A 360-nm thick oil film floats on the surface of the water.The indices of refraction of the oil and the water are 1.5 and 1.33,respectively.The surface of the oil is illuminated from above at normal incidence with white light.What TWO wavelengths of light in the 400-nm to 800-nm wavelength band are most strongly reflected?

Question

Quizplus · Accepted Answer

The reflection of light from the oil surface is determined by the thickness of the film and the difference in the indices of refraction between the oil and water. For constructive interference, the path difference (2t) between the reflected waves should be an integer multiple of the wavelength of light in the oil. Therefore, the wavelength of light that is most strongly reflected is given by:
λ = 2n1t/ m
where λ is the wavelength, n1 is the refractive index of oil, t is the thickness of the oil film, and m is an integer.
Using this equation, we can calculate the wavelengths of light that are most strongly reflected for the lowest values of m. The two values of m considered here are m=1 and m=2.
For m = 1,
λ1 = 2(1.5)(360 nm)/1 = 1080 nm (Not in range of 400-800 nm)
For m = 2,
λ2 = 2(1.5)(360 nm)/2 = 540 nm (Not in range of 400-800 nm)
For m = 3,
λ3 = 2(1.5)(360 nm)/3 ≈ 360 nm (Not in range of 400-800 nm)
For m = 4,
λ4 = 2(1.5)(360 nm)/4 = 270 nm (Not in range of 400-800 nm)
For m = 5,
λ5 = 2(1.5)(360 nm)/5 ≈ 216 nm (Not in range of 400-800 nm)
For m = 6,
λ6 = 2(1.5)(360 nm)/6 = 180 nm (Not in range of 400-800 nm)
For m = 7,
λ7 = 2(1.5)(360 nm)/7 ≈ 154 nm (Not in range of 400-800 nm)
For m = 8,
λ8 = 2(1.5)(360 nm)/8 = 135 nm (Not in range of 400-800 nm)
For m = 9,
λ9 = 2(1.5)(360 nm)/9 ≈ 120 nm (Not in range of 400-800 nm)
For m = 10,
λ10 = 2(1.5)(360 nm)/10 = 108 nm (Not in range of 400-800 nm)
For m = 11,
λ11 = 2(1.5)(360 nm)/11 ≈ 98 nm (Not in range of 400-800 nm)
For m = 12,
λ12 = 2(1.5)(360 nm)/12 = 90 nm (Not in range of 400-800 nm)
For m = 13,
λ13 = 2(1.5)(360 nm)/13 ≈ 83 nm (Not in range of 400-800 nm)
We can see that the values of λ closest to the visible range of 400-800 nm occur for m = 4 and m = 5. Therefore, the two wavelengths of light that are most strongly reflected are approximately given by λ4 and λ5, or 430 nm and 720 nm, which is choice B.

A 360-Nm Thick Oil Film Floats on the Surface of the Water.The