Question

notes{derivations } on combination of errors

Answer 1

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Answer 2

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Answer:
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Combination of Errors:-

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(a) Error of a sum or a difference:
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When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Z = A + B

We have by addition, 

Z ± ΔZ = (A ± ΔA) + (B ± ΔB).

The maximum possible error in Z

ΔZ = ΔA + ΔB

For the difference Z = A – B, we have

Z ± Δ Z = (A ± ΔA) – (B ± ΔB) = (A – B) ± ΔA ± ΔB

or, ± ΔZ = ± ΔA ± ΔB

The maximum value of the error ΔZ is again ΔA + ΔB.

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(b) Error of a product or a quotient:
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When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

Suppose Z = AB and the measured values of A and B are A ± ΔA and B ± ΔB. Then

Z ± ΔZ = (A ± ΔA) (B ± ΔB) = AB ± B ΔA ± A ΔB ± ΔA ΔB.

Dividing LHS by Z and RHS by AB we have,

1 ± (ΔZ/Z) = 1 ± (ΔA/A) ± (ΔB/B) ± (ΔA/A)(ΔB/B).

Since ΔA and ΔB are small, we shall ignore their product.

Hence the maximum relative error

ΔZ/ Z = (ΔA/A) + (ΔB/B).

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(c) Error in case of a measured quantity raised to a power
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The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.

Suppose Z = A^2,

Then,

ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A).

Hence, the relative error in A^2 is two times the error in A.

In general, if Z = (Ap Bq)/Cr

Then,

ΔZ/Z = p (ΔA/A) + q (ΔB/B) + r (ΔC/C).


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