Problem 17.26
A thin film of (MgF2) n= 1.38 coats a piece of glass. Constructive interference is observed for the reflection of light with wavelengths of 500nm and 625nm .
Part A -
What is the thinnest film for which this can occur?
906nm
To find the thinnest film for which constructive interference occurs, we can use the equation for the path difference of reflected light:
2nt = mλ
where:
- n is the refractive index of the film
- t is the thickness of the film
- m is the order of the interference
- λ is the wavelength of the light
In this case, since we are observing constructive interference, the order of the interference, m, is 1.
For the thinnest film, we want to find the smallest value of t that satisfies the equation for both wavelengths.
Let's solve for t using the given information:
For the first wavelength, λ = 500 nm = 500 × 10^(-9) m:
2nt = mλ
2n_1 * t = λ_1
Substitute the values:
2 * 1.38 * t = 500 × 10^(-9) m
Solve for t:
t = (500 × 10^(-9) m) / (2 * 1.38)
t ≈ 181 × 10^(-9) m
For the second wavelength, λ = 625 nm = 625 × 10^(-9) m:
2nt = mλ
2n_1 * t = λ_2
Substitute the values:
2 * 1.38 * t = 625 × 10^(-9) m
Solve for t:
t = (625 × 10^(-9) m) / (2 * 1.38)
t ≈ 227 × 10^(-9) m
The thinnest film for which constructive interference occurs is approximately 181 × 10^(-9) m or 227 × 10^(-9) m.
To determine the thinnest film for which constructive interference can occur, we can use the equation for constructive interference in thin films:
2nd = mλ,
where:
- d is the thickness of the film,
- n is the refractive index of the film,
- λ is the wavelength of light, and
- m is an integer representing the order of the interference fringe.
In this case, m = 1 since we want the first-order interference. Let's solve for d:
For λ = 500 nm (0.50 μm):
2nd = mλ
2nd = (1)(0.50 μm)
2nd = 0.50 μm
For λ = 625 nm (0.625 μm):
2nd = mλ
2nd = (1)(0.625 μm)
2nd = 0.625 μm
Since we want the thinnest film for which constructive interference can occur, we will consider the smaller value of 2nd, which is 0.50 μm.
Therefore, the thinnest film for which constructive interference can occur is 0.50 μm.