91Ó°ÊÓ

Bright colours reflected from thin oil on water and soap bubbles are a consequence of light interference. Due to the constructive interference of light reflected from the front and back surfaces of the thin film, these bright colours can be seen. For a perpendicular incident beam, the maximum intensity of light from the thin film satisfies the condition:

$2\mathrm{L}=\left(\mathrm{m}+\frac{1}{2}\right)\frac{\mathrm{λ}}{{\mathrm{n}}_2}\text{ â¶Ä‰â¶Ä‰}\mathrm{m}=0,\text{ }1,\text{ }2,\mathrm{...}$(Maxima—bright film in the air)

$2\mathrm{L}=\mathrm{m}\frac{\mathrm{λ}}{{\mathrm{n}}_2}\text{ â¶Ä‰â¶Ä‰}\mathrm{m}=1,\text{ }2,\text{ }3,\mathrm{..}$(Minima)

where $\mathrm{λ}$is the wavelength of the light in air,$L$ is its thickness, and$n_2$ is the film’s refractive index.

Here, in this case, light travels in a medium with${\mathrm{n}}_1=1.32$ and incident on the thin layer whose refractive index is$n_2=1.75$ and the reflected light has$180°$ phase change as the light is reflected off the denser medium. And then, the refracted light gets reflected of the back surface${\mathrm{n}}_3=1.39$ while traveling through the film. This results in no phase change. The phase difference between$r_1$ and$r_2$ is $180°$. As a result, the condition for constructive interference or maximum intensity is

$2\mathrm{L}=\left(\mathrm{m}+\frac{1}{2}\right)\frac{\mathrm{λ}}{{\mathrm{n}}_2}$ (Constructive)

The 3^rd least ($m=2$) thickness of the thin layer is

$\begin{array}{c}\mathrm{L}=\left(2+\frac{1}{2}\right)\frac{382\text{ }\mathrm{nm}}{2(1.75)}\\ =273\text{ }\mathrm{nm}\end{array}$

Hence the 3rd least thickness of the thin layer is $273\text{ }nm$.