Correct Answer - Option 1 : sum of relative errors in the multipliers
CONCEPT:
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Error: It is the uncertainty in measurement by any measuring instrument.
Error in measurement can be broadly classified
- Systematic errors
- Random errors.
- The relative error is the ratio of mean absolute error to the mean value of the quantity measured.
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Relative error: The relative error is the ratio of mean absolute error to the mean value of the quantity measured.
EXPLANATION:
- When two quantities are multiplied or divided the relative error in the result is the sum of relative errors in the multipliers.
- When two quantities are added or subtracted absolute error in the final result is sum of the absolute error of the individual quantities.
- Let the two physical quantities A and B have measured value A ± Δ A and B ± Δ B, where Δ A and Δ B are the absolute errors of A and B respectively:
Let product Z is given by:
⇒ Z = AB (i)
⇒ Z ± Δ Z = (A ± Δ A) (B ± Δ B), where Δ Z is the maximum error in the product (ii)
⇒ Z ± Δ Z = A B ± Δ A B ± A Δ B ± Δ A Δ B (iii)
Dividing LHS by Z and RHS by A B in (iii)
\(⇒ Z ± \frac {Δ Z}{Z} = 1 ± \frac{Δ A} {A} ± \frac { Δ B }{B}±\frac { Δ A Δ B}{AB} \)
Since Δ A and Δ B are small we ignore them
\(\Rightarrow \frac{Δ Z}{Z} = ± \frac{Δ A} {A} ± \frac { Δ B }{B}\)
Hence option 1) is correct.
