Lectures 20,21 (Ch. 32)Electromagnetic waves

1. Maxwell’s equations2. Wave equation3. General properties of the waves4. Sinusoidal waves5. Travelling and standing waves6. Energy characteristics: the Pointing

vector, intensity, power, energy7. Generation, transmission and

receiving of electromagnetic waves

Maxwell’s equations

James Clerk Maxwell (1831 –1879)

)(dt

dildB Eencl

dt

dldE B

enclq

AdE

0 AdB

Two Gauss’s laws + Faraday’s law +Amper’s law

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-magnetism1887 Hertz’s experiment1890 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

kxAtxy

tAtxy

cos)0,(

cos),0(

)cos(),(

0),(1),(

222

kxtAtxyt

xty

vx

xty

2

,2

kT

vkkT

v ,

TTkdt

dxvkdxdt

dconst

kxt

2

2,

0

0 AdE

0 AdB

dt

dldE B

dt

dldB E

Maxwell’s equations in the absence of charges and currents take particular symmetric form

Look for solution in the form:

To satisfy Gauss’s laws it is necessary to have:

!, vBvE

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.

Faraday’s law:

dt

vdtBaEa

Amper’s law:

1,

122

vvEvE

EBdt

vdtEaBa

s

mcvvacuumIn 8

00

1031

nKKv

cm

00

vBE

),1,( alwaysnotbutKnKtypically m

n

cv )1( alwaysnotbutcvntypically

Derivation of the wave equationLook for plane waves: Ey(x,t) and Bz(x,t)

Faraday’s law:

tx

B

x

E

t

B

x

E

xat

BxExxEa

zyzy

zyy

2

2

2

,

)]()([

Amper’s law:

2

22

,

)]()([

t

E

tx

B

t

E

x

B

xat

ExBxxBa

yzyz

yzz

01

01

2

2

22

2

2

2

22

2

t

B

vx

B

t

E

vx

E

zz

yy

1

vvBE

E and B in e.m. wave

)cos(

)cos(

0

0

kxtBB

kxtEE

z

y

)cos(

)cos(

0

0

kxtBkB

kxtEjE

or

)cos(

)cos(

0

0

kxtBB

kxtEE

z

y

)cos(

)cos(

0

0

kxtBkB

kxtEjE

or

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 v=c=3x108m/s

)(11

][

2

1

HertzHs

f

Tf

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,

vacuuminlengthwavef

c

nf

c

f

vvT

0

,

n0

Example. A carbon-dioxide laser emits a sinusoidal e.m. wave that travels in vacuum in the negative x direction. The wavelength is 10.6 μm 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.

sradmradsmck

mradm

radk

Tsm

mV

c

EB

kxtBjB

kxtEkE

/1078.1/1093.5)/103(

/1093.5106.10

17.322

105/103

/105.1

)cos(

)cos(

1458

56

38

60

0

0

0

NB1: Since B=E/c→B (in T) <<E (in V/m)NB2: in general, arbitrary initial phase may be added :

)cos(

)cos(

0

0

kxtBjB

kxtEkE

To find initial phase one needs to know either initial conditions E(x,t=0) or boundary condition E(t,x=0).

• Example. Nd:YAG 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?

10000103

103

3

30

)(3103/103

10062.1

15

12

158

6

s

s

fs

ps

TN

femtosfsssm

m

cT

pulse

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

nfL

vnfn

n

L

L

vvfLL

nn min

maxminmax

max

2...2,1,

2

22

2

Reflection from a perfect conductor. Standing wavesTotal 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.

tkxBtxBtxBtxB

kxtBtxB

kxtBtxB

tkxEtxEtxEtxE

kxtEtxE

kxtEtxE

refzinzz

refz

inz

refyinyy

refy

iny

coscos2),(),(),(

)cos(),(

)cos(),(

sinsin2),(),(),(

sinsincoscos)cos(

)cos(),(

)cos(),(

0

0

0

0

0

E(x)=0 at arbitrary moment of time in the positions where sinkx=0, that is kx=πn, n=0,1,2,3,..

)(,...2,2

3,,

2,0

,..2,1,0,2

Eofplanesnodalx

nn

x

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?

nfL

vnfn

n

L

L

vvfLL

nn min

maxminmax

max

2...2,1,

2

22

2

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

...3.18

2.12

1.62min

cmL

middletheinnodeonecmL

cmL

The Energy Characteristics of e.m. waves

The energy density:

1,,

2,

2

22

vvBEB

uE

u magel

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

,

,1

EBEBvS

uvdtAudVdUdt

dU

AA

PS

BE

S

EBB

EBE

u

2

222

22

Intensity is the power per unite area averaged over the period of oscillationsFor travelling waves:

22][][

m

W

sm

JIS

RMSRMSrms

BEI

EkxtEEE

BEkxt

BEkxt

BEI

,2

)(cos

2))(2cos1(

2)(cos

020

2

0000200

200BE

I

TA

PdtUAdSP ,

Standing waves do not transfer the energy:

02sincossin2

cossincossin4 0000 tkxkx

BEttkxkx

BEI

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?

Wmm

WIRAdSPsun

2622222

32 10410)5.1(14.34104.14

NB: the life on the earth is due to this power of radiation received from the sun!

kWmm

WIAAdSPpanels 6.54104.1 2

23

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.

27

210

4

21096.7

1028.6

105

2 m

W

mR

PI

Tc

EB

mVcIE

c

EBEI

1100

200

0

20

0

00

102.8

/105.22

22

E.m. waves are produced by oscillating charge or current

2)sin(~,

sin~~

rI

rBE

v

Richard Feynman ( 1918 – 1988)

Optimal position of antenna (maximizing the induced current in antenna) corresponds to the wire parallel to E

Optimal position of antenna (maximizing the induced current in antenna) corresponds to the loop perpendicular to B.

Optimal size of antenna~λ/2

Radiation Pressure

S

t

p

t

pF

p

S

S t

p

t

pF

2

Reflecting plane

Absorbing plane PaatmPam

NP

c

P

c

AIF

c

I

dt

Ud

Acdt

pd

AA

FP

c

Up

rad

rad

52

101,1][

11

c

IPrad

c

IPrad

2

Example. Find the force due to a radiation pressure on the solar panels. I=1.4kW/m2,A=1m2.

NAPF

Pasm

mW

c

IP

rad

rad

6

68

23

105

105/103

/104.1

However over long time it influences the satellite orbit!Comet tails, some stars formation

EMW carry both energy and momentum

r

Laser coolingNobel Prize,1997

Steven Chu,Claude Cohen –Tannoudji,Bill Phillips

s

mmvKT

s

kmvKT

kTmv

1~10~,1~300~

:~2

7

2

kv

laseratom

Photons: sJhh

kpE

2410626.6,2

,

Photon

Photon

Photon Photon

atom

atom

atom

atom

1E

2E

1E

2E

Polarization

Dichroism (dependence of absorption on polarization) is used for construction of the polarization filters for em wavesA grid of wires is a polarization flter for radio waves

When E in a radio wave is parallel to the wiresthe 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).

Unpolarized em wave (random polarization)

Ein

Eout

cosinout EE

Malus’s law (1809)

2cosinout II

In general case when linear plz wave goes through the filter only its projection on the axis of the filter goes through.

2cos2 in

inout

III

Sun, lamp and other thermal sources produce unpolarized light

NB: After the filter em wave is always linear polarized

along the axis of the filter.

How to check polaroid glasses?

Crossed polaroids do not transmit light

Circular polarization

x

y

z

Ey

Ez])sin()[cos(

)sin(

)cos(

0

0

0

kkxtjkxtEE

kxtEE

kxtEE

t

z

y

Ey

Ez

Left circular polarization

If elliptic polarization

oxoy EE

Birefrigent materials: refractive index depends on polarization: )(),( 21 nn

x

x

2)()( 2121

xnn

cxkk