Trace the path of a light ray with impact parameter d through a spherical water droplet.
Let l=656nm, “red” and the index of refraction for that wavelength be n=1.331. Assume the ray reflects at the back of the raindrop of radius r.
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(a) Show that the scattering angle f is given by f = 180o - 4q2 + 2q1, where q1 is the angle of incidence of the ray as it first meets the raindrop, and q2 is the angle of refraction. |
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(b) Express q1 and q2 in terms of d. |
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(c) Use a spreadsheet program to calculate q1 and q2 for values of d/r from 0 to 1. Calculate the scattering angle in each case and plot f versus d/r. You should find a minimum at about 138o. |
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(d) Repeat part (c) for l = 589nm, “yellow”, n = 1.333, and for l = 486nm, “blue”, n = 1.337. By how much does the angle of the minimum change between red and blue? |
f changes with d/r but it also depends on wavelength. You will see most of the scattered light at the minimum scattering angle. That minimum angle depends somewhat on the wavelength, so you see red light at a slightly different angle than blue light. You see a rainbow.
To earn extra credit, send your solutions for parts (a) through (d) (including the spreadsheet with the graphs of f versus d/r) to mbreinig@utk.edu.