JEE PYQ: Units & Measurements - Question ID 06c11ac25a42 (JEE Main 2020)

In experimental physics, when a derived quantity \( z \) is expressed as a product or quotient of several measured observables \( a, b, c, \dots \), the relative error (or percentage error) in \( z \) is determined using the propagation of errors formula. The key principle is:

• If \( z = k \cdot a^{x} \cdot b^{y} \cdot c^{z} \cdot d^{w} \) (where \( k \) is a dimensionless constant), then the relative error in \( z \) is: $\frac{\Delta z}{z} = |x| \frac{\Delta a}{a} + |y| \frac{\Delta b}{b} + |z| \frac{\Delta c}{c} + |w| \frac{\Delta d}{d}$
• The percentage error in \( z \) is then: $\% \text{ error in } z = \left( \frac{\Delta z}{z} \right) \times 100\%$

This formula arises from logarithmic differentiation, ensuring that errors in each observable contribute proportionally to their exponents.

Given the physical quantity:

Rewrite \( z \) in exponential form for clarity:

Let the percentage errors in \( a, b, c, d \) be:

• \( \frac{\Delta a}{a} \times 100\% = 2\% \) ⇒ \( \frac{\Delta a}{a} = 0.02 \)
• \( \frac{\Delta b}{b} \times 100\% = 1.5\% \) ⇒ \( \frac{\Delta b}{b} = 0.015 \)
• \( \frac{\Delta c}{c} \times 100\% = 4\% \) ⇒ \( \frac{\Delta c}{c} = 0.04 \)
• \( \frac{\Delta d}{d} \times 100\% = 2.5\% \) ⇒ \( \frac{\Delta d}{d} = 0.025 \)

Apply the error propagation formula:

Substitute the relative errors:

Compute each term:

• \( 2 \times 0.02 = 0.04 \)
• \( \frac{2}{3} \times 0.015 = 0.01 \)
• \( \frac{1}{2} \times 0.04 = 0.02 \)
• \( 3 \times 0.025 = 0.075 \)

Sum the contributions:

Convert to percentage error:

Thus, the correct option is B: 14.5%.

• Sign Confusion: Students often forget that exponents in the denominator (like \( \sqrt{c} = c^{-\frac{1}{2}} \)) contribute negatively to the exponent but positively to the error (due to absolute value). Always take the magnitude of the exponent.
• Fractional Exponents: Misapplying \( \frac{2}{3} \) as \( \frac{3}{2} \) or rounding it incorrectly (e.g., 0.666... vs 0.67) can lead to errors. Use exact fractions.
• Addition vs Multiplication: Errors are added, not multiplied. A common mistake is to multiply percentage errors, which is incorrect.
• Unit Consistency: While not directly relevant here, ensure all observables are in consistent units before applying error propagation in practical experiments.

Exam Tip: Always write down the formula for error propagation explicitly before substituting values. This avoids sign errors and ensures clarity.