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Sunday, March 22, 2009

ELECTROMAGNETIC WAVES

1. Before the investigations of James Clerk Maxwell around 1865, the known laws of
electromagnetism were:
a) Gauss' law of electricity: E dA q (integral is over any closed surface)
ε
∫ ⋅ =
􀁇 􀁇
b) Gauss' law of magnetism: 0 (integral is over any closed surface)
c) Faraday's law of induction: (integral is over any closed loop)
d) Ampere's
B
B dA
E ds d
dt
⋅ =
Φ
⋅ =−


􀁇 􀁇
􀁇 􀁇
0 law: (integral is over any closed loop)
2. But Maxwell realized that the above 4 laws were not consistent with the conservation of
charge, which is a fundamental principle. He argued
∫B⋅ds =μ I
􀁇 􀁇
(1,2,4) 3
that if you take the space between
two capacitors (see below) and take different surfaces 1,2,3,4 then applying Ampere's
Law gives an inconsistency: ⎡⎣∫B⋅ds⎤⎦ ≠⎡⎣∫B⋅ds⎤⎦ because obvio
􀁇 􀁇 􀁇 􀁇 usly charge cannot
flow in the gap between plates. So Ampere's Law gives different results depending upon
which surface is bounded by the loop shown!
( ) 0
0
Maxwell modified Ampere's law as follows: where the "displacement
current" is . Let's look at the reasoning that led to Maxwell's discovery of the
displacement cu
d
E
d
B ds I I
I d
dt
μ
ε
⋅ = +
Φ
=

􀁇 􀁇
( ) 0 0 0 0
rrent. The current that flows in the circuit is . But the charge on the
capacitor plate is . Hence, ( ) . In words, the
changing electric field in the gap
E
D
I dQ
dt
Q EA I d EA dEA d I
dt dt dt
ε ε ε ε
=
Φ
= = = = ≡
acts as source of the magnetic field in just the same way as
the current in the outside wires. This is really the most important point - a magnetic field
may have two separate reasons for existence - flowing charges or changing electric fields.
circuit
wavelength
wavelength
node
amplitude
400nm 5 00nm 600nm 70 0nm
0
0 0
3. The famous Maxwell's equations are as follows:
a)
b)
c) 0
d)
B
E dS Q
E d d
dt
B dS
B d I d
ε
μ ε
⋅ =
Φ
⋅ =−
⋅ =
Φ
⋅ = +




􀁇 􀁇
􀁇 􀁇
􀁁
􀁇 􀁇
􀁇 􀁇
􀁁
Together with the Lorentz Force ( v ) they provide a complete description
of all electromagnetic phenomena, including waves.
E
dt
F q E B
⎛ ⎞
⎜ ⎟
⎝ ⎠
= + ×
􀁇 􀁇 􀁇 􀁇
4. Electromagnetic waves were predicted by Maxwell and experimentally discovered many
years later by Hertz. Note that for these waves:
a) Absolutely no medium is required - they travel through vacuum.
b) The speed of propagation is for all waves in the vacuum.
c) There is no limit to the amplitude or frequency.
A wave is characterized by the amplitude and frequency, as illustr
c
ated below.
8
14
7
Example: Red light has = 700 nm. The frequency is calculated as follows:
3.0 10 / sec 4.29 10
7 10
By comparison, the electromagnetic waves inside a microwave
m Hertz
m
λ ν
ν −
×
= = ×
×
oven have wavelength of
6 cm, radio waves are a few metres long. For visible light, see below. On the other hand,
X-rays and gamma-rays have wavelengths of the size of atoms and even much smaller.
x
z
5. We now consider how electromagnetic waves can travel through empty space. Suppose
an electric field (due to distant charges in an antenna) has been created. If this changes
then this creates 0 0 a changing electric flux which, through ,
creates a changing magnetic flux . This, through , creates a changing
electric field. This chain of events in
E E
B B
d B ds d
dt dt
d E ds d
dt dt
μ ε
Φ Φ
⋅ =
Φ Φ
⋅ = −


􀁇 􀁇
􀁇 􀁇
free space then allows a wave to propagate.
0
0
In the diagram above, an electromagnetic wave is moving in the z direction. The electric
field is in the x direction, sin( ), and the magnetic field is perpendicular to
it, sin
x
y
E E kz t
B B
= −ω
=
0 0
( ). Here . From Maxwell's equations the amplitudes of the two
fields are related by . Note that the two fields are in phase with each other.
6. The production of electromagnetic wave
k z t kc
E cB
−ω ω=
=
s is done by forcing current to vary rapidly in a
small piece of wire. Consider your mobile phone, for example. Using the power from the
battery, the circuits inside produce a high frequency current that goes into a "dipole
antenna" made of two small pieces of conductor. The electric field between the two
oppositely charged pieces is rapidly changing and so creates a magnetic field. Both fields
propagate outwards, the amplitude falling as 1/ r.
2
2
2
The power, which is the square of the amplitude, falls off as ( ) sin . The sin
dependence shows that the power is radiated unequally as a function of direction. The
maximum power is a
I
r
θ
θ ∝ θ
microwave
oven
metal plate
polarization
7. The reception of electromagnetic waves requires an antenna. The incoming wave has an
electric field that forces the electrons to run up and down the antenna wire, i.e. it produces
a tiny electric current. This current is then amplified (increased in amplitude) electronically.
This is schematically indicated below. Here the variable capacitor is used to tune to
different frequencies.
8. As we have seen, the electric field of a wave is perpendicular to the direction of its motion.
If this is a fixed direction (say, ˆ), then we say that wave is polarized in the direction.
x x
Most sources - a candle, the sun, any light bulb - produce light that is unpolarized. In
this case, there is no definite direction of the electric field, no definite phase between the
orthogonal components, and the atomic or molecular dipoles that emit the light are randomly
oriented in the source. But for a typical linearly polarized plane electromagnetic wave
polarized along xˆ, 0
0
0 0
sin( ), sin( ) with all other components zero.
Of course, it may be that the wave is polarized at an angle relative to ˆ, in which case
cos sin( ), sin s
x y
x y
E E kz t B E kz t
c
x
E E kz t E E
ω ω
θ
θ ω θ
= − = −
= ⋅ − = ⋅ in( ), 0.
9. Electromagnetic waves from an unpolarized source (e.g. a burning candle or microwave
oven) can be polarized by passing them through a simple polarizer of the kind below. A
met
z k z−ω t E =
al plate with slits cut into will allow only the electric field component perpendicular
to the slits. Thus, it will produce linearly polarized waves from unpolarized ones.

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