JEE PYQ: Units & Measurements - Question ID 78daf4ca4eac (JEE Main 2024)

Core Formula & Concept:

To determine the refractive index ($n$) of a glass slab using a travelling microscope, we rely on the concept of apparent depth versus real depth. The refractive index is defined as:

$n = \frac{\text{Real depth}}{\text{Apparent depth}}$

The travelling microscope measures distances by combining a main scale (MS) and a vernier scale (VS). The least count (LC) of the vernier scale is crucial for precise measurements. The least count is calculated as:

$\text{LC} = \frac{\text{Value of 1 MSD}}{\text{Number of VS divisions}} = \frac{1 \text{ MSD}}{\text{VS divisions per MSD}}$

Given that 50 vernier divisions = 49 main scale divisions (MSD), and 20 MSD = 1 cm, we first compute the least count. Then, the total reading (TR) for any observation is:

$\text{TR} = \text{MSR} + (\text{VC} \times \text{LC})$

where MSR is the main scale reading and VC is the vernier coincidence.

Step-by-Step Derivation:

Step 1: Calculate the Least Count (LC) of the Vernier Scale

Given:

• 50 vernier scale divisions (VSD) = 49 main scale divisions (MSD)
• 20 MSD = 1 cm ⇒ 1 MSD = $\frac{1}{20}$ cm = 0.05 cm

The least count is:

$\text{LC} = \frac{1 \text{ MSD}}{50} = \frac{0.05 \text{ cm}}{50} = 0.001 \text{ cm}$

Step 2: Compute Total Readings for Each Observation

We have three observations:

1. Mark on paper (real depth reference):
   MSR = 8.45 cm, VC = 26
   TR₁ = 8.45 + (26 × 0.001) = 8.45 + 0.026 = 8.476 cm
2. Mark on paper seen through slab (apparent depth):
   MSR = 7.12 cm, VC = 41
   TR₂ = 7.12 + (41 × 0.001) = 7.12 + 0.041 = 7.161 cm
3. Powder particle on top surface of slab (reference for slab thickness):
   MSR = 4.05 cm, VC = 1
   TR₃ = 4.05 + (1 × 0.001) = 4.05 + 0.001 = 4.051 cm

Step 3: Determine Real and Apparent Depths

The real depth ($t$) of the slab is the distance from the mark on paper to the top surface of the slab:

$t = \text{TR}_1 - \text{TR}_3 = 8.476 - 4.051 = 4.425 \text{ cm}$

The apparent depth ($t'$) is the distance from the apparent position of the mark (seen through the slab) to the top surface:

$t' = \text{TR}_2 - \text{TR}_3 = 7.161 - 4.051 = 3.110 \text{ cm}$

Step 4: Calculate the Refractive Index

Using the refractive index formula:

$n = \frac{\text{Real depth}}{\text{Apparent depth}} = \frac{t}{t'} = \frac{4.425}{3.110} \approx 1.422$

Rounding to two decimal places, we get $n \approx 1.42$, which matches option D.

Common Traps & Exam Tip:

1. Incorrect Least Count Calculation: Students often confuse the relationship between vernier and main scale divisions. Ensure that 50 VSD = 49 MSD is correctly interpreted to find LC.
2. Misidentifying Real vs. Apparent Depth: The real depth is the actual thickness of the slab, while the apparent depth is the observed thickness through the slab. Mixing these up leads to incorrect $n$.
3. Ignoring Vernier Coincidence: Failing to add the vernier correction (VC × LC) to the MSR results in inaccurate total readings, affecting the final refractive index.
4. Rounding Errors: Premature rounding of intermediate values (e.g., TR₁, TR₂, TR₃) can propagate errors. Retain at least 3 decimal places until the final step.

Exam Tip: Always verify the least count first and double-check the order of subtractions for real/apparent depths. Cross-validate the final $n$ with the given options to catch calculation mistakes.