Rainbow Lab

Part 2 - Simulating a Rainbow

In this section we will build on our previous work to compute the rainbow angle. If rays of sunlight are traveling parallel to each other (which may be assumed on Earth), then the rainbow angle is the angle in the sky at which rainbows appear to observers on Earth's surface. Along the way we will examine what makes a rainbow. This simulation lets you follow rays of light as they are refracted (by obeying Snell's Law) and reflected (by obeying the Law of Reflection) in a raindrop.

In Figure 3 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 observer.


Figure 3: Ideal path of a light ray through a spherical water droplet.
Dashed lines represent normals.

To run this experiment you need to use the interactive Internet Web site http://www.geom.umn.edu/education/calc-init/rainbow/rainbow.cgi. There you are 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, w. In Figure 3, the central horizontal line represents w = 0, the top of the water droplet represents w = 1, and the bottom of the water droplet represents w = -1. Each time you run the simulation, it will calculate what angle the light leaves the water droplet at as measured from the horizontal. This angle is called the deflection angle, D. The deflection angle is measured as the (obtuse) angle between the light ray exiting the droplet and the central horizontal line.

Run the following experiments and answer the questions using the information you gather from simulations.

Experiment #1

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 wavelength later.)
  1. Run the Internet simulation by varying the impact parameter, w, ranging from 0 to 1, and recording the deflection angle.

  2. Make a graph of "The Deflection Angle versus The Impact Parameter" for your data.

  3. Numerically approximate the value of the impact parameter, w, for which the deflection angle is a minimum (to within an accuracy of 0.025). What is the minimum deflection angle?

  4. Note that each value of the impact parameter corresponds to a unique value of the angle of incidence, a. Geometrically determine an equation relating a and w. (Hint: Draw a radius in the water droplet and use a trig function.)

  5. What is the value of a that corresponds to the minimum deflection angle?

Experiment #2

The next activity will help you answer the question "When will incoming light be focused most intensely?" In other words, when will a range of incoming rays all leave the droplet at approximately the same angle?
  1. If an incoming ray has impact parameter w, we define D(w) to mean the deflected angle of that light ray. Send in 3 incoming beams of light at:
    • impact parameters 0.05, 0.10, 0.15
    • impact parameters 0.50, 0.55, 0.60
    • impact parameters 0.75, 0.80, 0.85

  2. Use your results to complete the chart below.

  3. For which set of impact parameters are the outgoing rays the most concentrated? Which set of impact parameters are the outgoing rays the most diffuse?

Impact Parameter

w0

Deflection at w0

D(w0)

Deviation from w0

deltaw

Deflection at w0+deltaw

D(w0+deltaw)

Relative Change in Deflection
{D(w0+[delta]w)-D(w0)}/{[delta]w}
0.10 _________ -0.05 _________ _________
0.10 _________ +0.05 _________ _________
0.55 _________ -0.05 _________ _________
0.55 _________ +0.05 _________ _________
0.80 _________ -0.05 _________ _________
0.80 _________ +0.05 _________ _________

Experiment #3

  1. Repeat Experiment #1 with the wavelength of the incoming light rays set to 400 nanometers (violet light). How does the angle of minimum deflection change?

  2. 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. 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 Part a of Experiment #3.)

  3. "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.

    Send in 5 incoming beams of light at impact parameters 0.9. The wavelengths of the incoming beams should be 400, 450, 525, 580, 625, and 700 nanometers.

    What are the corresponding angles of the outgoing rays?