a) To determine when the surface of the film becomes very bright, we need to consider the conditions for constructive interference. Constructive interference occurs when the change in phase between the two reflected rays is an integer multiple of 2π.
The phase change for a reflection at a thin film depends on the refractive indices and the thickness of the film. In this case, the film has a refractive index of nf and a thickness of t.
The phase change can be calculated using the formula:
Δφ = (2π/λ) * 2nt
Initially, nf = 4/3, so the initial phase change is:
Δφ_init = (2π/λ) * 2(4/3)t
When nf increases to 3.0, the phase change becomes:
Δφ_final = (2π/λ) * 2(3.0)t
For the surface to become very bright, the phase change should be an integer multiple of 2π:
Δφ_final - Δφ_init = 2π * n
Substituting the values, we have:
(2π/λ) * 2(3.0)t - (2π/λ) * 2(4/3)t = 2π * n
Simplifying the equation, we get:
(2π/λ) * 2(3/3)t - (2π/λ) * 2(4/3)t = 2π * n
(2π/λ) * 2t - (2π/λ) * (8/3)t = 2π * n
(2π/λ) * (6/3)t - (2π/λ) * (8/3)t = 2π * n
(2π/λ) * (-2/3)t = 2π * n
(-4/3) * (2/λ)t = n
Simplifying further, we get:
(8/9) * (1/λ)t = n
Since nf is increasing from 4/3 to 3.0, the surface becomes bright when the limiting value nf is reached. Therefore, we need to find the values of nf that satisfy:
nf >= (8/9) * (1/λ)t
b) When the refractive index is kept constant at nf = 2.0, we can calculate the intensity of the reflected light in terms of the intensity of the incident light, I0.
The intensity of the reflected light can be determined using the equation:
Intensity_reflected = (|(n - nf)/(n + nf)|)^2 * I0
Substituting the values, we have:
Intensity_reflected = (|(2.5 - 2.0)/(2.5 + 2.0)|)^2 * I0
= (|0.5/4.5|)^2 * I0
= (1/9)^2 * I0
= 1/81 * I0
Therefore, the intensity of the reflected light is (1/81) times the intensity of the incident light.