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Refraction

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?. n 1 sin  1 = n 2 sin  2.  2 = /2 ==> sin  c = n 2 / n 1. Water n=1.5 Air n=1.

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Refraction

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  1. Refraction • Minimize t with respect to x • dt/dx=0 using dL1/dx=x/L1 =sin1 and dL2/dx=(x-d)/L2 = -sin2 • dt/dx=(n1sin 1 - n2 sin 2)/c = 0 Time?

  2. n1 sin 1 = n2 sin 2 2=/2 ==> sin c = n2/n1 Water n=1.5 Air n=1. c= 41.80 Reflection refraction

  3. 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

  4. 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

  5. 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?

  6. 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- )1/2blue shift < 0

  7. Doppler Effect for Light = u/c • 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 f ` < f0 if separating Police radar uses microwaves => needs relativistic formula

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

  9. 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= (3x108x293/(4x109 ) m/s = 22 m/s = 79.2 km/h

  10. 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

  11. Transverse Doppler Effect

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