The Mysterious Vanishing Soap Film · AI Generator #GEN-F79A95F5

Physics · fluid-mechanics

Problem

A researcher studying soap films creates a vertical rectangular film of width w=5.0  cmw = 5.0\;\text{cm} suspended between two parallel wires. Due to gravity, the film naturally thins from top to bottom, with thickness varying as h(y)=h0e−αyh(y) = h_0 e^{-\alpha y}, where yy is measured downward from the top, h0=800  nmh_0 = 800\;\text{nm} is the thickness at the top, and α=15  m−1\alpha = 15\;\text{m}^{-1}. The film appears brilliant green (λ=540  nm\lambda = 540\;\text{nm} in air) when viewed perpendicularly due to constructive interference in the reflected light. The refractive index of the soap solution is n=1.35n = 1.35. Sudddenly, the researcher observes that the green reflection completely vanishes at a specific height ycy_c from the top, creating a dark horizontal band. Below this critical height, the film appears to have a different color. (a) Find the critical height ycy_c where the green reflection first vanishes. (b) The film continues to thin and eventually ruptures. Just before rupture, the film thickness becomes so small that it appears completely dark (black) when viewed in reflection. At what thickness does this occur? Take the wavelength of green light in air as λ=540  nm\lambda = 540\;\text{nm} and assume the refractive index of air is nair=1.00n_{\text{air}} = 1.00.

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