# Chapter 34. Electromagnetic Waves

Urban and Civil

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

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Electromagnetic Waves

Hertz’s Discoveries

Plane Electromagnetic Waves

Energy in an EM Wave

Momentum and Radiation Pressure

The Electromagnetic Spectrum

Maxwell:

Problem with Ampère’s Law

I
d
0
.

s
B
Ampère’s Law

For S
1
,

I
d
0
.

s
B
For S
2
,

0
.
0

I
d

s
B
(No current in the gap)

But there is a problem!

Consider a parallel plate capacitor

What can we do to fix
it?

Ampere’s Law works
well

most of the
time.

What is special about
the region between
the capacitors?

.
o o
Q AE
 
  
o o
Q
E
A

 
 
.
o
dQ d
I
dt dt

 

Let’s try to relate
E

to
the current
I.

Try fudging Ampere’s
Law:

0 0 0
..
E
d
d I
dt
 

 

B s
Solution
-

Displacement Current

Call
I

as the
conduction

current.

Specify a new current that exists when a change in the electric field occurs.
Call it the
displacement

current.

dt
d
I
E
d

0

Electrical flux through S
2
:

0
0

Q
A
A
Q
EA
E

I
dt
dQ
dt
d
I
E
d

0

Maxwell Gets
Excited!

dt
d
d
B

s
E
.
dt
d
I
d
E

0
0
0
.

s
B
Free Space:
I
=
0
,
Q
=
0

dt
d
d
E

0
0
.

s
B
t
E
x
B
t
B
x
E

0
0

2
2
0
0
2
2
0
0
2
2
t
E
x
E
t
E
t
x
B
t
t
B
x
x
E

2
2
0
0
2
2
t
B
x
B

t
kx
B
B
t
kx
E
E

cos
cos
max
max
Speed:
1
.
o o
v


Waves

Circular (or Spherical in 3D) Waves

Plane Waves

Ampère
-
Maxwell Law

Magnetic fields are produced by
both conduction currents and by
time
-
varying electrical fields.

dt
d
I
I
I
d
E
d

0
0
0
0
.

s
B
Now the generalized form of Ampère’s Law, or Ampère
-
Maxwell Lax becomes:

Maxwell’s Equations

dt
d
d
B

s
E
.
dt
d
I
d
E

0
0
0
.

s
B
0
.

A
B
d
0
.

Q
d

A
E

By supplying short voltage bursts from the coil
to the transmitter electrode we can ionize the
air between the electrodes.

In effect, the circuit can be modeled as a LC circuit,
with the coil as the inductor and the electrodes as
the capacitor.

A receiver loop placed nearby is able to receive
these oscillations and creates sparks as well.

EM Wave Oscillation

Some Important Quantities

2

k
Wavenumber

0
0
1

c
Speed of Light

f

2

Angular Frequency

f
c

Wavelength

c

k

t
kx
B
B
t
kx
E
E

cos
cos
max
max

t
kx
B
B
t
kx
E
E

cos
cos
max
max
t
E
x
B
t
B
x
E

0
0

c
k
B
E
B
E
B
kE

max
max
max
max
Properties of EM Waves

The solutions to Maxwell’s
equations in free space are
wavelike

Electromagnetic waves travel
through free space at the speed
of light.

The electric and magnetic fields
of a plane wave are
perpendicular to each other and
the direction of propagation (they
are transverse).

The ratio of the magnitudes of the
electric and magnetic fields is c.

EM waves obey the superposition
principle.

An EM Wave

MHz
f
40

a)

=?, T=?

s
f
T
m
f
c
8
6
6
8
10
5
.
2
10
40
1
1
5
.
7
10
40
10
3

b) If
E
max
=
750
N/C,
then

B
max
=?

T
c
E
B
6
8
max
max
10
5
.
2
10
3
750

Directed towards the z
-
direction

c)
E(t)=?

and
B(t)=?

)
10
51
.
2
838
.
0
cos(
10
5
.
2
cos
)
10
51
.
2
838
.
0
cos(
/
750
cos
8
6
max
8
max
t
x
T
t
kx
B
B
t
x
C
N
t
kx
E
E

s
f
m
k
/
10
51
.
2
10
40
2
2
/
838
.
0
5
.
7
2
2
8
6

Energy Carried by EM Waves

B
E
S

0
1

Poynting Vector (W/m
2
)

For a plane EM Wave:

2
0
0
2
0
B
c
c
E
EB
S

2
max
0
0
2
max
0
max
max
2
2
2
B
c
c
E
B
E
S
I
av

The average value of S
is called the
intensity
:

S is equal to the rate of EM energy flow per unit area
(power per unit area)

0
2
2

B
u
B

2
2
0
E
u
E

2
0
2
0
0
0
0
2
2
1
2
2
E
E
c
E
u
B

0
2
2
0
2
2
1

B
E
u
u
B
E

For an EM wave, the instantaneous
electric and magnetic energies are
equal.

0
2
2
0

B
E
u
u
u
B
E

0
2
max
2
max
0
2
0
2
2
1

B
E
E
u
av
av

av
av
cu
S
I

Total Energy Density of an EM Wave

The intensity is c times the total
average energy density

The Energy Density

Which gives the largest average energy density

at the distance specified and thus, at least
qualitatively, the best illumination

1
. a
50
-
W source at a distance
R.

2
. a
100
-
W source at a distance
2
R.

3
. a
200
-
W source at a distance
4
R.

Concept Question

Fields on the Screen

c
E
A
I
av
0
2
max
2

P
P
lamp
=
150
W
,
3
% efficiency

A =
15
m
2

m
V
A
c
E
av
/
42
.
18
2
0
max

P

W
lamp
av
5
.
4
03
.
0

P
P
T
c
E
B
8
max
max
10
14
.
6

A

Pressure

c
U
p

Momentum for complete absorption

dt
dp
A
A
F
P
1

A
dt
dU
c
c
U
dt
d
A
P
1
1

c
S
P

Pressure on surface

For complete reflection:

c
U
p
2

c
S
P
2

Pressure From a Laser Pointer

P
laser
=
3
mW
,
70
% reflection,
d =
2
mm

2
2
/
955
)
001
.
0
(
003
.
0
m
W
A
S

P
2
6
8
/
10
4
.
5
10
3
955
7
.
1
7
.
1
7
.
0
m
N
P
c
S
c
S
c
S
P

Review of key points:

Partial derivatives

Relationship between E and B

Poynting vector

Omit section
34.4
.

Section
34.5
: only responsible for ideas
presented next.

t
kx
B
B
t
kx
E
E

cos
cos
max
max
t
E
x
B
t
B
x
E

0
0

c
k
B
E
B
E
B
kE

max
max
max
max
Properties of EM Waves

The solutions to Maxwell’s
equations in free space are
wavelike

Electromagnetic waves travel
through free space at the speed
of light.

The electric and magnetic fields
of a plane wave are
perpendicular to each other and
the direction of propagation (they
are transverse).

The ratio of the magnitudes of the
electric and magnetic fields is c.

EM waves obey the superposition
principle.

Energy Carried by EM Waves

B
E
S

0
1

Poynting Vector (W/m
2
)

For a plane EM Wave:

2
0
0
2
0
B
c
c
E
EB
S

2
max
0
0
2
max
0
max
max
2
2
2
B
c
c
E
B
E
S
I
av

The average value of S
is called the
intensity
:

S is equal to the rate of EM energy flow per unit area
(power per unit area)

Antennas

The fundamental mechanism for electromagnetic radiation is
an accelerating charge

Half
-
Wave Antenna

FM and TV broadcast
antennas?

Problem:

Very roughly, what is the frequency of cell
phones?

Are CB radio frequencies higher or lower?

Dorm room FM Antenna

http://www.wfu.edu/~matthews/misc/dipole.
html

Electromagnetic Spectrum