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: The situation.
Two coherent sources make bright and dark bands on a screen. Point A holds a dark band lying right between the 4th and 5th bright bands. The light has wavelength $6000$ angstrom. We want the path difference of the two waves reaching A.
Step 2: How bands are numbered.
The 4th bright band has path difference $4\lambda$ and the 5th bright band has path difference $5\lambda$. The dark band sitting between them is the 5th dark band.
Step 3: The dark band rule.
For the $n$-th dark band the path difference is \[ \Delta x = \left(n - \frac{1}{2}\right)\lambda \]
Step 4: Put n = 5.
\[ \Delta x = \left(5 - \frac{1}{2}\right)\lambda = 4.5\,\lambda \]
Step 5: Convert the wavelength.
\[ \lambda = 6000\ \text{angstrom} = 6000 \times 10^{-8}\ \text{cm} = 6 \times 10^{-5}\ \text{cm} \]
Step 6: Compute the path difference.
\[ \Delta x = 4.5 \times 6 \times 10^{-5} = 27 \times 10^{-5} = 2.7 \times 10^{-4}\ \text{cm} \] This is option (4). \[ \boxed{\Delta x = 2.7 \times 10^{-4}\ \text{cm}} \]