An oil film `(n = 1.45)` floating on water is illuminated by white light at normal incidence. The film is 280 nm thick. Find (a) the color or the light in the visible specturm most strongly reflected and (b) the color of the light in the specturm most strongly transmitted. Explain your reasoning.

Explanation

The light reflected from the top of the oil film undergoes phase reversal. Since `1.45 gt 1.33`, the light reflected from the bottom undergoes no reversal. For constructive interference of reflected light, we then have
`2 n t = (m + (1)/(2)) lambda`
or `lambda_(m) = (2 n t)/(m + (1)/(2)) = (2(1.45)(280nm))/(m + (1)/(2))`
substituting for m gives:
`m = 0, lambda_(0) = 1620 nm` (infrared)
`m = 1, lambda_(1) = 54 nm` (green)
`m = 2 lambda_(2) = 325 nm` (ultraviolet)
Both infrared and ultraviolet lights are invisible to the human eye, so the dominant color in reflected light is green.
b. The dominant wavelengths in the transmitted light are those that produce destuctive interference in the reflected light The condition for destructive interference upon reflection is
`2 nt = m lambda`
or `lambda_(m) = (2nt)/(m) = (812 nm)/(m)`
substituting for m gives
`m = 1, lambda_(1) = 812 nm` (near infrared)
`m = 2, lambda_(2) = 406 nm` (violet)
`m = 3, lambda_(3) = 271 nm` (ultraviolet)
Of these, the only wavelength visible to the human eye (and hence the dominant wavelength observed in hte transmitted light) is 406 nm. Thus, the dominant color in the transimitted light is violet.