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;