Question

What are the three smallest non-zero thicknesses of soapy water (n=1.33) on Plexiglas if it appears green (constructively reflecting 520-nm light) when illuminated perpendicularly by white light?

Answer 1

The three smallest non-zero thicknesses of soapy water on Plexiglas to reflect 520-nm light constructively are approximately 195.5 nm, 391 nm, and 586.5 nm, calculated using the formula for constructive interference in thin films.

Step-by-step explanation:

To solve for the thickness of soapy water (n=1.33) that appears green when reflecting 520-nm light, we use the formula for constructive interference in thin films:

2nt = mλ, where:
n is the index of refraction of the film,
t is the thickness of the film,
m is the order number (m = 0, 1, 2, ...),
and λ is the wavelength of light in the film (which is the wavelength in vacuum divided by the index of refraction).

For the first non-zero thickness where m=1:
2nt = λ
t = λ / (2n)
t = 520 nm / (2 × 1.33)
t ≈ 195.5 nm

For the second thickness where m=2, we double the thickness:
t = 2× 520 nm / (2 × 1.33)
t ≈ 391 nm

For the third thickness where m=3, we triple the thickness:
t = 3× 520 nm / (2 × 1.33)
t ≈ 586.5 nm

The three smallest non-zero thicknesses of soapy water on Plexiglas that will reflect green light constructively are approximately 195.5 nm, 391 nm, and 586.5 nm.