Maxwell’s Equations Differential forms electromagnetic waves y x z E ⊥ B, and E ⊥ k, B ⊥ k,...

Maxwell’s Equations

0

d

qAE

0d AB

L

sE

d At

B d

isBL

0d

At

E d00

Differential forms

D

t

BE

0 B

t

DjH

e

At

Ei emm

d000

DEe

0

qAD

d

HB

m

0

At

DisdH

d

0 ,0 iqif

0

d00 At

E

electromagnetic waves

y

x

z

E⊥B, and E⊥k, B⊥k, k is the direction of the wave.

Polarization of electromagnetic waves

])(sin[ v

ztBB m

])(sin[ v

ztEE m

E and B are in phase

EB

C

The speed of electromagnetic wave00

1

c

Energy Transport and the Poynting Vector

The direction of the propagation of the electromagnetic wave is given by: BE

This wave carries energy. This energy transport is defined by the Poynting vector S as:

BES

0

1

EBS0

1

2

0

1E

cS

2

0

or Bc

S

x

y

EB

C

Find the Poynting vector at point Pparallel-plate capacitor of radius R. a conducting current i

Example:

Q O

ruR

rqiBES ˆ

2

142

00

do iR

rrB

2

2

2

20 R

qE

iR

rB

20

2

0 q

E dt

di Ed

0dt

dq E

BS

Q OE

B S

charge increase field increaseenergy increase

(1)Find the E and B at point P(2) Find the Poynting vector at point P

A conducting current i in a wire with radius R and resistivity .

Example:

BES

0

1

2

1

RijE

iRB 02 PR

iB

2

0

i

BE

EBS0

1

32

2

2 R

iS

SL

2

22

Lip RLS

R

2Ri

Example:

BES

0

1

LE

irB 02

B

rRr

iB

2200

E

EBS0

1

rRLS

2

2

SFind the Poynting vector R

On wire E=0, S=0

On resistance

On wire On resistance On

On Rrr

iB

'2'200

'L

E

E

BS

EBS0

1

BES

0

1

''2

2

RLrS

RrLSp

2

2

RSLrp

2

''2

Example:

r

iB

2

0

SdAP

Er

Bz

E

B1r2r

1

210

10 ln2

1 r

rrEdr

r

rEEdr

r

r

1

2ln rrr

E

1

2ln2

12

2

0 rrRr

EBS

?,S

2

1

d2r

r

rrS 2

1 1

2d2

ln2 2

2r

r rr rr

Rr

R

2

R 0122 lErrlE

r

ErE 01

rR

2 0

i

1

2ln10

rrr

E

The Doppler effect for light

uv

vff

0

v

uvff

0

1. Sound wave, observer fixed, source moving away

2. Sound wave, source fixed, observer moving away

3. Light wave, source and observer separating

?f

* o u

22 2

2 2 2

/'

1 /

t ux ct

u c

1

1

0

0

x

t

S

1 /'

1 /

u cf f

u c

Source or observer leaving

Source or observer approaching1 /

'1 /

u cf f

u c

2

2

x cT

t T

2 2

/

1 /

T uT c

u c

2 2

1 /

1 /

u cT

u c

2

1 /'

1 /

u ct T

u c

'T

1

'f 1

f

Chapter 39 Light Waves

Radio waves

Microwaves Infrared radiation

visible region

Ultraviolet

X rays Gamma rays

The sensitivity of the human eye as a function of wavelength

The Wavelength vs Temperature

Thermal radiation—the sun light

Luminescence—cool light source

Glass

ITO

Matel Organic

The speed of light

c

L2L

c2

s/m103 8c

in vacuum

1

v00

1

em

em

c

matterin ])(sin[ v

rtEE m

Propagation of light in matter

The speed of light in a material depends on the the frequency or wavelength.

The phase change

1

00

Reflection and refraction of light waves

Reflection:11

Refraction:

2211 sinsin nn

Index of refraction: nv

c

em

cv

n

c

emn eemn )(n

rainbow In second rainbow pattern is reversed

Reflection and refraction of electromagnetic waves

Reflection and refraction of electromagnetic waves

Total Internal Reflection

021 90sinsin nn c

1

21sinnn

c

In case of ni larger than nR

c

Examples: refraction at water/air interface

Diver sees all of horizonrefracted into a 97°cone.

Diver’s illusion

Optical fiber:1

21c sin

nn

At each contact the glass air interface, if the light hits at greater than the critical angle, it undergoes total internal reflection and stays in the fiber.

Total Internal Reflection only works if noutside < ninside

ninside

noutside

Optical fiber

n2

n1

d

d

d d

n2

n1

Apparent depth:

Apparent Depth

50

actual fish

apparent fish

B

P

a

d’d

θ2

θ1

n2

n1

21 ' tgddtg

2

1'

tg

tg

d

d

2

1

sin

sin

1

2

n

n

θ2

θ1

ACT: RefractionAs we pour more water into bucket, what will

happen to the number of people who can see the ball?

ACT: RefractionAs we pour more water into bucket, what will

happen to the number of people who can see the ball?

Huygens’ Principle:

All points on a wavefront can be considered as point sources for the production of spherical secondary wavelets. After a time t the new position of a wavefront is the surface tangent to these secondary wavelets.

Huygens’ Principle:

All points on a wavefront can be considered as point sources for the production of spherical secondary wavelets. After a time t the new position of a wavefront is the surface tangent to these secondary wavelets.

1sinACtcBC

21 sinsin

2sinACtcAD

21

21

Deriving the law of refraction

1

2

n1

n2

n1 n2 v1 v2

1 cannot equal 2 !

1212 , vvnn

11 sinACtvBC

C

22 sinACtvAD

1

2

2

1

2

1

sin

sin

n

n

v

v

2211 sinsin nn

1

2

Fermat’s Principle:

A

B

2222 )( xdbxaL

0d

d

x

t 0

d

)/d(

x

cL

0)(

2222

xdb

xd

xa

x

sinsin 11 11

A light ray traveling from one fixes point to another fixed point follows a path such that, compared with nearby paths, the time requires is either a minimum or a maximum or remains unchange (that is, stationary).

0d

d

x

L

c

Lt

AB

Pd

x d-x

θ1’1a b

A

B

P

a

b

d

xθ1

θ2

n1

n2

2211 sinsin nn

0d

)(d

/

1

d

d

/

1 22

2

22

1

x

xdb

ncx

xa

nc

0d

d

x

t2

22

1

22 )(

v

xdb

v

xat

0/

1

/

122

222

1

x)(db

d-x

ncxa

x

nc

2

22

1

22

/

)(

/ nc

xdb

nc

xat

Snell’s Law

1 1

1

2

L n1

n2

1

2

2 2 2

The two triangles above each have hypotenuse L

1

1

2

2

sinsin

L2

1

2

1

sin

sin

2211 sinsin nn

fv

fv

/

/

2

1

2

1

1

2

n

n

2

1

v

v

1

2

n

n

Why is the sky blue?• Light from Sun scatters off of air particles––Rayleigh scattering is wavelength-dependent, More scatter for shorter wavelengths (blue end of the visible spectrum)Less scatter for longer wavelengths (red end of the visible spectrum)

Sun lightred and orange

longer wavelengths scatter lessShorter wavelengths scatter more

blueThis is also why sunsets are red

Example1n

2n

3n321 )( nnnA

312 )( nnnB

231 )( nnnC

213 )( nnnD

1

2

2

1

sin

sin

n

n

1

2

1221 , nn

2

3

2323 , nn

1313 , nn 3

2

2

3

sin

sin

n

n

3

1

1

3

sin

sin

n

n

Example

n

1

2

11 ?2

n

sin

sin 1

090

n )90sin(

sin0

2

221

2 sinsin n

220222

2 cos)90(sinsin nn

22

21

2 sinsin n

'22

Example

cosc

Lt

L

Example

cosc

Lt

L

n1sin

sin

n

?'t

1 1cos'

c

nLt t

n

1cos

cos

)sin

arcsin(1 n

t

n

nt

)]sin

(cos[arcsin

cos'

Exercises:P906-908 19, 36, 44ProblemsP910-911 7, 12