Light exhibits a Doppler effect
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Remember, the Doppler effect is an
apparent change in frequency due to the
motion of an observer or the source
Since there is no medium required for
light waves, only the relative speed, v,
between the source and the observer
can be identified
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Doppler Effect, cont.
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The equation also depends on the laws of
relativity
cv
f f
cv
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v is the relative speed between the source
and the observer
c is the speed of light
ƒ’ is the apparent frequency of the light seen
by the observer
ƒ is the frequency emitted by the source
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Doppler Effect, final
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For galaxies receding from the Earth, v
is entered as a negative number
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Therefore, ƒ’<ƒ and the apparent
wavelength, l’, is greater than the actual
wavelength
The light is shifted toward the red end of
the spectrum
This is what is observed in the red shift
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