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  • Percentage error in the measurement of rho, length and ra is 1%, 2% and 3% respectively. Then the percentage error in the measurement of R i

The resistance R of a wire is given by the relation R=ρl / πr2. Percentage error in the measurement of ρ,l and r is 1 % , 2% and 3% respectively. Then the percentage error in the measurement of R is

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\\\text{Resistance of a wire } R=\frac{pl}{\pi r^{2}} \ \ .....(i)\\ \text{Take log to base e on both sides of equation } (i), \text{ we get } \\ \ln R=\ln p+\ln l-\ln \pi-2\ln r \ \ .....(ii) \\ \begin{aligned} &\text { Differentiate both sides of equation (ii), we get }\\ &\frac{\Delta R}{R}=\frac{\Delta R}{p}+2 \frac{\Delta l}{l}-2 \frac{\Delta r}{r} \ldots ... (iii)\end{aligned}

\begin{aligned} &\text { Errors are always added, (for maximum error) }\\ &\therefore \frac{\Delta R}{R} \times 100=\frac{\Delta P}{P} \times 100+\frac{\Delta l}{l} \times 100+\left|\frac{-2 \Delta r}{r} \times 100\right|\\ &=1 \%+2 \%+2(3 \%)=9 \% \end{aligned}

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