Simulating a Rainbow
In this section we will build on our previous work to compute the
rainbow angle. If sunlight is traveling parallel to the
earth, then the rainbow angle is the angle in the sky at which
rainbows appear. Along the way we will examine what makes a rainbow.
This simulation lets you follow rays of light as they are refracted
(by Snell's law) and reflected in a raindrop.
In Figure 4 we depict a light ray hitting a water droplet at point A.
Some of the light will be reflected and some of the light will enter
the droplet. The light that enters will be refracted as discussed in
the previous section. It then hits the other side of the droplet at
point B where some of it will exit and the rest will reflect back.
Finally at point C some of the light will exit and be seen by an
Figure 4: Ideal path of a
light ray through a spherical water droplet.
At the end of this page is a button. When you click on this button you will be
able to experiment with "shooting" rays of light into a water droplet.
The beam of light enters the water droplet at some height as measured from
the droplet's center. This height is called the impact parameter.
Each time you run the simulation, you will be told what angle the light
leaves the water droplet (as measured from the horizontal). The angle
at which light leaves the droplet is called the deflection angle.
Answer the following questions using the information you gather from the
For the purpose of the first two experiments, you may ignore
the "wavelength" of light rays. For best "experimental data," set the
"wavelength" to be 700 nanometers (red light). (We will look at the effect of
- Graph the deflection angle as a function of the impact parameter for
the impact parameter ranging from 0 to 1 on the
graph provided. (Note: w=0
is the center of the droplet whereas w=1 is the top of
- Numerically approximate the value of the impact parameter for
which the deflection angle is a minimum (to within an accuracy
- Note that each value of the impact parameter corresponds to a unique
value of the angle of incidence (a).
What is the value of a that corresponds to the
minimum deflection angle.
The next activity will help you answer the question "When will incoming
light be focussed most intensely?" In other words, when will a range
of incoming rays all leave the droplet at approximately the same angle?
- Send in 3 incoming beams of light at:
- impact parameters 0.05, 0.1, 0.15
- impact parameters 0.5, 0.55, 0.6
- impact parameters 0.75, 0.8, 0.85
- Sketch the results
- For which set of impact parameters are the outgoing rays the most
concentrated? The most diffuse?
- If an incoming ray has impact parameter w, we define
D(w) to mean the deflected angle of that light ray.
Use the results of the previous experiment to complete
the chart you were given.
Repeat Experiment #1 with the wavelength of
the incoming light rays set to 400 nanometers (violet light).
The color of light corresponds to its wavelength. Light with a
wavelength of 400 nanometers is violet; a wavelength of 450 nanometers
is blue. Light with a wavelength of
525 nanometers appears green, 580 nanometers is yellow,
625 nanometers corresponds to orange, and 700 nanometers is red.
- How does the angle of minimum deflection change?
"White" light is composed of light of all wavelengths. Therefore a
"true" simulation of light passing through a water droplet is to send
in several rays of differing wavelengths at the same
impact parameter and to see how these rays disperse into a rainbow.
- Conjecture how the minimum angle of deflection varies according to
the wavelength of light. Specifically, as the wavelength of light
decreases, does the minimum angle of deflection increase or
decrease? (Make sure your conjecture is supported by your
results from Experiment #1 and the first portion of Experiment #3.)
Send in 5 incoming beams of light at impact parameters 0.9.
The wavelengths of the incoming beams should be
400, 450, 550, 600, and 700 nanometers.
- What are the corresponding angles of the outgoing rays?
Frederick J. Wicklin <firstname.lastname@example.org>
Stuart Levy <email@example.com>
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 30 13:07:05 1995