JEE PYQ: Units & Measurements - Question ID 24b2da39dc07 (JEE Main 2024)

Core Formula & Concept:

In experiments, every measured quantity carries an uncertainty (error). When quantities are combined through mathematical operations (like division, multiplication, etc.), their errors propagate. The key concept here is error propagation in division.

Given:

• Resistance $R = \frac{V}{I}$, where $V$ is voltage and $I$ is current.
• $V = (200 \pm 5) \text{ V}$, so the absolute error in $V$ is $\Delta V = 5 \text{ V}$.
• $I = (20 \pm 0.2) \text{ A}$, so the absolute error in $I$ is $\Delta I = 0.2 \text{ A}$.

The percentage error in a quantity $X$ is defined as: $\text{Percentage Error} = \left( \frac{\Delta X}{X} \right) \times 100\%$

For a quotient $R = \frac{V}{I}$, the relative error (fractional error) in $R$ is given by: $\frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I}$ This is because errors in division add up in quadrature for independent measurements, but in basic error propagation (for small errors), we approximate by adding the relative errors.

Step-by-Step Derivation:

Step 1: Compute the nominal value of $R$.
$R = \frac{V}{I} = \frac{200 \text{ V}}{20 \text{ A}} = 10 \text{ } \Omega$

Step 2: Compute the relative errors in $V$ and $I$.
$\frac{\Delta V}{V} = \frac{5}{200} = 0.025 \quad (2.5\%)$ $\frac{\Delta I}{I} = \frac{0.2}{20} = 0.01 \quad (1\%)$

Step 3: Compute the relative error in $R$.
Since $R = \frac{V}{I}$, the relative errors add: $\frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I} = 0.025 + 0.01 = 0.035$

Step 4: Convert the relative error to percentage error.
$\text{Percentage Error in } R = \left( \frac{\Delta R}{R} \right) \times 100\% = 0.035 \times 100\% = 3.5\%$

Conclusion: The percentage error in $R$ is $3.5\%$, which corresponds to option D.

Common Traps & Exam Tip:

Trap 1: Adding absolute errors instead of relative errors.
Students often mistakenly add $\Delta V$ and $\Delta I$ directly, leading to incorrect results like $\Delta R = 5 + 0.2 = 5.2$, which is meaningless. Always convert to relative errors first.

Trap 2: Using the product rule for division.
Some students confuse the error propagation for division with multiplication. For $R = \frac{V}{I}$, the relative errors add, not multiply or subtract.

Trap 3: Ignoring significant figures.
The given errors ($\pm 5$ and $\pm 0.2$) have one significant figure, so the final percentage error should be reported to one decimal place (3.5%, not 3.50%).

Exam Tip:
For error propagation in division or multiplication, always:

1. Compute the relative errors ($\frac{\Delta X}{X}$) for each quantity.
2. Add the relative errors for division/multiplication.
3. Convert the final relative error to percentage error.

This method ensures accuracy and avoids common pitfalls.