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Lectures 20,21 (Ch. 32) Electromagnetic waves

- Maxwells equations
- Wave equation
- General properties of the waves
- Sinusoidal waves
- Travelling and standing waves
- Energy characteristics the Pointing vector,

intensity, power, energy - Generation, transmission and receiving of

electromagnetic waves

Maxwells equations

Two Gausss laws Faradays law Ampers law

James Clerk Maxwell (1831 1879)

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 Maxwells theory of

electro-magnetism 1887 Hertzs experiment 1890

Marconi radio (wireless communication)

Mechanical waves

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.

Wave equation and major characteristics of the

wave

Maxwells equations in the absence of charges and

currents take particular symmetric form

Look for solution in the form

To satisfy Gausss laws it is necessary to have

If there is a component of E or B parallel to v

Gausss laws are not satisfied . It may be

verified choosing the front of the Gaussian

surface ahead of the wave front.

Faradays law

Ampers law

Derivation of the wave equation Look for plane

waves Ey(x,t) and Bz(x,t)

Faradays law

Ampers law

E and B in e.m. wave

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).

The frequency range (spectrum) of e/m. waves

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 vc3x108m/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 vc/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 BE/c?B (in T) ltltE (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,t0) or boundary condition

E(t,x0).

- Example. NdYAG laser emits IR radiation in

vacuum at the wavelength 1.062µm. - The pulse duration is 30ps(picos). How many

oscillations of E does the pulse contain?

The shortest pulses (100 as (attos),1as10-18s)

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

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Reflection from a perfect conductor. Standing

waves

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 sinkx0, that is kxpn,

n0,1,2,3,..

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 Characteristics of e.m. waves

The energy density

The Poynting vector is the energy transferred

per unite time per unite cross-section, i.e.

power per unite areathe 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 its 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!

Example

- 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

v

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?/2

Optimal position of antenna (maximizing the

induced current in antenna) corresponds to the

loop perpendicular to B.

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Radiation Pressure

EMW carry both energy and momentum

Absorbing plane

Example. Find the force due to a radiation

pressure on the solar panels. I1.4kW/m2,A1m2.

Reflecting plane

However over long time it influences the

satellite orbit! Comet tails, some stars formation

r

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Laser cooling Nobel Prize,1997

Steven Chu,Claude Cohen Tannoudji,Bill Phillips

atom

Photon

atom

Photon

atom

Photon

atom

Photon

Photons

Polarization

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).

Maluss law (1809)

In general case when linear plz wave goes through

the filter only its projection on the axis of the

filter goes through.

Eout

Ein

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

How to check polaroid glasses?

Crossed polaroids do not transmit light

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Circular polarization

y

Ey

Ez

Ey

x

z

Ez

Left circular polarization If elliptic

polarization

Birefrigent materials refractive index depends

on polarization

x