PHYSICS 3 (ELECTRICITY AND MAGNETISM) - CHAPTER 7 pps

Electricity and Magnetism
65

Chapter 7 ELECTROMAGNETIC WAVES

7.1 Maxwell’s Equations
1) Maxwell’s Equations
Gauss’law for electricity
∫
surfaceclosed
A.dE
r
r
=
o
q
ε
(7.1)

Gauss’law for magnetism
∫
surfaceclosed
A.dB
r
r
= 0 (7.2)

Faraday ‘s law
∫
pathclosed
s.dE

r
r
= -
t
B
∂
Φ∂
(7.3)

Ampere-Maxwell Law
∫
pathclosed
s.dB
r
r
=
t
εµ
E
oo
∂
Φ∂
+
µ
o
i (7.4)

2) Vector calculus (or vector analysis)

del operator :

z
i
y
i
x
i
zyx
∂
∂
+
∂
∂
+
∂
∂
=∇
r
r
r
(7.5)


•
gradient : grad(V) =
z
V
i
y
V
i

x
V
iV
zyx
∂
∂
+
∂
∂
+
∂
∂
=∇
r
r
r
(7.6)

Maps scalar fields to vector fields.
Measures the rate and direction of change in a scalar field.


•
divergence : div(
F
r
) =
(
)
zzyyxxzyx

FiFiFi.
z
i
y
i
x
iF
rrrrrr
r
++








∂
∂
+
∂
∂
+
∂
∂
=∇

=
z

F
y
F
x
F
z
y
x
∂
∂
+
∂
∂
+
∂
∂
(7.7)

Maps vector fields to scalar fields.
Measures the magnitude of a source or sink at a given point in a vector field.

Property :
∫
∫
=∇
surfaceclosedvolume
A.dF.dVF
r
r
r

(7.8)


•
curl: curl(
F
r
) = rot(
F
r
) =
(
)
zzyyxx
x
zyx
x
FiFiFi
z
i
y
i
x
iF
rrrrrr
r
++









∂
∂
+
∂
∂
+
∂
∂
=∇

Electricity and Magnetism
66

=
zyx
zyx
FFF
zyx
iii
∂
∂
∂
∂
∂
∂

r
r
r
(7.9)

Maps vector fields to vector fields.
Measures the tendency to rotate about a point in a vector field.

Property :
∫
∫
=∇
pathclosedsurface
s.dFA.dF
x
r
r
r
r
(7.10)


•
Laplacian:
))div(grad(VV VV
2
=∇∇=∇=∆

=
V

z
i
y
i
x
i.
z
i
y
i
x
i
zyxzyx








∂
∂
+
∂
∂
+
∂
∂









∂
∂
+
∂
∂
+
∂
∂
rrrrrr

=
2
2
2
2
2
2
z
V
y
V
x
V

∂
∂
+
∂
∂
+
∂
∂
(7.11)

Maps scalar fields to scalar fields.
A composition of the divergence and gradient operations.



F
r
∆
=
F
zyx
2
2
2
2
2
2
r









∂
∂
+
∂
∂
+
∂
∂
=
zzyyxx
iFiFiF
r
r
r
∆+∆+∆
(7.12)


Property :
F
r
∆
=
F

2
r
∇
-
∇
x
∇
x
F
r
(7.13)

3) Maxwell’s Equations in term of del operator
Gauss’law for electricity
∫
∇
volume
.dVE
r
=
∫
surfaceclosed
A.dE
r
r
=
o
q
ε
=

∫
volume
.dV
ε
ρ
o


It follows that
E
r
∇
=
o
ε
ρ
(7.14)


Gauss’law for magnetism
∫
∇
volume
.dVB
r
=
∫
surfaceclosed
A.dB
r

r
= 0

It follows that
B
r
∇
= 0 (7.15)


Faraday ‘s law
∫
∇
surface
A.dE
x
r
r
=
∫
pathclosed
s.dE
r
r
= -
t
B
∂
Φ∂
= -

Ad.
t
B
r
r
∫
∂
∂
surface

Electricity and Magnetism
67

It follows that
E
x
r
∇
= -
t
B
∂
∂
r
(7.16)


Ampere-Maxwell Law
∫
∇

surface
A.dB
x
r
r
=
∫
pathclosed
s.dB
r
r
=
dt
d
E
oo
Φ
εµ
+
µ
o
i
=
∫









+
∂
∂
surface
AdJµ
t
E
εµ
ooo
r
r
r
=

It follows that
B
x
r
∇
=
Jµ
t
E
εµ
ooo
r
r
+

∂
∂
(7.17)


4) Wave equation
Applying (7.13) yields


E
r
∆
=
E
2
r
∇
-
∇
x
∇
x
E
s
(7.18)

With J = 0 and
ρ
= 0

E
r
∇
= 0 and
∇
x
E
s
= -
t
B
∂
∂
r
(7.19)
We have

E
r
∆
=
∇
x
t
B
∂
∂
r
=
t

∂
∂
B
x
r
∇
=

t
E
εµ
2
2
oo
∂
∂
r
(7.20)

Inserting (7.19) and (7.20) into (7.18) we have the wave equation

E
r
∆
-

t
E
εµ
2

2
oo
∂
∂
r
= 0
(7.21)


7.2 Electromagnetic Waves
An electromagnetic wave consists of oscillating electric and magnetic fields. The various possible
frequencies of electromagnetic waves form a spectrum, a small part of which is visible light. An
electromagnetic wave traveling along an x axis has an electric field
E
r
and a magnetic field
B
r
with
magnitudes that depend on x and t

E = E
m
sin(kx-
ω
t) (7.22)

B = B
m
sin(kx-

ω
t) (7.23)

where
ω
: angular frequency of the wave, k : angular wave number of the wave. These two components
can not exist independently. The two fields continuously create each other via induction : the time varying
magnetic field induces the electric field via Faraday ‘s law of induction, the time varying electric field
induces the magnetic field via Maxwell ‘s law of induction.
Electricity and Magnetism
68



a) b)
Fig. 7.1 a) Electric field induced by magnetic field b) Magnetic field induced by electric field

The key features of an electromagnetic wave
- The electric field
E
r
is always perpendicular to the magnetic field
B
r
. The electric field
E
r
and the
magnetic field
B

r
are always perpendicular to the direction in which the wave is travelling (the wave
is a transverse wave). The cross product
E
r
x
B
r
always gives the direction in which the wave travels.
- The fields always vary sinusoidally with the same frequency and in phase with each other.
- All electromagnetic waves, including visible light, have the same speed c (3x10
8
m/s) in vacuum. The
electromagnetic wave requires no medium for its travel. It can travel through a medium such as air or
glass. It can also travel through vacuum.

c
k
1
B
E
oo
m
m
=
ω
=
εµ
==
B

E
(7.24)



Fig. 7.2 : The electromagnetic spectrum


7.3 Energy Flow
The rate per unit area at which energy is transported via an electromagnetic wave is given by the Poynting
vector
Electricity and Magnetism
69


o
1
S= E x B
µ
r
r r
(7.25)

Fig. 7.3 : Electromagnetic wave

The direction of
S
r
(and thus of the wave’s travel and the energy transport) is perpendicular to the
direction of both

E
r
and
B
r
. Since
E
r
and
B
r
are perpendicular


o
2
o
c
EEB
S
µ
=
µ
=
=
2
m
o
E
c

µ
sin
2
(kx-ωt) [W/m
2
] (7.26)
The time-averaged of S is called the intensity I of the wave

2
m
o
E
I =
2c
µ
[W/m
2
] (7.27)
A point source of electromagnetic waves emits the wave isotropically (i.e. with equal intensity in all
directions). The intensity of the waves at distance r from a point source of power P
s
is

s
2
P
I =
4
πr
[W/m

2
] (7.28)

7.4 Radiation pressure
When a surface intercepts electromagnetic radiation, a force and a pressure are exerted on the surface. If
the radiation is totally absorbed by the surface, the force is

IA
F=
c
(7.29)
where I is the intensity of the radiation and A is the area of the surface perpendicular to the path of the
radiation. If the radiation is totally reflected back along its original path, the force is

2IA
F=
c
(7.30)
The radiation pressure p
r
is the force per unit area

r
F
p =
A
(7.31)

Electricity and Magnetism
70

Problems
7.1) An electromagnetic wave with frequency 4x10
14
Hz travels through vacuum in the positive direction of an
x axis. The wave has its electric field directed parallel to the y axis with amplitude E
m
. At time t = 0, the
electric field at point P on the x axis has a value of E
m
/4 and is decreasing with time. What is the distance
along the x axis from point P to the first point with E = 0 if we search in
a) the negative direction of the x axis
a) the positive direction of the x axis

7.2) An airplane flying at a distance of 10km from a radio transmitter receives a signal of intensity 10
µW/m
2
.
What is the amplitude of the electric and magnetic component of the signal at the airplane ? If the
transmitter radiates uniformly over a hemisphere, what is the transmission power ?

7.3) The maximum electric field 10m from an isotropic point source of light is 2V/m. What are the maximum
value of the magnetic field and the average intensity of the light there ? What is the power of the source ?

7.4) Sunlight just outside earth’s atmosphere has an intensity of 1.4 kW/m
2
. Calculate the amplitude of the
electric and magnetic field there, assuming it to be a plane wave.

7.5) A plane electromagnetic wave, with wave length 3m, travels in vacuum in the positive direction of an x
axis. The electric field, of amplitude 300V/m, oscillates parallel to the y axis. What are the frequency,
angular frequency and angular wave number of the wave ? What is the amplitude of the magnetic field
component ? Parallel to which axis does the magnetic field oscillates ? What is the time-averaged rate of
energy flow associated with this wave ? The wave uniformly illuminates a surface of area 2m
2
. If the
surface totally absorbs the wave, what are the rate at which momentum is transferred to the surface and the
radiation pressure on the surface ?

7.6) An isotropic point source emits light at wavelength 500nm, at rate of 200W. A light detector is positioned
400m from the source. What is the maximum rate
B
t
∂
∂
at which the magnetic component of the light
changes with time at the detector’s location ?

7.7) The basic equations of electromagnetism is called Maxwell’s equations which are given in the vacuum
(J = 0,
ρ = 0) as below:

D
ρ
∇ =
r
(Gauss’s law for magnetism)

0

B
∇ =
ur
(Gauss’s law for electricity)

B
E
t
∂
∇× = −
∂
ur
ur
(Faraday’s law)

D
H j
t
∂
∇× = +
∂
r
r
r
(Ampere-Maxwell’s law)
Where
j E
σ
=
r

r
,
o
D E
εε
=
r r
,
HB
o
r
r
µµ
=
. Show that from Maxwell’s equation the following wave
equation can be derived.

2
2
0
o o
E
E
t
ε µ
∂
∆ − =
∂
r
r

Electricity and Magnetism
71

Homeworks 7
H7.1 A plane electromagnetic wave, with wave length λ [m], travels in vacuum in the positive direction of an x
axis. The electric field, of amplitude E [V/m], oscillates parallel to the y axis. What are the frequency,
angular frequency and angular wave number of the wave ? What is the amplitude of the magnetic field
component ? Parallel to which axis does the magnetic field oscillates ? What is the time-averaged rate of
energy flow associated with this wave ? The wave uniformly illuminates a surface of area 2m
2
. If the
surface totally absorbs the wave, what are the rate at which momentum is transferred to the surface and the
radiation pressure on the surface ?

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
λ
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
E 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850

n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
λ
2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5
E 25 50 75 100 150 200 250 300 350 400 450 500 550 600 650 700

n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
λ
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5

E 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950

n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
λ
4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 4.5 5 5.5 6 6.5
E 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95


Appendix I
Factor Prefix Symbol

Factor Prefix Symbol
10
24
yotta- Y

10
-24
yocto- y
10
21
zetta- Z

10
-21
zepto- z
10
18
exa- E

10
-18
atto- a
10
15
peta- P

10
-15
femto- f
10
12
tera- T

10
-12
pico- p
10
9
giga- G

10
-9
nano- n
10
6
mega- M

10

-6
micro- µ
10
3
kilo- k

10
-3
milli- m
10
2
hecto- h

10
-2
centi- c
10
1
deka- da

10
-1
deci d

Appendix II
Surface of a sphere of radius R : S = 4πR
2

Volume of a sphere of radius R : V = 4
πR

3
/3
Electricity and Magnetism
72

Circumference of a circle of radius R : C = 2πR
( ) ( )
2/1
222
2/3
22
axa
x
ax
dx
+
=
+
∫


Appendix III
Dot product of two vectors
is a scalar

A
r
•
B
r

= |
A
r
|.|
B
r
|.cos(α) = A
x
B
x
+ A
y
B
y
+ A
z
B
z




Cross product of two vectors
is a vector

C
r
=
A
r

x
B
r

where |
C
r
| = |
A
r
|.|
B
r
|.sin(α) and the direction of
C
r
is determined by the right hand rule.

The line integral
of the vector
F
r
along the curve L from A to B is a scalar

∫
•
B
A
LdF
rr

=
∫
B
A
dLF
)cos(||
α
=
∫
x
x
B
A
xx
dLF +
∫
y
y
B
A
yy
dLF
+
∫
z
z
B
A
zz
dLF

The surface integral
of the vector
F
r
through the surface A is a scalar

∫
•
A
dAnF
r
r
=
∫
A
dAF )cos(||
α
=
(
)
∫
++
A
zzyyxx
dAnFnFnF

The volume integral

of the scalar F over the volume V

∫
V
FdV