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:

$2\mathrm{L}=\left(\mathrm{m}+\frac{1}{2}\right)\frac{\mathrm{λ}}{{\mathrm{n}}_2}\text{ â¶Ä‰â¶Ä‰}\mathrm{m}=0,\text{ }1,\text{ }2,...$(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${\mathrm{n}}_1=1.60$ and incident on the thin layer whose refractive index is${\mathrm{n}}_2=1.40$ and the reflected light has no phase change as the light is reflected off the rarer medium. And then, the refracted light gets reflected of the third surface${\mathrm{n}}_3=1.80$ while traveling through the film. This results result in $180°$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 thickness of the thin layer is

$\begin{array}{c}\mathrm{L}=\left(1+\frac{1}{2}\right)\frac{632\text{ }\mathrm{nm}}{2\left(1.40\right)}\\ =339\text{ }\mathrm{nm}\end{array}$

Hence the 2nd least thickness of the thin layer is $339\text{ }nm$.