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
\( P = \frac{a^2b^2 }{ c\sqrt{d}} .\)
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
Show 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.
- 11%
- 12%
- 9%
- 13%
The Correct Option is D
Solution and Explanation
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%.
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