Batch 3 - Class 243 - Rainbow Formation

• Dudeney - 381: Can you construct a card frame, one as shown below, so that each side adds up to the same number
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• Answer: 
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• Lets students try and explain the phenomenon. Break it into the several pieces
• What is diffraction? Why does light change direction when it enters a denser medium?
• Light has different speed in different media. Why? ( https://www.youtube.com/watch?v=CUjt36SD3h8)
• One explanation is that light is bouncing around amongst atoms and hence effectively taking a longer path. But, scattering is not a precise pattern - so light could be much slower, and there is no guarantee that it will land up back on the original path
• Second idea is that light is absorbed and emitted by atoms, and that slows it down. But atoms dont remember direction of photons, and can re-emit in any direction. So same problem as before.
• One needs to embrace the idea that light is a wave. Wavelength is the distance between peaks. Second key idea is superposition, i.e. you can add waves. When light enters a medium, its electric field makes the electrons move, thereby creating another electric field. This is like two waves moving at different speeds, and the net resultant is sum of the two - this sum moves slower than the original wave (See video above from 7 mins onwards)
• A simplistic way to think about it is also that the higher electron density creates electric fields that slow down the orignal electric field.
• The lifeguard analogy - Light takes the shortest time path between any two points. (Fermat's Principle)
• Explain through a diagram (https://www.youtube.com/watch?v=NLmpNM0sgYk)
• But why?
• Marching Soldiers Analogy - Consider a set of soldiers marching side by side, and if their speed will slow down on entering a denser medium, trace their steps. They will bend.
• However, why should the direction of soldiers change - the marching line may bend, but the soldiers will still continue in the same direction.
• Huygen's principle - depends on wave nature of light. When a wave enters a small aperture, it passes and then expands. (Diffraction). With more than one gap, waves from each gap can interfere, with waves adding or subtracting each other. 
• This leads to multiple wavefronts, so doesn't give a unique answer
• Light is actually oscillating electromagnetic fields (Maxwell's equations)
• Electric field is perpendicular to the direction of light. On hitting the surface, the component of electric field parallel to the surface remains the same, but the one perpendicular is opposed by alignment of particles in the denser medium. Hence the net electrical field in glass is lower in perpendicular direction. Now if you add back the components, the electric field has bent, and hence light has bent.
• Another way to think about this is the probabilistic nature of light - it is actually taking all paths with different probabilities, and we need to add all the photons reaching through different paths. The ones towards the edges cancel out, and the ones reaching are mostly contributed by the least time paths. ( https://www.youtube.com/watch?v=cep6eECGtw4) - what if we wanted to equalize the time - the ones passing straight would need most glass, and the ones at edges will need least glass. What does this look like
• Convex lens! which converges all the paths, and hence focuses light!
• What is total internal reflection? Why does it happen?
• In most explanations, it is not explained "why" total internal reflection happens, but only "when" it happens (critical angle). 
• While the complete explanation is complex (pun intended), one way to think about it as follows. At any angle there is a reflected component and a refracted component. The total energy must get preserved. When the critical angle is reached, the sin(angle) in the less denser medium must be greater than 1. This is possible by making the angle complex. Physically, think of this as a decaying wave forming in the less dense medium, which decays the refracted wave. Due to conservation of energy, now the reflected wave must acquire most of the amplitude of incoming wave.
• As an experiment, think about two slabs of glass that you are bringing together, with a thin air gap. Initially, total internal reflection happens. But as you bring the other slab closer, the distance in which this evanescent wave is present reduces, till such point that it may reduce so much that the refracted component "jumps" to the other slab. At this point, it becomes like one slab and no total internal reflection happens. ( https://www.researchgate.net/post/Why_does_Total_Internal_Reflection_behave_the_way_it_does)
• Ok, now lets see what is happening in rainbow.
• Violet light bends the most, red the least. Trace the path of violet and red lights. ( https://www.youtube.com/watch?v=xkDhQGXqwCM)

• If violet light refracts more, it should land up exiting at the top. Why do we see red light at the top of a rainbow
• Look at it from an observer's standpoint and what will reach her eyes. It will be the red light from the droplets above, and violet from droplets below. And hence we see red color above.

• Given that different droplets are sending different colors, why don't they all cancel out?
• From the topmost layer (angle ~42 degrees), only the red light can reach us. No violet light from any layer can cancel that. Similarly at different angles, different colors can reach us. For a given layer, one band of color dominates corresponding to the angle.
• Why is the rainbow circular
• The angle between the sun and our eyes at the droplet matters. Show that this 40-42% angle will follow a circular arc.
• Can you see a full circle? Yes, if earth doesn't get in the way. Or if you are high up in the sky where a 40-42 degree angle is available all around (eg: flying in an airplane)
• Why does the sky above the rainbow look dark, and below the rainbow lighter?
• Note that for the droplets slightly above the red layer, nothing can reach us through total internal reflection (the normal scattering still works, so its not completely dark)
• Below the violet layer, we are getting all the different colors, hence lighter
• Have you seen a double rainbow? What causes that?
• In this case the light does two total internal reflections. That is the reason that on this one, violet appears at the top (inverted relative to first rainbow), and it is less bright than the first rainbow.

• The Same Court Time - Basketball is one of the most popular team sports in this summer’s Olympic Games. The game is played with five players from each side on the court at the same time, and is played for four 10-minute quarters. The coach of the Brazil team has decided to give every one of her eight players the same court time. What scheme should she use to ensure this?
• Many ways to solve for this. One easy solution is 12345, 23456, 34567, 45678, 56781, 67812, 78123, 81234
• Another funny one is to forfeit!
• Another one is to have the coach play all through and then give each player two quarters!