JEE PYQ: Units & Measurements - Question ID 1ef7854c8696 (JEE Main 2024)

Core Formula & Concept:

In a Vernier instrument (such as a traveling microscope), the Vernier constant (also called the least count) is the smallest length that can be measured accurately. It is defined as the difference between one main-scale division and one Vernier-scale division.

Key formula: $\text{Vernier constant} = \text{Value of one main-scale division} - \text{Value of one Vernier-scale division}$

If $n$ Vernier divisions coincide with $(n-1)$ main-scale divisions, then: $n \times \text{Vernier division} = (n-1) \times \text{Main-scale division}$ Rearranging gives: $\text{Vernier division} = \frac{(n-1)}{n} \times \text{Main-scale division}$ Hence, $\text{Vernier constant} = \text{Main-scale division} - \text{Vernier division} = \frac{1}{n} \times \text{Main-scale division}$

Step-by-Step Derivation:

Given:

• 50 Vernier divisions = 49 main-scale divisions
• One main-scale division = $0.5 \text{ mm}$

Step 1: Express the value of one Vernier division.
Since 50 Vernier divisions = 49 main-scale divisions, $50 \times \text{Vernier division} = 49 \times \text{Main-scale division}$ $\text{Vernier division} = \frac{49}{50} \times \text{Main-scale division}$

Step 2: Substitute the value of one main-scale division.
$\text{Vernier division} = \frac{49}{50} \times 0.5 \text{ mm} = \frac{49}{50} \times \frac{1}{2} \text{ mm} = \frac{49}{100} \text{ mm} = 0.49 \text{ mm}$

Step 3: Calculate the Vernier constant.
$\text{Vernier constant} = \text{Main-scale division} - \text{Vernier division} = 0.5 \text{ mm} - 0.49 \text{ mm} = 0.01 \text{ mm}$

Step 4: Match with the given options.
The Vernier constant is $0.01 \text{ mm}$, which corresponds to option A.

Common Traps & Exam Tip:

• Unit confusion: Students often mix up millimeters and centimeters. Always double-check the units in the question and options.
• Incorrect formula application: Some students mistakenly use the formula $\text{Vernier constant} = \frac{\text{Main-scale division}}{n}$ without understanding the underlying principle. Always derive it from first principles to avoid errors.
• Misinterpretation of divisions: Ensure that the number of Vernier divisions and main-scale divisions are correctly matched (here, 50 Vernier divisions = 49 main-scale divisions).

Exam Tip: When solving Vernier constant problems, always write down the given data clearly and derive the value step-by-step. This minimizes errors and ensures full marks.