Chapter 25

Lecture 12April 28, 2005

Electromagnetic waves are transverse

0

0

0 0

0 0 0

cos( )

cos( )

B

P S

E kx t

B kx t

d dd d

dt dt

kE B

E B cBk

E x

B z

E s B A

^

^y

PoyntingVector

0

1S E B

S

2

0 0

~ / : E=cBEB E

Power area notec

S

Poynting Vector and Intensity

• Points in the direction of the wave• the magnitude is the rate of

energy transfer per unit area carried by the wave

20

02avg

EPowerI S

Area c

Average{ [cos(kx-wt)]2 } = 1/2

Traveling Waves

Traveling Waves

Standing Waves between two flat mirrors

EB

My total energy output per unit time is constant

My energy output per unit time and area drops as the

distance2

RNear Earth:P~0.00001 N/m2

Radiation pressureThe theory of relativity: anything moving at the speed of light will carry momentum p=E/c

Light can push stuff!

2

( ) 1

4Sun

Sail Sail Sail Sail

PowerI RF PA A A

c c R

c

I

A

Power

ct

cE

At

p

AArea

Force

1/11

Pressure

Polarization

When E points in one direction the wave is

linearly polarized

There are materials that absorb waves when E points in one direction

E

… but not in the other

●

E

Points out of the screen

There are also other polarizations for which

E changes direction

Polarizers at angles reduce the intensity

Selects one polarization

Only the projection onto the transmission

axis gets through I=I1 cos2

I=I1=I0/2

I=I0

Crossed polarizers transmit approximately what fraction of an electromagnetic wave?

1. 0%

2. 25%

3. 50%

4. 75%

5. 100%

What is light? I believe light is a stream of fast moving particles. This explains why and how light

reflects and refracts.

I can also understand how and why light reflects and refracts if I assume

it is a wave.HUYGENS’ PRINCIPLE

If light is a wave it can should be able to go around small obstacles…and it

does!YOUNG’S INTERFERENCE EXPT.

My equations predicted that light is a high frequency electromagnetic wave in1865.

So, is light a wave or a particle?

Since it sometimes behaves like one and sometimes like the other it is neither.

Instead of trying to force it into some label convenient to us we should find out its

properties.

In many cases light behaves like a wave, but sometimes (when quantum effects are

important) it behaves like a particle.

The ray approximation

When light propagates its wave nature is hidden if

• We never look at distances of the order of (or smaller)• All obstacles have typical sizes much larger than

d

The wave nature of light is not important

for d >>

Light behaves as a ray. In uniform media it travels in a straight

line

The ray approximation

d ~d d

For smaller distances (d ~ ) the wave nature begins to show

up

For d << the wave nature is central in

understanding light’s behavior

When looking at features smaller than the interference of

light waves shows up

The shortest time principle – FERMAT’S PRINCIPLE

When light behaves as a ray and travels from point A to point B it follows the path that gets it to B in the shortest time possible

In a uniform medium where the light speed is c1 …

A

B

This path is longer

This is the shortest path

… for constant speed the shortest path takes the least amount of time

In uniform media light rays travel in straight lines

A B

L

When light behaves as a ray and travels from point A to point B it follows the path that gets it to B in the shortest time possible

Let us look at a reflected ray

mirror

h

x

L - x

x is such that it takes the least amount of time to go from A to B

22 hx 22 hxL

2222 11)( :time hx

vhxL

vxt

110 0

2222 hx

x

vhxL

xL

vdx

dt(x)

'

sin sin

sin1

sin1

0

'

'

'vv

Speed of light in the medium

I. The law of reflection

II. The path of a light ray is reversible.

III. The path of a light ray in vacuum defines what is meant by “a straight line”.

1

25.17

• The reflecting surfaces of two intersecting flat mirrors are at an angle of θ. If a light ray strikes the horizontal mirror, show that the emerging ray will intersect the incident ray at an angle of β=180º-2

Exercise 25.17

This looks like an application to the reflection formula and a bit of geometry

When light behaves as a ray and travels from point A to point B it follows the path that gets it to B in the shortest time possible

Look at a ray going form one medium with v1 to another with v2

v1

v2

h

h

A

B

x

L - x

22 hx

22 hxL

22

2

22

1

11)( hxL

vhx

vxt

22

11

222

221

sin1

sin1

1

1

: 0

vv

hxL

xL

vhx

x

v

dx

dt(x)

1

2

Define the index of refraction:

v

cn

mediumin light of speed

in vacuumlight of speed

2211

22

11

sin sin

sin sin

nn

v

c

v

c

v1

v2

A

B

1

2

Then under refraction,

Index of Refraction

Snell’s law of refraction

n2 sinn1sin1

n2

n1

Given two slabs of transparent material of equal thickness, Fermat’s principle means that the part of a ray passing through the medium with the higher index of refraction is ______ the part passing through the lower index medium.

1. longer than

2. equal to

3. shorter than

4. perpendicular to

5. parallel to

25.13

• When the light in the figure passes through the glass block, it is shifted laterally by the distance d. If n = 1.76, find the value of d.

Exercise 25.13

This looks like an application to the refraction formula and a bit of geometry

2

21

2

21

212

cos

sin

cos

sin

sin cos

θ

θθhd

h

dL

d

L

h

h

L

1122 sin sin nn

d

cm 39.09428.0

1827.02

34.0

5.01sin 5.1

6/ ,51 ,1

2

2

121

d

.nn

Larger than 1

When light goes from medium 1 to medium 2 with n1 > n2

12

12 sin sin

n

n

If we increase 1 the right hand side grows

Eventually, when sin1 = n2/n1 we get sin2=1

If we increase 1 beyond that the wave in medium 2 disappears

The ray suffers total internal reflection

θc is when sinθ2=1 i.e. n1/n2 sin θc =1

Total internal reflection

Find the critical angle for a ray of light in glass

Critical Anglen1 sin1 n1 sinC

n2 sin2 n2 sin(90o)

sinc n2

n1

for n1 n2

Air: n2 = 1 Glass: n1 = 1.5 --> c = 42o

Two Rt Angle PrismsNo Loss of Light; use inoptical instruments

Fiber Optics

core, n1

clad, n2 < n1

n1

n2

Huygen’s principle (1678)

Each point on a wave front is a source of secondary spherical wavelets.

Constructive interference creates the new wave front.

Endoscope

"Foreign Body" in the Stomach Swallowed QuarterHere is a quarter which a young man swallowed and which is lying in the stomach. These are easily removed with a wire snare or device for grasping a coin.

Why is the sky blue?

why are sunsets red?

If n depends on we get dispersion

Dispersion

blue bent more than red

Iscattered=λ-4

longer wavelengthless scattered more scattered

Rainbows

400

420

Primary

520

Secondary

Colors Reversed

Examples

24.22

• At what distance from a 100W electromagnetic wave point source does Emax=15V/m

24.22

24.26

• A possible means of space flight is to place an absorbing sheet into orbit around the Earth and then use the light from the Sun to push this “solar sail.” Suppose a sail of area 6105 m2 and mass 6000kg is placed in orbit facing the Sun.

a) what is the force exerted on the sail? b) What is the sail’s acceleration?c) How long does it take the sail to reach the Moon,

3.84108 m away? Ignore all gravitational effects, assume that the acceleration calculated in part b) remains constant, and assume a solar intensity of 1340 W/m2

24.26

24.35

• An important news announcement is transmitted by radio waves to people sitting next to their radios, 100 km from the station, and by sound waves to people sitting across the news room, 3M from the newscaster. Who receives the news first? Explain. Take the speed of sound in air to be 343 m/s.

Exercise 24.35

• the sound and radio waves start at the same time• sound covers a distance d in a time d/v• radio waves cover a distance L in a time L/c

3sound

34

light 8

3 m s 8.75 10 s

343 m/s

100 103.33 10 s

3 10 /

t

mt

m s

Light wins

24.41

In the figure, suppose that the transmission axes of the left and right polarizing disks are perpendicular to each other.

Also, let the center disk be rotated on a the common axis with angular speed . Show that if unpolarized light is incident on the left disk with an intensity Imax, the intensity of the beam emerging from the right disk is

This means that the intensity of the emerging beam is modulated at a rate four times the rate of rotation of the center disk. Hint: Use the trig. identities

cos2=(1+ cos2)/2 and sin2=(1- cos2)/2

max

1(1 cos 4 )

16I I t

Exercise 24.41

• c is so large that the polarizers appear frozen to a “bit” of light ... At time t the rotation angle will bet• the intensity is decreased by (cos )2

Polarization after the 1st polarizer

Polarization after the 2nd polarizer

Polarization after the 3rd polarizer

0 max

1

2I I 2

01 cos II

Imax

2

2 1

2

0

cos / 2

cos cos / 2

I I

I

24.21

2

12

F=-kx x=Acos( t+ )

1

2spring

f fT

kSHO

m

gPendulum

L

PE kx Energy PE KE

total 0

0

2 2 2 20

2 20

sin( )

/

( / ) ( )

/tan

ma F kx bv F t

F mA

b m

b m

2

)2/(

2

)cos(

m

b

m

k

tAex tmb

/0

t mE E e where

b

( , ) sin( )

2

D x t A kx t

k vk

dampled SHO

driven and damped SHO

traveling waves

2 2

2 2 2

1d d

dt c dx

E E 8

0 0

13 10 m/sc

0

0

0 0

cos( )

cos( )

E kx t

B kx t

E cB

E x

B z

^

^2

av 0 0

1

2u E

0

1S E B

2

0 0

~ /EB E

Power areac

S2

0

02avg

EPowerI S

Area c

p=E/c

1 1 / 1 PowerP

F p E c I

A A t A t A c c

I=I0 cos2

LL S

S

v vf f

v v

positive is from L to Spositive v will use (+), negative v (-)

sin(a+a) + sin(a-a)=2 sin(a) cos(a)

string

Tv