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Chapter 29
Maxwell’s Equations and
Electromagnetic Waves
Maxwell’s Theory
• Electricity and magnetism were originally thought to be
unrelated
• Maxwell’s theory showed a close relationship between
all electric and magnetic phenomena and proved that
electric and magnetic fields play symmetric roles in
nature
• Maxwell hypothesized that a changing electric field
would produce a magnetic field
• He calculated the speed of light – 3x108 m/s
– and concluded that light and other
electromagnetic waves consist of
James Clerk Maxwell
fluctuating electric and magnetic fields
1831-1879
Maxwell’s Theory
• Stationary charges produce only electric fields
• Charges in uniform motion (constant velocity) produce
electric and magnetic fields
• Charges that are accelerated produce electric and
magnetic fields and electromagnetic waves
• A changing magnetic field produces an electric field
• A changing electric field produces a magnetic
field
• These fields are in phase and, at any point,
they both reach their maximum value at the
James Clerk Maxwell
same time
1831-1879
Modifications to Ampère’s Law
• Ampère’s Law is used to analyze magnetic fields
created by currents
• But this form is valid only if any electric fields present
are constant in time
• Applying Ampère’s law to a circuit with a changing
current results in an ambiguity
• The result depends on which surface is used to
determine the encircled current.
 
B

d
s


I
0

Modifications to Ampère’s Law
• Maxwell used this ambiguity, along with symmetry
considerations, to conclude that a changing electric
field, in addition to current, should be a source of
magnetic field
• Maxwell modified the equation to include time-varying
electric fields and added another term, called the
displacement current, Id
• This showed that magnetic fields are produced both by
conduction currents and by time-varying electric fields
 
d E
B

d
s


I
  0 0
0

dt
d E
Id  0
dt
Maxwell’s Equations
• In his unified theory of electromagnetism, Maxwell
showed that the fundamental laws are expressed in
these four equations:
 
 B  dA  0
 
d E
B

d
s


I
  0 0
0

dt
  q
 E  dA   0
 
d B
E

d
s



dt
Maxwell’s Equations
• Gauss’ Law relates an electric field to the charge
distribution that creates it
• The total electric flux through any closed surface equals
the net charge inside that surface divided by o
 
 B  dA  0
 
d E
B

d
s


I
  0 0
0

dt
  q
 E  dA   0
 
d B
E

d
s



dt
Maxwell’s Equations
• Gauss’ Law in magnetism: the net magnetic flux
through a closed surface is zero
• The number of magnetic field lines that enter a closed
volume must equal the number that leave that volume
• If this wasn’t true, there would be magnetic monopoles
found in nature
 
 B  dA  0
 
d E
B

d
s


I
  0 0
0

dt
  q
 E  dA   0
 
d B
E

d
s



dt
Maxwell’s Equations
• Faraday’s Law of Induction describes the creation of an
electric field by a time-varying magnetic field
• The emf (the line integral of the electric field around any
closed path) equals the rate of change of the magnetic
flux through any surface bounded by that path
 
 B  dA  0
 
d E
B

d
s


I
  0 0
0

dt
  q
 E  dA   0
 
d B
E

d
s



dt
Maxwell’s Equations
• Ampère-Maxwell Law describes the creation of a
magnetic field by a changing electric field and by
electric current
• The line integral of the magnetic field around any closed
path is the sum of o times the net current through that
path and oo times the rate of change of electric flux
through any surface bounded by that path
 
 B  dA  0
 
d E
B

d
s


I
  0 0
0

dt
  q
 E  dA   0
 
d B
E

d
s



dt
Maxwell’s Equations
• Once the electric and magnetic fields are known at
some point in space, the force acting on a particle of
charge q can 
be found



F  qE  qv  B
• Maxwell’s equations with the Lorentz Force Law
completely describe all classical electromagnetic
interactions
 
 B  dA  0
 
d E
B

d
s


I
  0 0
0

dt
  q
 E  dA   0
 
d B
E

d
s



dt
Maxwell’s Equations
• In empty space, q = 0 and I = 0
• The equations can be solved with wave-like solutions
(electromagnetic waves), which are traveling at the
speed of light
• This result led Maxwell to predict that light waves were
a form of electromagnetic radiation
 
 B  dA  0
 
d E
B

d
s


I
  0 0
0

dt
  q
 E  dA   0
 
d B
E

d
s



dt
Electromagnetic Waves
• From Maxwell’s equations applied to empty space, the
following relationships can be found:
 E
 E
  0 0 2
2
x
t
2
2
 B
 B
  0 0 2
2
x
t
2
2
• The simplest solutions to these partial differential
equations are sinusoidal waves – electromagnetic
waves:
E  E max cos( k x   t ); B  B max cos( k x   t )
• The speed of the electromagnetic wave is:

1
E E max
8
v 
 c  2 .9 9 7 9 2  1 0 m/s  
k
B Bmax
 0 0
Plane Electromagnetic Waves
• The vectors for the electric and magnetic
fields in an em wave have a specific spacetime behavior consistent with Maxwell’s
equations
• Assume an em wave that travels in the x
direction
• We also assume that at any point in space,
the magnitudes E and B of the fields depend
upon x and t only
• The electric field is assumed to be in the y
direction and the magnetic field in the z
direction
Plane Electromagnetic Waves
• The components of the electric and
magnetic fields of plane electromagnetic
waves are perpendicular to each other and
perpendicular to the direction of
propagation
• Thus, electromagnetic waves are
transverse waves
• 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
Poynting Vector
• Electromagnetic waves carry energy
John Henry Poynting
can 1852 – 1914
• As they propagate through space, they
transfer that energy to objects in their path
• The rate of flow of energy in an em wave is
described by a vector, S, called the
Poynting vector defined as: S  1 E  B
μo
• Its direction is the direction of propagation
and its magnitude varies in time
• The SI units: J/(s.m2) = W/m2
• Those are units of power per unit area
Poynting Vector
• Energy carried by em waves is shared equally by the
electric and magnetic fields
• The wave intensity, I, is the time average of S (the
Poynting vector) over one or more cycles
• When the average is taken, the time average of cos2(kx ωt) = ½ is involved
I  S avg
2
2
E max Bmax E max
c Bmax



2 μo
2 μo c
2 μo
Chapter 29
Problem 29
What would be the average intensity of a laser beam so strong
that its electric field produced dielectric breakdown of air (which
requires Ep = 3 MV/m)?
Polarization of Light
• An unpolarized wave: each atom
produces a wave with its own orientation
of E, so all directions of the electric field
vector are equally possible and lie in a
plane perpendicular to the direction of
propagation
• A wave is said to be linearly polarized if
the resultant electric field vibrates in the
same direction at all times at a particular
point
• Polarization can be obtained from an
unpolarized beam by selective
absorption, reflection, or scattering
Polarization by Selective Absorption
• The most common technique for polarizing light
• Uses a material that transmits waves whose electric
field vectors in the plane are parallel to a certain
direction and absorbs waves whose electric field
vectors are perpendicular to that direction
Polarization by Selective Absorption
• The intensity of the polarized beam transmitted
through the second polarizing sheet (the analyzer)
varies as S = So cos2 θ, where So is the intensity of the
polarized wave incident on the analyzer
• This is known as Malus’ Law and applies to any two
polarizing materials whose transmission axes are at
an angle of θ to each other
Étienne-Louis Malus
1775 – 1812
Chapter 29
Problem 40
A polarizer blocks 75% of a polarized light beam. What’s the angle
between the beam’s polarization and the polarizer’s axis?
Electromagnetic Waves Produced by
an Antenna
• Neither stationary charges nor steady currents can
produce electromagnetic waves
• The fundamental mechanism responsible for this
radiation: when a charged particle undergoes an
acceleration, it must radiate energy in the form of
electromagnetic waves
• Electromagnetic waves are radiated by any circuit
carrying alternating current
• An alternating voltage applied to the wires of an
antenna forces the electric charge in the antenna to
oscillate
Electromagnetic Waves Produced by
an Antenna
• Half-wave antenna: two rods are connected to an ac
source, charges oscillate between the rods (a)
• As oscillations continue, the rods become less charged,
the field near the charges decreases and the field
produced at t = 0 moves away from the rod (b)
• The charges and field reverse (c) and the oscillations
continue (d)
Electromagnetic Waves Produced by
an Antenna
• Because the oscillating charges in the rod
produce a current, there is also a magnetic
field generated
• As the current changes, the magnetic field
spreads out from the antenna
• The magnetic field lines form concentric
circles around the antenna and are
perpendicular to the electric field lines at
all points
• The antenna can be approximated by an
oscillating electric dipole
The Spectrum of EM Waves
• Types of electromagnetic
waves are distinguished
by their frequencies
(wavelengths): c = ƒ λ
• There is no sharp
division between one
kind of em wave and the
next – note the overlap
between types of waves
The Spectrum of EM Waves
• Radio waves are used in
radio and television
communication systems
• Microwaves (1 mm to 30
cm) are well suited for
radar systems +
microwave ovens are an
application
• Infrared waves are
produced by hot objects
and molecules and are
readily absorbed by most
materials
The Spectrum of EM Waves
• Visible light (a small
range of the spectrum
from 400 nm to 700 nm) –
part of the spectrum
detected by the human
eye
• Ultraviolet light (400 nm
to 0.6 nm): Sun is an
important source of uv
light, however most uv
light from the sun is
absorbed in the
stratosphere by ozone
The Spectrum of EM Waves
• X-rays – most common
source is acceleration of
high-energy electrons
striking a metal target,
also used as a diagnostic
tool in medicine
• Gamma rays: emitted by
radioactive nuclei, are
highly penetrating and
cause serious damage
when absorbed by living
tissue
Answers to Even Numbered Problems
Chapter 29:
Problem 14
3.9 μA
Answers to Even Numbered Problems
Chapter 29:
Problem 22
(a) 3 m
(b) 6 cm
(c) 500 nm
(d) 3 Å
Answers to Even Numbered Problems
Chapter 29:
Problem 32
(a) 160 W/m2
(b) 350 V/m
(c) 1.2 μT
Answers to Even Numbered Problems
Chapter 29:
Problem 36
3.1 cm