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Unformatted text preview: nteger multiple of 2π . |φAB − φACD | = 2πN .
(a) Write an expression for the phase diﬀerence in terms of Θi , n1 , n2 , d, and λ0 .
(b) If the oil ﬁlm is 1 µm thick, at what angles do we see the ﬁrst strong reﬂections for λ = 450nm,
λ = 530nm, and λ = 630nm?
(c) Using the results from above, why do we see a rainbow of colors on an oil slick?
(d) For TM polarized light, what happens when the angle for constructive interference is equal to Brewster’s
Angle for the air-oil interface? 4 6.007 Spring 2011 Problem Set 8: Electromagnetic Waves at Boundaries Problem 8.4 – Rainbows A very narrow beam of unpolarized red light of intensity Io is incident (at A) on a spherical water drop (see
ﬁgure above). At A, some of the light is reﬂected and some enters the water drop. The refracted light reaches
the surface of the drop at B where some of the light is reﬂected back into the water, and some emerges into
the air. The light that is reﬂected back into the water reaches the surface of the drop at C where some of
the light is reﬂected back into the drop, and some emerges into the air. The index of refraction of water for
the red light is nred =1.331.
(a) Using the data form the ﬁgure, what is the angle α? (You can leave this answer in the form of an
(b) What is the intensity of light that refracts into the drop at A? (Take into account both TE and TM
(c) For what value of angle would you ﬁnd that the light reﬂected at A is entirely TE polarized ? (You can
leave this answer in the form of an expression.)
(d) What is the dominant polarization of light that emerges at B? Explain. 5 6.007 Spring 2011 Problem Set 8: Electromagnetic Waves at Boundaries (e) For blue light, the index of refraction in water is 1.343. The speed of blue light in water is therefore about
1% slower than that of red light. In the ﬁgure above, showing the red-light path, assume that the incoming
narrow beam of light also contains blue-light and draw the trajectory that the blue-light beam would take
after it enters the water droplet at A.
(f) If you look at the rainbow in the sky you will notice that the blue band of color is closest to the ground
and the red color band is highest up. Using the ﬁgure below explain why the colors are ordered in this
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- Spring '11