Analysis of the Experiment

You experimented with rays striking the droplet at various positions; each one is scattered at a different angle. You found some position where the angle is an extremum (in this case, a minimum). Let's examine what this means.

Question 6

On the graph of the impact parameter versus deflection angle (from Experiment #1)
Now let's attempt to analyze the data using calculus. If a light ray enters the droplet with impact parameter w, we want to find the angle through which the ray is rotated when it leaves the droplet. Although the experimentally relevant quantity in so-called scattering experiments is the impact parameter, w, the analysis of this problem is a little easier if we instead think of the angle of incidence, a, as the important quantity. Since w=sin(a), then as w increases from 0 to 1, a increases from 0 to Pi/2, so it makes little difference whether we use w or a as the independent variable in this problem.

If an incoming ray enters the droplet at angle of incidence a, let D(a) be the angle through which the ray is rotated upon leaving the droplet.

Question 7

Let's see if there is an algebraic way to derive the information in Experiment #1.

The beam will be at its most concentrated when small changes in the input angle have the least effect on the output angle. That is, if we let a_0 be the value of the angle of incidence that corresponds to the critical value of the impact parameter computed in Experiment #1. If da is a very small angle, then D(a_0+da) should be very close to D(a_0).

Question 8:

So now we know how to compute the value of a_0 which gives us the most concentrated beam, and hence the impact parameter that will produce the brightest light. But, we still need to know what D(a) is in terms of a.

Question 9

Question 10

We now have a formula for D(a) and a formula for beta as a function of a. Use all this information and your answer to Question 8 to find the angle w_0 which will result in the brightest ray of light. (Implicit differentiation will prove helpful here.) This angle is called the rainbow angle.
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Frederick J. Wicklin <>

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

Last modified: Mon Oct 23 15:08:17 1995