Lectures 20,21 (Ch. 32) Electromagnetic waves. Maxwell’s equations Wave equation General properties of the waves Sinusoidal waves Travelling and standing waves Energy characteristics: the Pointing vector, intensity, power, energy
Two Gauss’s laws + Faraday’s law +Amper’s law
James Clerk Maxwell
Maxwell introduced displacement current, wrote these four equations together, predicted the electromagnetic waves propagating in vacuum with velocity of light and shown that light itself is e.m. wave.
1865 Maxwell’s theory of electro-magnetism
1887 Hertz’s experiment
1890 Marconi radio (wireless communication)
Transverse waves: oscillation is in the direction perpendicular to the propagation direction (waves on the rope, on the surface of water)
Longitudinal waves : oscillation in the direction of the propagation (sound, spring)
E.M. waves are transverse waves
In mechanical waves there is collective oscillations of particles.
E and B oscillate in e.m. waves. Matter is not required. E.M waves may propagate in vacuum.
Look for solution in the form:
To satisfy Gauss’s laws it is necessary to have:
If there is a component of E or B parallel to v Gauss’s laws are not satisfied . It may be verified choosing the front of the Gaussian surface ahead of the wave front.
This is y-polarized wave. The direction of E oscillations determines polarization of the wave. Do not confuse polarization of the wave with polarization of dielectric (i.e.separation of charges in E).
Radio waves, microwaves, IR radiation, light, UV radiation, x-rays and gamma-rays are e/m waves of different frequencies. All of them propagate in vacuum with v=c=3x108m/s
Frequency of e.m.wave does not depend on the medium where it propagates. It is determined by the frequency of charge oscillations. Both the speed of propagation and the wavelength do depend on the medium: v=c/n,
Example. A carbon-dioxide laser emits a sinusoidal e.m. wave that travels in vacuum in the negative x direction. The wavelength is 10.6μm and the wave is z-polarized. Maximum magnitude of E is 1.5MW/m. Write vector equations for E and B as functions of time and position. Plot the wave in a figure.
NB1: Since B=E/c→B (in T) <<E (in V/m)
NB2: in general, arbitrary initial phase may be added :
To find initial phase one needs to know either initial conditions E(x,t=0) or boundary condition E(t,x=0).
The shortest pulses (~100 as (attos),1as=10-18s) obtained today consist of less then 1 period of E oscillations.They allow to visualize the motion of e in atoms and molecules.
Ends of string are fixed→nodes on the ends
Max possible wavelength is determined by the length of string
Total E is the superposition of the incoming and reflected waves. On the surface of the conductor E total parallel to the surface should be zero. Perfect conductor is a perfect reflector with E in ref. wave oscillating in opposite phase.
E(x)=0 at arbitrary moment of time in the positions where sinkx=0, that is kx=πn, n=0,1,2,3,..
If two conductors are placed parallel to each other the nodes of E should be on the ends just as on the string with fixed ends
Example.In a microwave oven a wavelength 12.2cm (strongly absorbed by a water) is used. What is the minimum size of the oven? What are the other options? Why in the other options rotation is required?
The energy density:
The Poynting vector is the energy transferred per unite time per unite cross-section, i.e. power per unite area=the energy flow rate in the direction of propagation
Intensity is the power per unite area averaged over the period of oscillations
For travelling waves:
Standing waves do not transfer the energy:
Example. The distance from the sun to the earth is 1.5x1011m.1) What is the power of radiation of the sun if it’s intensity measured by the earth orbiting satellite is 1.4 kW/m. 2) If the area of the panels of the satellite is 4m and is perpendicular to the radiation of the sun, what is the power received by satellite?
NB: the life on the earth is due to this power of radiation received from the sun!
A radio station on the surface of the earth radiates a sinusoidal wave with an average total power 50kW. Assuming that transmitter radiates equally in all directions, find the amplitudes of E and B detected by a satellite at a distance 100km.
E.m. waves are produced by oscillating charge or current
Richard Feynman ( 1918 – 1988)
Optimal position of antenna (maximizing the induced current in antenna) corresponds to the wire parallel to E
Optimal size of antenna~λ/2
Optimal position of antenna (maximizing the induced current in antenna) corresponds to the loop perpendicular to B.
EMW carry both energy and momentum
Example.Find the force due to a radiation pressure on the solar panels. I=1.4kW/m2,A=1m2.
However over long time it influences the satellite orbit!
Comet tails, some stars formation
Steven Chu,Claude Cohen –Tannoudji,Bill Phillips
Dichroism (dependence of absorption on polarization) is used for construction of the polarization filters for em waves
A grid of wires is a polarization flter for radio waves
When E in a radio wave is parallel to the wires
the currents are induced in the wires and wave is absorbed.
Long molecules play a role of wires for light and used for building of polarization filters (polaroids)
Linear polarized, namely, y-plz e.m.wave
Axis of the filter. If em wave is polarized along this axis it goes through without asborption. Linear plz em wave with orthoginal to this axis in not transmitted (fully absorbed by the filter).
Malus’s law (1809)
In general case when linear plz wave goes through the filter only its projection on the axis of the filter goes through.
Unpolarized em wave (random polarization)
NB: After the filter em wave is always linear polarized along the axis of the filter.
Sun, lamp and other thermal sources produce unpolarized light
Crossed polaroids do not transmit light
Left circular polarization
Birefrigent materials: refractive index depends on polarization: