A soap film (n = 1.33) is 375 nm thick and lies on a glass plate (n = 1.52). Sun

Question

A soap film (n = 1.33) is 375 nm thick and lies on a glass plate (n = 1.52). Sunlight, whose wavelengths (in vacuum) extend from 380 to 750 nm, travels through air and strikes the film perpendicularly.
(a1) What is the algebraic expression for the vacuum-wavelength ?vacuum at which destructive interference causes the film to look dark in reflected light? Express your answer in terms the refractive index n of the film, the thickness t of the film, and the integer variable m(m = 0, 1, 2, 3,...)..
(b1) What is the vacuum wavelength ?vacuum in the visible range (380 nm to 750 nm) at which destructive interference causes the film to look dark in reflected light?

Explanation / Answer

(a) since the indices of refraction go 1.00 to 1.33 to 1.52 (i.e. increasing order) we get

    2 t n / wavelength = (m +1/2)   or

   wavelength = 2tn / (m+1/2)

(b) now plug in numbers:

wavelength = 2 * 375 * 1.33 / (m+1/2) =

                = 997.5 / (m+1/2)

    calculate wavelength for each m until you find all wavelengths between 380 and 750

     m        wavelength

      0         1995

      1           665

      2            399

      3          285

So there are two answers within the given range:    665 nm and 399 nm

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