Where Do Rainbows Come From?

1
Where Do Rainbows Come From?

• A Tour of Differential Calculus

2
Where would you rather be?
3
(No Transcript)
4
(No Transcript)
5
Supernumerary Rainbows
6
Rainbows in Mythology

• Greek Path to heaven made by Iris.
• Chinese Goddess Nuwa uses 5 stones of different
  colors to patch a slit in the sky
• Indian Bow of Indra, God of thunder
• Irish Leprechauns hide their pot of gold at the
  end of rainbow
• Persian Reflection of light off of holy sites

7
Rainbows in Bible

• The Deluge (Genesis 9.13-15)
• I have set my bow in the clouds, and it shall be
  a sign of the covenant between me and the earth.
  When I bring clouds over the earth and the bow is
  seen in the clouds, I will remember my covenant
  that is between me and you and every living
  creature of all flesh and the waters shall never
  again become a flood to destroy all flesh.

8
Rainbows in Religion
9
Where do rainbows come from?Attempts by the
scientists

• Ibn Sahl, Al-Biruni 990
• Al-Hacen, Al-Shirazi, Al-Farisi 1280
• Freiberg 1300
• Snell, Descartes, Huygens, Newton, Fermat 1650
• Young, Airy 1830
• Mie 1910
• You 2020

10
Unweaving the Rainbow

• Do not all charms
  fly
• At the mere touch of cold philosophy?
• There was an awful rainbow once in heaven
• We know her woof, her texture she is given
• In the dull catalogue of common things.
• Philosophy will clip an Angel's wings,
• Conquer all mysteries by rule and line,
• Empty the haunted air, and gnomed mine -
• Unweave a rainbow Keats, 1820 Lamia

11
Where Do Rainbows Come From? Basics Reflection
and Refraction
12
Let Us Unweave the Rainbow

• Q- How does light choose a path to go from point
  A to point B?
• A- Light is like an intelligent prescient
  being.
• Light chooses a path that takes
  the
• least time to go from A to B!
• This is Fermats Principle.
• Depending on your point of view it unweaves or
  camouflages some of the mysteries of rainbows.
• Lets see how it explains reflection and
  refraction.

13
How does light reflect?

• Suppose you are to run from A to B and touch the
  wall L on the way. What path is the shortest?

14
Geometry to the rescue!

• Angle of IncidenceAngle of Reflection

15
So we understand the reflection part.How about
refraction?
16
Refraction Shallow deep pools, broken straight
lines
17
How Does Light Refract?
18
Calculus to the Rescue!
19
Draw a Picture
20
(No Transcript)
21
Time Minimization
22
Enter the Dragon
23
But where do rainbows come from?

• Different colors of light travel at different
  speeds in water, so they refract differently.
  This gives the spectrum of colors.
• Intensity is highest when deflection is
  stationary. So we need to find angle of
  deflection and differentiate it.
• Now Enters the Real Dragon.

24
Which ray causes the rainbow?
25
Angle of Deviation
26
Where do rainbows come from?
27
42 degrees above suns line
28
(No Transcript)
29
References

• James Stewart, Calculus
• Richard Dawkins, Unweaving the Rainbow
• Wikipedia (for pictures)

30
(No Transcript)