Chapter 29

clappergappawpawUrban and Civil

Nov 16, 2013 (3 years and 10 months ago)

76 views

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
a form of electromagnetic radiation

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
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 μ
A

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/m
2

(b)

350 V/m

(c)

1.2 μ
T

Answers to Even Numbered Problems


Chapter 29:


Problem
36


3.1 cm