In an experiment for determination of the focal length of a thin convex lens, the distance of the ob

Question

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 ________\

Professor PC Thomas Classes · Answer

(1) : Given, $u=-(10 \pm 0.1) \mathrm{cm}$, $$ v=(20 \pm 0.2) \mathrm{cm} $$ By lens equation :\ $\frac{1}{f}=\frac{1}{v}-\frac{1}{u} \Rightarrow \frac{1}{f}=\frac{1}{20}+\frac{1}{10}=\frac{1+2}{20}$\ $f=\frac{20}{3} \mathrm{~cm}$\ Now differentiate lens equation $$ \frac{-1}{v^{2}} d v+\frac{1}{u^{2}} d u=-\frac{1}{f^{2}} \cdot d f $$For errors :\ $\frac{1}{f^{2}} d f=\frac{1}{u^{2}} d u+\frac{1}{v^{2}} d v$\ $\frac{1}{f}\left(\frac{d f}{f}ight)=\left(\frac{1}{20^{2}} 	imes 0.2+\frac{1}{10^{2}} 	imes 0.1ight)$\ $\frac{3}{20} \cdot \frac{d f}{f}=\frac{0.2}{400}+\frac{0.1}{100}$\ $100 	imes \frac{d f}{f}=\left(\frac{0.2}{400}+\frac{0.1}{100}ight) 	imes \frac{20}{3} 	imes 100$\ % error in $f=\left(\frac{0.2}{4}+0.1ight) 	imes \frac{20}{3}=1 \%$\