Question

A physical quantity P is given as
\( P = \frac{a^2b^2 }{ c\sqrt{d}} .\)
The percentage error in the measurement of a,b, c and d are1%, 2%, 3% and 4% respectively. The percentage error in the measurement of quantity P  will be

• A. 11%
• B. 12%
• C. 9%
• D. 13%

Hint:

Use the formula for propagation of errors to find the percentage error in P. The percentage error in xn is n times the percentage error in x. The percentage error in √x is half the percentage error in x.

Answer 1

The Correct Option is D

Calculation of Percentage Change in \( P \):

Formula:

The percentage change in \( P \) is given by: \[ \frac{\Delta P}{P} \times 100 = 2 \frac{\Delta a}{a} + 3 \frac{\Delta b}{b} + \frac{\Delta c}{c} + \frac{1}{2} \frac{\Delta d}{d} \]

Step 1: Substitute the Given Values:

Substituting the provided values for the terms:

• \( 2 \frac{\Delta a}{a} = 2 \times 1 = 2 \)
• \( 3 \frac{\Delta b}{b} = 3 \times 2 = 6 \)
• \( \frac{\Delta c}{c} = 3 \)
• \( \frac{1}{2} \frac{\Delta d}{d} = \frac{1}{2} \times 4 = 2 \)

Step 2: Add the Results:

\[ \frac{\Delta P}{P} \times 100 = 2 + 6 + 3 + 2 = 13\% \]

Conclusion:

The percentage change in \( P \) is 13%.