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In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10 \pm 0.1 \mathrm{~cm}$ and the distance of its real image from the lens is $20 \pm 0.2 \mathrm{~cm}$. The error in the determination of focal length of the lens is $n \%$. The value of $n$ is. . . . . . .
$1$
$5$
$7$
$10$
Solution
$u=10 \pm 0.1 \mathrm{~cm}, \quad v=20 \pm 0.2 \mathrm{~cm}$
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f} \Rightarrow \frac{1}{v^2} d v+\frac{1}{u^2} d u=-\frac{1}{f^2} d f$
$\frac{1}{20}+\frac{1}{10}=\frac{1}{f} \Rightarrow \frac{1}{f}=\frac{3}{20} \Rightarrow f=\frac{20}{3} \mathrm{~cm}$
$\Rightarrow \frac{1}{(20)^2}(0.2)+\frac{1}{(10)^2}(0.1)=\frac{9}{400} d f$
$\text { df }=\frac{1}{9}\left(\frac{400}{400} \times 0.2+\frac{400}{100} \times 0.1\right)$
$d f=\frac{1}{9}(0.2+0.4) \Rightarrow \mathrm{df}=\frac{0.6}{9}$
$\frac{\mathrm{df}}{\mathrm{f}}=\frac{0.6}{9} \times \frac{3}{20}=\frac{1}{100}$
$\% \text { error }=1 \%$
