Question

In a YDSE experiment, a transparent slab of refractive index 1.4 is placed in front of one slit. The fringe pattern shifts by 0.3 cm on the screen.
Given:
Screen distance = 60 cm
Slit separation = 1.5 mm
Thickness of the slab (in 碌m) is:

• A. 6
• B. 8
• C. 10
• D. 12

Hint:

The formula for fringe shift, $\Delta y = \frac{D}{d}(\mu - 1)t$, is crucial for YDSE problems involving thin films. Always ensure all units are consistent (preferably SI units) before calculation. Be aware that exam questions can sometimes contain typos; if your result is far from the options, re-check your calculations and consider if a simple typo (like a misplaced decimal or a wrong digit) could explain the discrepancy.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
In a Young's Double Slit Experiment (YDSE), introducing a transparent slab in the path of one of the beams introduces an extra path difference, causing the entire interference pattern to shift. We need to find the thickness of the slab given the shift and other parameters.
Step 2: Key Formula or Approach:
The path difference introduced by a slab of thickness 't' and refractive index '碌' is given by $\Delta x = (\mu - 1)t$.
The shift of the fringe pattern on the screen, $\Delta y$, is related to this path difference by:
\[ \Delta y = \frac{D}{d} \Delta x = \frac{D}{d}(\mu - 1)t \] where $D$ is the screen distance and $d$ is the slit separation.
Step 3: Detailed Explanation:
Given values:
Fringe shift, $\Delta y = 0.3 \text{ cm} = 0.3 \times 10^{-2} \text{ m}$
Refractive index, $\mu = 1.4$
Screen distance, $D = 60 \text{ cm} = 0.6 \text{ m}$
Slit separation, $d = 1.5 \text{ mm} = 1.5 \times 10^{-3} \text{ m}$
Let's assume the slit separation was intended to be $d=0.5$ mm ($0.5 \times 10^{-3}$ m).
Rearranging the formula to solve for thickness 't':
\[ t = \frac{\Delta y \cdot d}{D(\mu - 1)} \] Substitute the values (with the assumed correction for d):
\[ t = \frac{(0.3 \times 10^{-2} \text{ m}) \cdot (0.5 \times 10^{-3} \text{ m})}{(0.6 \text{ m})(1.4 - 1)} \] \[ t = \frac{0.15 \times 10^{-5}}{0.6 \times 0.4} = \frac{0.15 \times 10^{-5}}{0.24} = 0.625 \times 10^{-5} \text{ m} \] To convert the thickness to micrometers (碌m), multiply by $10^6$:
\[ t = 0.625 \times 10^{-5} \times 10^6 \text{ 碌m} = 6.25 \text{ 碌m} \] This value is very close to 6 碌m.
Step 4: Final Answer:
Based on the assumption of a typo in the slit separation, the thickness of the slab is approximately 6 碌m.