# Chapter 29

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Nov 16, 2013 (4 years and 7 months ago)

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Maxwell’s Equations and
Electromagnetic Waves

Chapter 29

James Clerk Maxwell

1831
-
1879

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

3x10
8

m/s

and concluded that light and other
electromagnetic waves consist of
fluctuating electric and magnetic fields

James Clerk Maxwell

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
same time

Modifications to Amp
ère’s Law

Amp
ère’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.

I
s
d
B
0

Modifications to Amp
ère’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
, I
d

This showed that magnetic fields are produced both by
conduction currents and by time
-
varying electric fields

I
s
d
B
0

dt
d
E

0
0

dt
d
I
E
d

0

Maxwell’s Equations

In his unified theory of electromagnetism, Maxwell
showed that the fundamental laws are expressed in
these four equations:

0

q
A
d
E

0

A
d
B

dt
d
s
d
E
B

I
s
d
B
0

dt
d
E

0
0

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

0

q
A
d
E

0

A
d
B

dt
d
s
d
E
B

I
s
d
B
0

dt
d
E

0
0

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

0

q
A
d
E

0

A
d
B

dt
d
s
d
E
B

I
s
d
B
0

dt
d
E

0
0

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

0

q
A
d
E

0

A
d
B

dt
d
s
d
E
B

I
s
d
B
0

dt
d
E

0
0

Maxwell’s Equations

Amp
ère
-
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

0

A
d
B

0

q
A
d
E

I
s
d
B
0

dt
d
E

0
0

dt
d
s
d
E
B

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

Maxwell’s equations with the Lorentz Force Law
completely describe all classical electromagnetic
interactions

0

A
d
B

I
s
d
B
0

dt
d
E

0
0

0

q
A
d
E

dt
d
s
d
E
B

B
v
q
E
q
F

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

0

A
d
B

dt
d
s
d
E
B

I
s
d
B
0

dt
d
E

0
0

0

q
A
d
E

Electromagnetic Waves

From Maxwell’s equations applied to empty space, the
following relationships can be found:

The simplest solutions to these partial differential
equations are sinusoidal waves

electromagnetic
waves
:

The speed of the electromagnetic wave is:

2
2
0
0
2
2
t
E
x
E

2
2
0
0
2
2
t
B
x
B

0
0
1

k
v
m/s

10

2.99792

8

c
)
cos(
);
cos(
max
max
t
k x
B
B
t
k x
E
E

max
max
B
E
B
E

Plane Electromagnetic Waves

The vectors for the electric and magnetic
fields in an em wave have a specific space
-
time 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

John Henry Poynting
1852

1914

Poynting Vector

Electromagnetic waves carry energy

As they propagate through space, they can
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:

Its direction is the direction of propagation
and its magnitude varies in time

The SI units: J/(s
.
m
2
) = W/m
2

Those are units of power per unit area

1
o
μ
 
S E B
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 cos
2
(kx
-

ω
t) = ½ is involved

2 2
max max max max
avg
2 2 2
o o o
E B E c B
I S
μ μ c μ
   
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 E
p

= 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 = S
o

cos
2

θ
, where
S
o

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
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

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

are used in
communication systems

Microwaves

(1 mm to 30
cm) are well suited for
microwave ovens are an
application

Infrared waves

are
produced by hot objects
and molecules and are
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
highly penetrating and
cause serious damage
when absorbed by living
tissue

Chapter 29:

Problem
14

3.9 μ
A

Chapter 29:

Problem
22

(a)

3 m

(b)

6 cm

(c)

500 nm

(d)

3 Å

Chapter 29:

Problem
32

(a)

160 W/m
2

(b)

350 V/m

(c)

1.2 μ
T