1. 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 Time?

2. n 1 sin  1 = n 2 sin  2  2 =  /2 ==> sin  c = n 2 /n 1 Water n=1.5 Air n=1.  c = 41.8 0 Reflection refraction

4. 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+v s /v)] < f source fixed and observer moving away: f ` =f ( 1- v d /v) < f note: f ` and f ` are different even if v s =v d

5. Doppler Effect at Low Speeds f ` = f [(v  v D ) /(v v S )] 1/(1+x) ~ 1 - x + … 1/(1-x) ~ 1 + x +... if v S <<v and v D <<v, then f ` ~ f ( 1  u/v) where u = | v S  v D | is relative speed of source with respect to detector

6. Doppler Effect for Light can we use the same result for light by replacing v by c ? c=3.00 x 10 8 m/s f `= f ( 1 ± u/c) higher if approaching! u<<c in astronomy we measure wavelengths c = f = `f ` `= / ( 1 ± u/c) ( `- )/ ~ u/cDoppler shift decrease => blue shift => f ` increase=>approach increase => red shift => f ` decrease => receding light from all distant galaxies is red shifted => moving away?

7. Doppler Effect for Light  =u/c For source and detector separating f = f 0 (1-  2 ) 1/2 /(1+  ) red shift > 0 = f 0 (1-  ) 1/2 /(1+  ) 1/2 For source and detector approaching f = f 0 (1+  ) 1/2 /(1-  ) 1/2 blue shift < 0

8. 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  = u/c f ` < f 0 if separating Police radar uses microwaves => needs relativistic formula

9. Doppler Effect Car approaching: light (radar) travels at speed c Same as for sound but involves different shifts!

10. 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 = f 0 (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, f rec = f (1-  ) 1/2 /(1+  ) 1/2 = f 0 (1-  )/(1+  )  f 0 (1- 2  ) since v << c. 4. Solve for v: v/c =  f/2f 0 v = (3x10 8 x 293/(4x10 9 ) m/s = 22 m/s = 79.2 km/h

11. 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

12. Transverse Doppler Effect