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Question: When a plastic thin film of refractive index 1.45 is placed in the path of one of the interfering wa...

When a plastic thin film of refractive index 1.45 is placed in the path of one of the interfering waves then the central fringe is displaced through a width of five fringes. The thickness of the film, if the wavelength of light is 5890  A˙5890\;{\rm{\dot A}}, will be:
A. 6.554×10−4  cm6.554 \times {10^{ - 4}}\;{\rm{cm}}
B. 6.554×10−4  m6.554 \times {10^{ - 4}}\;{\rm{m}}
C. 6.54×10−4  cm6.54 \times {10^{ - 4}}\;{\rm{cm}}
D. 6.5×10−4  cm6.5 \times {10^{ - 4}}\;{\rm{cm}}

Explanation

Solution

For the calculation of the thickness of the film, use the expression of path difference and rearrange it according to the desired result. The expression of path difference is,
Xo=nλ(μ−1)t{X_o} = \dfrac{n}{\lambda }\left( {\mu - 1} \right)t

Complete step by step answer:
Given:
The refractive index of the thin film is μ=1.45\mu = 1.45.
The displacement of the central fringe is Xo=5×fringe  width{X_o} = 5 \times {\rm{fringe}}\;{\rm{width}}.
The wavelength of the light is λ=5890  A˙\lambda = 5890\;{\rm{\dot A}}.
Write the expression of the path difference and explain each term.
Xo=nλ(μ−1)t{X_o} = \dfrac{n}{\lambda }\left( {\mu - 1} \right)t
Here, Xo{X_o} is the displacement of the central fringe, nn is the width of the fringe, μ\mu is the refractive index, λ\lambda is the wavelength and tt is the time.
We will rearrange the above equation to obtain the expression of thickness.
t=Xoλn(μ−1)t = \dfrac{{{X_o}\lambda }}{{n\left( {\mu - 1} \right)}}
Substitute the values in the above equation.
Therefore, we get
t=5n×5890  A˙×10−10  m1  A˙n(1.45−1)   ⟹  t=6.544×10−6  m×100  cm1  m   ⟹  t=6.544×10−4  cmt = \dfrac{{5n \times 5890\;{\rm{\dot A}} \times \dfrac{{{{10}^{ - 10}}\;{\rm{m}}}}{{1\;{\rm{\dot A}}}}}}{{n\left( {1.45 - 1} \right)}}\\\ \implies t = 6.544 \times {10^{ - 6}}\;{\rm{m}} \times \dfrac{{100\;{\rm{cm}}}}{{1\;{\rm{m}}}}\\\ \implies t = 6.544 \times {10^{ - 4}}\;{\rm{cm}}

So, the correct answer is “Option A”.

Note:
In these types of questions, always remember the expression of path difference and rearrange this equation according to the demand of the question. Like here, in this question, we need to determine the thickness of the film, so we rearranged the path difference expression according to the need of the question. The calculation and solution become easy and clear when we rearrange the equation.