Problem 7 - Chapter 17 problem 80 The optical Doppler shift for an approaching source (with small velocities compared to c) is given by f0 u =1+ : f c Re‡ected waves undergo a Doppler shift twice. Hence for an approaching velocity of u = 2 10 10 m= sec and a laser frequency of f = 5 1014 Hz, the observed frequency of the re‡ected light is f0 = 1 + 2 u f= c 1+ 4 10 10 3 108 5 1014 = 5 1014 Hz +6:667 10 4 Hz: (a) Adding two sine functions yields, sin + sin = 1 sin 2 + 2 cos 2 : Now there are phase issues etc., but the net result is that the beat frequency, fB ; is simply fB = f1 f2 or twice ( ) =2: Thus the observed beat frequency is fB = 6:667 10 4 Hz: (b) The fraction of the frequency from the original frequency is 2u=c or u 4 10 10 f =2 = = 1:333 f c 3 108 10 18 : Problem 8 - Chapter 35 problem 62 As shown in the …gure, assume that point A is located a distance xA from the interface between a medium with an index of refraction of nA and a medium of index nB . Further assume, as shown in the …gure, that point B is located a distance xB into medium B from this interface. Additionally, point B is a distance yB as measured parallel to the interface from point B. If we assume that a light ray intersects the interface at a distance y < yB while traveling from A to B. The transit time along this path is simply the distance traveled divided by the velocity along each path in A and B or q p 2 2 x2B + (yB y)2 xA + y T = + vA vB q q 1 2 2 T = nA xA + y + nB x2B + (yB y)2 : c 1 (x ,y ) B B B θ θ B (0,y) A A 0 (-x ,0) (x ,0) A B 0 Figure 1: Snell’s law and Fermat’s principle According to Fermat’s principle, light will take the path of least transit time. Thus, to …nd the minimum we simply …nd dT =dy and set it equal to zero or 0 1 dT 1 y (yB y) A = 0: = @n A p 2 nB q dy c 2 xA + y 2 2 xB + (yB y) q p Now the quantities y= x2A + y 2 and (yB y) = x2B + (yB y)2 are the sines of the angles of incidence of the light ray in medium A and B respectively. Thus the above expression reduces to nA sin A = nB sin which is the desired result. 2 B;