Lesson12 Em Waves

Lecture 19 1

Lecture 12

Maxwells Equations and Electromagnetic Waves

Maxwells Equations - Waves DAlembert

equation - Electromagnetic Waves The e.m.

spectrum -Energy of Electromagnetic Waves -

Poynting vector - Refractive index - Spherical

waves - Electromagnetic Radiation as Particles

Lecture 19 2

t

EjB

t

BE

B

E

000

0

0

MAXWELLs EQUATIONS

Lecture 19 3

WAVES

Lecture 19 4

WAVE: a transfer of energy from one point to another via

a traveling disturbance

Lecture 19 5

WAVES on a string:

pulses and harmonic wave

Lecture 19 6

WAVE:1D

WAVE:2D

Lecture 19

3D cylindrical wave

3D spherical wave

Lecture 19 8

2

22

2

2

x

yv

t

y

)(),( vtxftxy )()(),( 21 vtxfvtxftxy

Differential equation of wave

motion (DAlembert):

General solution:

Lecture 19

Plane harmonic wave

)(),( 0 tkxsenEtxE

kv

P

k

2

2

Lecture 19 10

From Maxwells equations to e.m. waves

Lecture 19

t

EB

t

BE

B

E

00

0

0

IN VACUUM:

NO charges

NO current

Supposing that E and B are transeverse waves

orthogonal to the x-axis, which is the propagation direction


x

E

t

B

x

E

t

B

t

B

yz

zy

x 0

!!!!zero! are derivative z andy The

x

B

t

E

x

B

t

E

t

E

yz

zy

x

00

00

1

1

0

2

2

002

2

dt

Bd

dx

Bd yy

2

2

002

2

dt

Bd

dx

Bd zz

Lecture 19 12

2

2

002

2

dt

Ed

dx

Ed yy

2

2

002

2

dt

Ed

dx

Ed zz

2

2

002

2

dt

Bd

dx

Bd yy

2

2

002

2

dt

Bd

dx

Bd zz


2

2

00

2 dt

EdE

2

2

00

2 dt

BdB

Both magnetic and electric fields

satisfy DAlembert equation

Lecture 19

x

y

z

Campo

elettrico

Campo

magnetico

Electric

field

Magnetic

field

1. Ex=Bx=0 that is: e.m. waves are transverse-wave

2. The e.m. wave propagation velocity in vacuum is constant and given by:

3. Ez, Bz , Ey, By components satisfy the wave-equation, and harmonic

solutions can be given by:

4. The simple solution for E and B fields propagation in vacuum along x-axis

is to have E//y-axis and B//z-axis:

smcv

v /1031

1

1 8

00

200

00

plane in the lying axis, with )sin(),(

plane in the lying axis, with )sin(),(

00

00

(y,z)x-BtkxBtxB

(y,z)x-EtkxEtxE

FIRST IMPORTANT RESULTS:

axis// with )sin(),(

axis// with )sin(),(

00

00

zBtkxBtxB

yEtkxEtxE y

Lecture 19

2. And making the derivatives::

cuc

EBE

c

EB

BE

0

2

OTHER VERY IMPORTANT RESULTS:

c

EB Demonstration of

1. Assuming a plane wave with E//y and B//z:

t

B

x

E

t

BE

utkxBB

utkxEE

zy

zz

yy

)sin(

)sin(

0

0

00

00

000

0

0

)cos(

)cos(

cBE

cBk

BEBkE

tkxBt

B

tkxkEx

E

zz

yzy

zz

y

y

Lecture 19 15

E.M. WAVES PROPAGATION:

-Polarization

-Rays and wavefronts

Lecture 19 16

e.m. WAVES PROPAGATION

Lecture 19 17

POLARIZATION

Circular

polarization:the electric

field of the passing

wave does not change

strength but only

changes direction in a

rotary manner.

The shape traced out in a fixed

plane by the electric vector as such a

plane wave passes over it (a

Lissajous figure) is a description of

the polarization state.

Waves can oscillate with more than

one orientation.

Electromagnetic waves, such as

light exhibit polarization; sound waves

in a gas or liquid do not have

polarization because the medium

vibrates only along the direction in

which the waves are travelling.

Lecture 19 18

POLARIZING FILTERS

Lecture 19

PHASE of THE Harmonic WAVE: tkx

WAVE-FRONT: is composed by all the points where the field has the

same value. A wavefront is the locus of points having the same

phase: a line (or a curve) in 2-D, or a surface in 3-D.

PLANE WAVE

It propagates along the x axis, the electric field E(x,t)

has to have nul x components: Ex(x,t)=0

E has the same value in a plane that is orthogonal to the

x axis, that is: in the planes // (y,z)

Plane

wave-front

The simplest form of a wavefront is the plane wave, where the rays are parallel to one another. The light from this type of wave is referred to as collimated light. The plane wavefront is a good model for a surface-section of a very large spherical wavefront; for instance, sunlight strikes the earth with a spherical wavefront that has a radius of about 150 million kilometers (1 Angstrom). For many purposes, such a wavefront can be considered planar.

Lecture 19

BE

00

1

c

e.m. WAVES PROPAGATION

The wave velocity propagation in vacuum is:

The propagation direction is given by the direction of :

Wave front

Lecture 19

Huygens' principle provides a quick method to predict the propagation of a wavefront: for

example, a spherical wavefront will remain spherical as the energy of the wave is carried

away equally in all directions. Such directions of energy flow, which are always

perpendicular to the wavefront, are called rays creating multiple wavefronts.

HUYGENS PRINCIPLE: any point on a wave front of light may be regarded as the source

of secondary waves and that the surface that is tangent to the secondary waves can be

used to determine the future position of the wave front.

RAYS and WAVEFRONTS

Lecture 19 22

E.M. WAVES GENERATION

Lecture 19 23

Electromagnetic Waves generation

Electric and magnetic fields are coupled through Ampres and

Faradays laws

Once created they can continue to propagate without further input

Only accelerating charges will create electromagnetic waves

Lecture 19 24

Electromagnetic Waves generation

Lecture 19 25

Electromagnetic Waves

With the changing current restricted to a line, the fields

propagate with cylindrical symmetry outward from the current line.

The electric field is aligned parallel to the current and the

magnetic filed is aligned perpendicular to both the electric field

and to the direction of propagation. These are general features

of electromagnetic waves.

The current must change in time if it is to give rise to propagating

fields (as a steady current merely produces a static magnetic

field). We can translate this into a statement about the charges

whose flow gives rise to the current: The charges that give rise

to the propagating electric and magnetic fields must be

accelerating. Harmonically varying currents will give rise to

harmonically varying electric and magnetic fields.

Lecture 19 26

THE

ELECTRO-

MAGNETIC

SPECTRUM:

Lecture 19 27

Images taken of the Whirlpool galaxy recordiung radiation in

different frequency ranges (and a s consequensce different details

are revealed)

ELECTROMAGNETIC WAVES ARE REAL

Lecture 19 28

Lecture 19 29

The refractive index

Lecture 19 30

Electromagnetic Waves in a medium

Electromagnetic waves travel more slowly through a

medium by a factor n:

This defines n, the index of refraction.

Except for ferromagnets, the speed can be written:

r

rr

nc

vv

1

00

Or, using r instead of k:

Lecture 19 31

Energy and Poynting vector

Lecture 19 32


The energy in an electromagnetic wave must be shared equally

between the electric and magnetic fields:

This energy is transported by the wave; u is the energy density [J/m3]; it varies with time because E and B fields depend on time, so

we consider a mean value

The Energy flux is given by the energy/(unit area * unit time):

Energy flux = cu

Lecture 19

BEcucEcES

cSES

tcSEtS

Ew

BwEw

c

em

BE

2

0

2

0

2

0

2

0

2

0

2

0

2

0

2

0

: vectorPoynting

:ugh power thro

:in ough energy thr

2

1 ;

2

1


More precisely, the energy flowing through a given surface can be

expressed as the flux through the same surface of the vector:

This vector is called the Poynting vector.

Lecture 19

)cos(),(

)cos(),(

0

0

tkxBtxB

tkxEtxE

sm

J

2

1

:be willS of valueaverage the

)(cos),(

2

2

00

22

00

IcES

tkxcEtxS

mean


The direction of the Poynting vector is the

direction in which the energy flow propagates

The module of the Poynting vector represents

the energy per unit time per unit area (assuming

a surface orthogonal to the propagation direction)

The flux of the Poynting vector through a

surface gives the power through the surface

The average value of the Poynting vector is the

intensity of the wave

An electromagnetic wave also carries

momentum, and can exert pressure (called

radiation pressure) on objects

Lecture 19 35

SPHERICAL WAVES

Lecture 19

SPHERICAL WAVES

sm

J

2

12

2

00 IcES

22

00 4)(2

1rrcEPower

The power has to be constant, for each value of the radius r, because the radiation

emitted must pass through any sphere that surrounds the source

The value of the amplitudes E0(r) and B0(r) will be E0/r, B0/r (they decrease as 1/r)

Lecture 19 37

Electromagnetic Radiation as

Particles

Lecture 19 38

In physics, a black body is an object

that absorbs all electromagnetic

radiation that falls onto it. No radiation

passes through it and none is reflected.

Despite the name, black bodies are not

actually black as they radiate energy as

well, since every object with non-zero

temperature radiates electromagnetic

waves, due to the motion of electrons

and protons. How much electromagnetic

radiation they give off just depends on

their temperature.

Black body radiation

Lecture 19 39

Prediction of classical mechanics

Ultraviolet disaster

Lecture 19 40

Black body radiation 1900

hvE Energy=Plank constant times frequency

Plancks crazy idea: the energy can not be divided into smaller and smaller amounts. It is

emitted in discrete packet, called quantum (Nobel Prize 1918)

h=6.6310-34Js

Electromagnetic radiation is not only a kind of wave, but also a kind of particle: wave-particle duality

Lecture 19 41

Electromagnetic Radiation as

Particles

Electromagnetic radiation can act as though it is

made of individual particles, called photons.

The energy of each individual photon depends on

the frequency of the radiation:

with

The photon also has

momentum:

Lecture 19 42

DIPOLE RADIATION

Lecture 19 43

DIPOLE RADIATION

Lecture 19 44

DIPOLE RADIATION

Lecture 19 45

DIPOLE RADIATION: angular distribution

Lecture 19 46

Summary

Using Maxwells equations to find an

equation for the electric field:

This is a wave equation, with solution:

And propagation speed:

Lecture 19 47

This is the speed of light, c!

The magnetic field obeys the same wave equation.

The amplitude of the magnetic field is related to the

amplitude of the electric field:

Also, the two fields are everywhere orthogonal:

Summary

Lecture 19 48

Electromagnetic waves are transverse the E

and B fields are perpendicular to the direction of

propagation

The E and B fields are in phase

Summary

Lecture 19 49

Summary

In the absence of free charges, electric and

magnetic fields obey a wave equation:

These waves propagate at the speed

of light:

Lecture 19 50

Summary

The waves:

Vary sinusoidally

Are transverse

Have related electric and magnetic field

intensities: E = cB

Have perpendicular electric and magnetic

fields

Carry energy and momentum

Lesson12 Em Waves

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