# 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)
- Mark the set of impact parameters that correspond to deflection
angles in the range:
- 135 - 137.5 degrees
- 137.5 - 140 degrees
- 140 - 142.5 degrees
- 142.5 - 145 degrees
- 145 - 147.5 degrees
- 147.5 - 150 degrees

- Of the sets of impact parameters that you have marked, which is the longest?

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

- What is
*D(0)*?
- Sketch the function
*D(a)* for *0 < a < Pi/2*.
- Find the limit of
*D(a)* as *a* approaches
*Pi/2*.
- What is the significance of your answer to Experiment #1 for this graph?

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:

- Write down an approximation for
*D(a_0+da)*
using *D'(a_0)* that is good if *da* is small.
- If we want this to produce the tightest possible output beam, what
does that say about the value of
*D'(a_0)*.
- Interpret this result in terms of the experimental data you gathered earlier.

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

- Referring to Figure 4, decide the amount that a light
ray is deflected (clockwise) at the following locations:
- Across Point A
- At Point B
- Across Point C

- Use your answers to the previous question to write
down an equation for
*D(a)* in terms
of *a* and *beta*.
- If we let
*c_a* be the speed of light in
air and *c_a* be the speed of light in water find a formula
relating *b* to *a* by using Snell's law.
(Again, you may need to look
up the indices of refraction for *c_a* and *c_w*.)

## 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*.

**Next: **Conclusions

**Previous: **Rainbows: Exploration

**Return to: **Outline

Frederick J. Wicklin <fjw@geom.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: Mon Oct 23 15:08:17 1995