Refraction • Minimize t with respect to x • dt/dx=0 using dL 1 /dx=x/L 1 =sin 1 and dL 2 /dx=(x-d)/L 2 = -sin 2 • dt/dx=(n 1 sin 1 - n 2 sin 2 )/c = 0 1 2 1 1 2 2 1 2 L L v v nL nL t c c 2 2 2 1 2 2 2 2 ( ) L a x L b d x Time?
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