A thin transparent film with refractive index 1.4 is held on a circular ring of radius 1.8 cm. The fluid in the film evaporates such that transmission through the film at wavelength 560 nm goes to a minimum every 12 seconds. Assuming that the film is flat on its two sides, the rate of evaporation is:
The rate of evaporation is related to the thickness change that causes a shift in the interference pattern.
Using the given data and the wavelength for minimum transmission, the rate of evaporation can be calculated as: \[ {Rate of evaporation} = \pi \times 10^{-13} \, {m}^3/{s} \]
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: