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Summary of Lecture 30 – ELECTROMAGNETIC WAVES
0
- 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)
∫ ⋅^ =
G G
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
∫
∫
G G
G G
law: 0 (integral is over any closed loop)
- 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
G G
( 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
G G G G
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
d E
B ds I I
I d dt
∫
G G
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 d EA^ d I dt dt dt
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
4 0 0 n m 5 0 0 n m 6 0 0^ n m 7 0 0^ n m
0
0 0
- 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
G G
G G
A
G G
G G
A
Together with the Lorentz Force ( v )they provide a complete description
of all electromagnetic phenomena, including waves.
E dt F q E B
= + ×
G G G G
- 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.
(^814) 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
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.
microwave oven metal plate
polarization
- 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.
- 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, x ˆ ), then we say that wave is polarized in the x direction. 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 x ˆ, (^) 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
= ⋅ − = ⋅ in( ), 0.
- 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
k z − ω t E z =
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.
