Question

Two coherent sources 'P' and 'Q' produce interference at point 'A' on the screen, where there is a dark band which is formed between 4th and 5th bright band. Wavelength of light used is $6000\text{ \r{A}}$. The path difference PA and QA is

• A. $3.6 \times 10^{-4}\text{ cm}$
• B. $3.2 \times 10^{-4}\text{ cm}$
• C. $2.4 \times 10^{-4}\text{ cm}$
• D. $2.7 \times 10^{-4}\text{ cm}$

Hint:

To easily find the multiplier without formulas, just average the integer orders of the surrounding bright bands: the point halfway between the 4th bright ($4\lambda$) and 5th bright ($5\lambda$) must be exactly $4.5\lambda$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the spatial path difference ($\Delta x = |PA - QA|$) at a specific location on an interference fringe screen. The position is described as a dark fringe situated exactly between the 4th and 5th bright fringes.

Step 2: Key Formula or Approach:
The central bright band is designated as $n=0$.
The 4th bright band corresponds to a path difference of $4\lambda$.
The 5th bright band corresponds to a path difference of $5\lambda$.
The dark band lying directly between them is the

5th dark band. The general path difference formula for the $n$-th dark fringe is given by: $$\Delta x = \left(n - \frac{1}{2}\right)\lambda$$ Substituting $n = 5$ gives $\Delta x = 4.5\lambda$.

Step 3: Detailed Explanation:
Identify the parameters from the problem: Wavelength, $\lambda = 6000\text{ \r{A}} = 6000 \times 10^{-8}\text{ cm} = 6 \times 10^{-5}\text{ cm}$ Dark fringe index order, $n = 5$ Calculate the path difference: $$\Delta x = \left(5 - \frac{1}{2}\right)\lambda = 4.5\lambda$$ Substitute the value of $\lambda$ in centimeters to align with the options: $$\Delta x = 4.5 \times (6 \times 10^{-5}\text{ cm})$$ $$\Delta x = 27 \times 10^{-5} = 2.7 \times 10^{-4}\text{ cm}$$

Step 4: Final Answer:
The path difference between the two waves is $2.7 \times 10^{-4}\text{ cm}$, matching option (D).