91Ó°ÊÓ

Radius of circular film $r=1.8\text{cm}$

Index of refraction of film ${\text{n}}_2\text{=1.4}$

Wavelength of light $\mathrm{λ}=550nm$

The thin liquid film is a phase of small thickness, in which the two interfacial layers overlap to form a unified non-homogeneous structure of specific properties.

In the figure at $t=0$, intensity is minimum and again at $t=\left(6\right)\left(2\text{s}\right)\text{= 12s}$

it is minimum.

The change in time from one minimum to next minimum

$\mathrm{Δ}t=12\text{s - 0}\text{s}$

But we have condition for minima

$\begin{array}{rcl}2L& =& m\frac{\mathrm{λ}}{n_2}\\ L& =& m\frac{\mathrm{λ}}{2n_2}\end{array}$

Change in thickness from one minimum to next minimum is

$\mathrm{Δ}L=\mathrm{Δ}m\frac{\mathrm{λ}}{2n_2}$

Here, $\mathrm{Δ}m=1$

Therefore $\mathrm{Δ}L=\frac{\mathrm{λ}}{2n_2}$

But change in volume $\mathrm{Δ}v=\left(\mathrm{π}{r}^2\right)\mathrm{Δ}L$

(Since the film is circular)

$\mathrm{Δ}L=\frac{\mathrm{λ}}{2n_2}$

Rate of change of volume $\frac{dv}{dt}=\frac{\mathrm{π}{r}^2\mathrm{λ}}{2n_2\mathrm{Δ}t}$

Given radius of circular film $r=1.8\text{cm}$

$$\begin{gathered} =\left(1.8\right)\left({10}^{-2}\text{m/cm}\right) \\ =0.018\text{m} \end{gathered}$$

Index of refraction of film $n_2=1.4$

Wavelength of light

$$\begin{gathered} =\text{550}\text{nm} \\ =\left(\text{550}\text{nm}\right)\left({10}^{-9}\text{m/nm}\right) \\ =550×{10}^{-9} \end{gathered}$$

Rate of change of volume:

$\begin{array}{rcl}\frac{dv}{dt}& =& \frac{\left(\mathrm{π}\right){\left(0.018\text{m}\right)}^2\left(\text{550}×{\text{10}}^{-9}\text{m}\right)}{\left(2\right)\left(1.4\right)\left(12s\right)}\\ & =& 0.0166684×{10}^{-9}{\text{m}}^3/\text{s}\\ & =& \text{1.67}×{\text{10}}^{-11}{\text{m}}^3/\text{s}\end{array}$

Therefore, the rate is $\text{1.67}×{\text{10}}^{-11}{\text{m}}^3/\text{s}$.