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