Refraction

• Minimize t with respect to x

• dt/dx=0 using dL1/dx=x/L1 =sin1

and dL2/dx=(x-d)/L2 = -sin2

• dt/dx=(n1sin 1 - n2 sin 2)/c = 0

1 2 1 1 2 2

1 2

L L

v v

n L n Lt

c c

2 2 21

2 2 22 ( )

L a x

L b d x

Time?

n1 sin 1 = n2 sin 2

2=/2 ==>

sin c = n2/n1

Water n=1.5Air n=1.

c= 41.80

Reflectionrefraction

Doppler Effect for Light

• Recall for mechanical waves that all speeds are with respect to a “medium”

• detector fixed and source moving away: f ` =f [ 1/(1+vs/v)] < f

• source fixed and observer moving away: f ` =f ( 1- vd/v) < f

• note: f ` and f ` are different even if vs=vd

LNM

OQP

L

NMMM

O

QPPP

f f fdv v

v v

vv

1 vv

s

d

s

1

Doppler Effect at Low Speeds

• f ` = f [(v vD) /(v vS)]

• 1/(1+x) ~ 1 - x + …

• 1/(1-x) ~ 1 + x + ...

• if vS <<v and vD <<v , then f ` ~ f ( 1 u/v)where u = | vS vD | is relative speed

of source with respect to detector

Doppler Effect for Light• can we use the same result for light by replacing v by c ?

c=3.00 x 108 m/s

• f `= f ( 1 ± u/c) higher if approaching! u<<c

• in astronomy we measure wavelengths

• c = f = `f `

• `= / ( 1 ± u/c)

• ( `- )/ ~ u/c Doppler shift

• decrease => blue shift => f ` increase=>approach

• increase => red shift => f ` decrease => receding

• light from all distant galaxies is red shifted

• => moving away?

Doppler Effect for Light

=u/c

• For source and detector separating

• f = f0 (1-2)1/2/(1+) red shift > 0

• = f0 (1-)1/2/(1+ )1/2

• For source and detector approaching

• f = f0 (1+ )1/2/(1- )1/2 blue shift < 0

Doppler Effect for Light• For light, v=c

• no medium is needed

• both cases should be the same

• Doppler effect for light depends only on the relative velocity of the source and detector

• time dilation is important 1/ 2 1/ 2

0 0

2

0

1 1 /

1 1 1+u/c

1 u cf f f f

= u/c

f ` < f0 if separating

Police radar uses microwaves => needs relativistic formula

2 20 / 1 /t t u c

Doppler Effect• Car approaching: light (radar) travels at speed c

0

1 /'

1 /

u cf f

u c

0

1 /1 : '

1 /

1 /2 : " '

1 /

u cst shift f f

u c

u cnd shift f f

u c

0

(1 / )"

(1 / )

u cf f

u c

Same as for sound but involvesdifferent shifts!

Problem • A radar device emits microwaves with a frequency of 2.00 GHz.

When the waves are reflected from a car moving directly away from the emitter, a frequency difference of 293 Hz is detected. Find the speed of the car.

• 1. The frequency f received by the car is given by f = f0 (1- )1/2/(1+ )1/2 = v/c

• 2. The car now acts as the source, sending signals of frequency f to the stationary radar receiver.

• 3. Consequently, frec = f (1- )1/2/(1+ )1/2 = f0 (1- )/(1+ ) f0 (1- 2)

since v << c.

• 4. Solve for v: v/c = f/2f0 v = (3x108x 293/(4x109 ) m/s = 22 m/s = 79.2 km/h

Transverse Doppler Effect

• In previous cases, the relative motion was along the line connecting the source and receiver

• in general the relative velocity could be at some angle to this line

• time dilation only depends on the magnitude of

u

u

0

2

0

2 21 1 ( / )

1 cos 1 ( / ) cos

/ 21

1

f fu c

f whe

u c

n

Transverse Doppler Effect