Note that Fermat's Principle would still hold if the mirror in Figure 1 were completely submerged in water.
Consider, then, what happens to the path of a light ray when a portion
of the light path is in the water and a portion in the air. In Figure
3, a light ray leaves point *A* and passes through *B*.
Instead of a mirror, consider what happens as the light ray passes
from air into water. Now, the air/water interface, like the mirror,
occurs at the point *O*. Recall that Fermat's Principle states
that light follows a path that minimizes total travel
time---regardless of the speed of the light ray. Therefore, unlike
our investigation with the mirror, our analysis of where the point
*O* occurs must consider both the speed of light in the air and
in the water.

- Write down the equation that x will satisfy if light minimizes
the time from
*A*to*B*. - Use this to derive an equation relating the sines of the angles
*a*and*b*to the speeds*c_w*and*c_a*.

Referring to Figure 3 the angle that the path

The speed of light in air depends on the temperature and pressure of the
air, and similarly for water and other substances. By contrast the speed
of light in a vacuum is an absolute constant, which we represent by
*c*.

The *index of refraction* for substances is the ratio
*c*/*v* where *v* is the speed of light in that substance.
Tables have been compiled for the ratio of the
speed of light in a vacuum to that of various other substances.

- If a medium has a large index of refraction, what does that say about the speed of light in that medium?
- If light travels from one medium to another of higher refractive index, what can you say about the way the light ray bends in relation to the perpendicular (or the "normal"), to the surface between the media?
- What happens when light travels to a medium of lower refractive index?

- Given a "slab" of crystal of thickness
*T*, draw a diagram that depicts the passage of light through the crystal. - What is the relationship between the angle of incidence and the angle at which the light ray leaves the crystal?
- Design a method to experimentally determine the angle of refraction in the crystal.

- Suppose now we slightly change the direction of the oncoming light
so that the angle of
incidence is now
*a=a0+da*. Estimate the new angle of refraction,*b*. Hints:- Think of
*b*as a function of*a*and use your knowledge of calculus to write down an approximation to*b(a0+da)*. Your answer should involve the quantity*b'(a0)*. - Compute
*b'*by implicitly differentiating an equation that relates*b*and*a*. - Your final answer will be in terms of
*a0*,*da*,*b0*and the index of refraction of the crystal.

- Think of
- Use the above to estimate the angle of refraction when
*a0=45*degrees,*da=5*degrees, and the crystal is diamond. (You will need to look up the index of refraction for diamond in a table.) - Compute the actual angle of refraction for diamond for a 50 degree angle of incidence. How close was your estimate to the actual answer?
- Repeat the previous two questions for
*da*=20 degrees. Is your estimate better or worse in the case? Why?

Frederick J. Wicklin <fjw@geom.umn.edu> Paul Edelman <edelman@math.umn.edu>

This lab is based on a module developed by Steven Janke and published
in *Modules in Undergraduate Mathematics and its Applications*, 1992.

Last modified: Tue Oct 24 15:00:24 1995