Question

In a biprism experiment, fifth dark fringe is obtained at a point. A thin transparent film of refractive index '\(μ\)' is placed in one of the interfering paths. Now \(7^{th}\) bright fringe is obtained at the same point. If '\(λ\)' is the wavelength of light used, the thickness of film is equal to

• A. \(1.5(μ-1)λ\)
• B. \(\frac{1.5\,λ}{(μ-1)}\)
• C. \(2.5(μ-1)λ\)
• D. \(\frac{2.5\,λ}{(μ-1)}\)

Hint:

Fifth dark fringe has path difference 4.5 lambda. Seventh bright fringe has 7 lambda. The film adds (mu - 1) t.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In a biprism (or double slit) experiment, a point is dark when the path difference is an odd multiple of \(\lambda/2\), and bright when it is a whole multiple of \(\lambda\). A thin film of index \(\mu\) and thickness \(t\) adds an extra optical path of \((\mu - 1)t\) in the beam it covers.

Step 2: Key Formula or Approach:
1. \(n\)th dark fringe: \(\Delta = (2n - 1)\dfrac\lambda2\).
2. \(n\)th bright fringe: \(\Delta = n\lambda\).

Step 3: Detailed Explanation:
Before the film, the point has the fifth dark fringe:
\[ \Delta_1 = (2\times5 - 1)\frac\lambda2 = \frac{9\lambda}{2} = 4.5\lambda \]
After the film, the same point has the seventh bright fringe:
\[ \Delta_2 = 7\lambda \]
The film changes the path difference by
\[ (\mu - 1)t = 7\lambda - 4.5\lambda = 2.5\lambda \]
\[ t = \frac{2.5\lambda}{\mu - 1} \]
Options (A) and (C) have \(t\) proportional to \((\mu - 1)\), which has the wrong dependence: a thicker film needs a lower index, not a higher one, to produce the same shift. Option (B) 1.5 would result from \(7 - 5.5\).

Final Answer:
The thickness of the film is \(\dfrac{2.5\lambda}{\mu - 1}\), option (D). \[ \boxed{\frac{2.5\lambda}{\mu-1} \text{ (D)}} \]