Rainbows Lab – Falinski – Sp ‘11

Rainbows Lab

“The rainbow is a bridge between two cultures: poets and scientists alike have long been challenged to describe it… Some of the most powerful tools of mathematical physicswere devised explicitly to deal with the problem of the rainbow and with closely related problems. Indeed, the rainbow has served as a touchstone for testing theories of optics. With the more successful of those theories, it is now possible to describe the rainbow mathematically, that is, to predict the distribution of light in the sky. The same methods can also be applied to related phenomena, such as the bright ring of color called the glory , and even to other kinds of rainbows, such as atomic and nuclear ones.” - H.M. Nussenzveig

Introduction:

In this lab, we set out to answer the following questions about rainbows. Many of you have seen rainbows (perhaps all of you), but have you really seen a rainbow? Consider, can you remember:

Rainbows Lab – Falinski – Sp ‘11

  • Is the red on the outside?
  • What is the radius of the arc, in degrees?
  • What is the radius of the arc, from an imaginary center?
  • What is the width of the bow?
  • How does the light intensity of the bow compare inside to outside?
  • What time of day do you see rainbows?
  • Are they north, east, south or west?
  • Are there two bows?
  • If two, where is the second in relation to the first?
  • If two, what is the color sequence – red to violet or reverse?
  • What is the radius of this second bow?
  • Are the bows polarized?

Rainbows Lab – Falinski – Sp ‘11

Table 1: Indices of Refraction at Various Wavelengths

Red / Orange / Yellow / Green / Blue / Violet
(nm) / 660 / 610 / 580 / 550 / 470 / 410
Water / 1.3318 / 1.332 / 1.333 / 1.335 / 1.338 / 1.3435
Air / 1.000 / …

Snell’s law:

Figure 1: Geometry of ray paths for (a) primary rainbow and (b) secondary rainbow

Materials:

  • Fine misting hose, spray bottle or mister
  • Ruler or tape measure
  • Protractor

There will be two parts to this lab report: first, we will calculate the relationship between the angle of the incident ray and the angle of the rainbow; next, we will create rainbows using misters or spray bottles.

Part I:

1)Experiment with this applet: to investigate the properties of primary rainbows.

2)Using geometry, calculate the angle of the refracted ray at point C in Figure 1a above. We will call this angle δ, and it should be a function of i and r.

a)How do you do this? Be sure to clearly redraw the refraction diagrams - a big piece of paper works the best.

b)The angle δ is the sum of the deviations. What is a deviation? A deviation is the change in angle that happens every time the ray changes direction. For instance, if I am going west, and then shift 45° to the south, my deviation is 45°. Do this for our raindrop in terms of i and r.

3)Using Snell’s law, substitute r so that you get δ(i). Although you can leave n a variable, you can substitute 1.331 to start (the value for red).

4)Calculate the minimum value for δ. If you have taken calculus, you can take the derivative and set it equal to zero. If you have a graphing calculator or program, you can plot δ(i) and find a minimum value. You can also use Microsoft Excel to get a numerical solution. Make sure your answer is valid to 3 sig figs, minimum. Your values for the primary rainbow should be about 140°.

5)δ is an interesting value, but we are really interested in it’s supplement  ψ = 180º - δ; What is the maximum value of ψ? This the maximum angle that you, as a viewer, can see the red of the rainbow.

6)Repeat your calculations for all of the colors in the spectrum (using the different values for n found in Table 1) and place your values for ψ and δ in the table at the end.

7)Figure 1b provides the same type of geometry. Repeat steps 2 through 6 for secondary rainbows. What is the maximum value of ψ now? Is it more or less than the answer you got in (5)? Place your answers in Table 3.

Part II:

For this part, we will validate the answers we calculated above. In order to do this, we will have to create rainbows. If for whatever reason, there is no sun to create rainbows, we will use some photographs of rainbows to check our results.

  1. Before you go out, make a diagram of you standing in relation to the sun. Where do expect the rainbow to be? What should you measure so that you can verify your calculations for part I?
  1. Make a rainbow! Use the mister or hose to outline the full circle of the rainbow. It may be below you. Find a place where there is no wind, and preferably, a dark background on which to see the rainbow.
  1. Measure your shadow and height.Calculate the angle of the sun. Be sure that this is the maximum shadow you can produce – in other words, that the sun is behind you.
  1. With a lab partner (or two), establish where in space the top of the rainbow is. Measure the distance from your eyes where the rainbow is, and the distance from your eyes to where the rainbow is.
  1. Experiment to see what kind of rainbows you can make. Are they only at one angle? Are the colors always the same?
  1. With a partner doing the measuring (because they won’t be able to see the rainbow):
  2. measure the radius of the rainbow
  3. measure the width of the rainbow
  4. note the color order of the rainbow
  5. is there a secondary rainbow?
  6. measure the angle of red color of the rainbow from the horizontal
    Analysis:

Table 2: Angles for Primary Rainbow

Red / Orange / Yellow / Green / Blue / Violet
δ
ψmax

Table 3: Angles for Secondary Rainbow

Red / Orange / Yellow / Green / Blue / Violet
δ
ψmax
  1. Begin by answering the questions in the Introduction.
  2. Could there be moon rainbows? When would you see them?
  3. Why would it be very unlikely to see rainbows at true noon? How could you see a rainbow at this time (be creative)?
  4. Explain Newton’s drawing in Fig 2. Is it correct?
  5. Fig 3 is a rainbow picture from Kauai. Can you tell how high in the sky is by looking at the rainbow? What is the angle of the sun?
  6. How was the picture in Fig 4 taken? Is it real?

Figure 2: Newton's Drawing of a Rainbow

Figure 3: Note the time of day and the bright light coming from within the rainbow arc.

Figure 4: How was this picture taken?