A light beam with wavelength 500 nm is reflected constructively from a thin layer of oil having index of refraction 1.25.The oil floats on the top of water of index of refraction 1.33.What is the minimum thickness of the layer of oil?
A)100 nm
B)150 nm
C)175 nm
D)200 nm
E)225 nm

Question

Quizplus · Accepted Answer

For constructive interference to occur, the path difference between the two reflected rays must be an integer multiple of the wavelength. In this case, we have:

2nt = mλ

where n is the refractive index of the oil, t is the thickness of the oil layer, m is an integer, and λ is the wavelength of the light. We want to find the minimum value of t that satisfies this condition.

Using Snell's law, we can find the angle of incidence at the air/oil interface:

sin θ1 = n2 sin θ2
sin θ1 = (1.33/1.25) sin 90° = 1.064
θ1 = 44.1°

Since the oil layer is thin, we can assume that the angle of reflection at the oil/water interface is approximately equal to the angle of incidence at the air/oil interface. Thus, we have:

2t = (m + 1/2)λ/ sin θ1

For the minimum value of t, we take m = 0. Thus,

2t = λ/2 sin θ1

t = λ/4 sin θ1

Plugging in the values, we get:

t = (500 nm)/(4 sin 44.1°) ≈ 201 nm

Therefore, the minimum thickness of the layer of oil is 200 nm, which corresponds to choice D.

A Light Beam with Wavelength 500 Nm Is Reflected Constructively