W9_parallel_resonance.eps

Week 11: Light 379

If the speed of light is a constant, this condition minimizes both distance and hence time

t=H/v. Thusθi=θl, and we see that the Law of Reflection can be derived from

Fermat’s principle. What about Snell’s Law?

y

y

H

x

D−x

2

θ

θ

1

1

1

2

H^2

B

A

Figure 147: The path withn 1 sin(θ 1 ) =n 2 sin(θ 2 ) is the one with the minimal time when the tra-
jectory goes between median 1 andn 2 where light has distinct speeds. As suggested, one minimizes
the time by choosing a trajectory that trades off more distance in the faster medium against less
distance in the slower one.

To derive Snell’s Law, we need a figure like that one drawn in figure 147. As was the

case for reflection, we only need consider straight line trajectoriesina given medium,

but we allowxto (again) be a variable that we adjust to find the trajectory with the

minimum time.

The major difference this time is that the speeds in the two media aredifferent. When

we right down the times required for the trajectories in media 1 and 2, we have to include

the indices for refraction for those media, that is:

t 1 =

√

y 12 +x^2

v 1

=

n 1

√

y^21 +x^2

c

(928)

and

t 2 =

√

y^21 + (D−x)^2

v 2 =

n 2

√

y^21 + (D−x)^2

c (929)

as the times it takes for the light to travel in a straight line 1) fromAtoxand 2) from

xtoB.

The total time is thus:

t=t 1 +t 2 ==

n 1

√

y^21 +x^2

c

+

n 2

√

y 22 + (D−x)^2

c

(930)

Differentiating and setting the result equal to zero recapitulates thesame algebra as used

aboveto derive the law of reflection, except that there is an extra factor ofn 1 andn 2

on each side. The details are thus left as a (simple) exercise that youshould attempt

without looking back; the result is:

n 1 sin(θ 1 ) =n 2 sin(θ 2 ) (931)

and we see that Snell’s law can be derived from Fermat’s principle as well!