An optical bench has 1.5 m long scale having four equal divisions… - Vidyadip

MCQ

An optical bench has $1.5 m$ long scale having four equal divisions in each $cm$. While measuring the focal length of a convex lens, the lens is kept at $75 cm$ mark of the scale and the object pin is kept at $45 cm$ mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at $135 cm$ mark. In this experiment, the percentage error in the measurement of the focal length of the lens is. . . . .

✓
$0.69$
B
$0.75$
C
$0.80$
D
$0.85$

✓

Answer

Correct option: A.

$0.69$

a
For the given lens

$u =-30 cm$

$v =60 cm$

$\& \frac{1}{ f }=\frac{1}{ v }-\frac{1}{ u }$ on solving $: f =20 cm$

also $\frac{1}{ f }=\frac{1}{ v }-\frac{1}{ u }$

on differentiation

$\frac{d f}{f^2}=\frac{d v}{v^2}+\frac{d u}{u^2}$

$\frac{d f}{f}=f\left[\frac{d v}{v^2}+\frac{d u}{u^2}\right]$

$\& \frac{d f}{f} \times 100=f\left[\frac{d v}{v^2}+\frac{d u}{u^2}\right] \times 100 \%$

$f=20 cm , d u=d v=\frac{1}{4} cm$

Since there are $4$ divisions in $1 cm$ on scale

$\therefore \frac{ df }{ f } \times 100=20\left[\frac{1 / 4}{(60)^2}+\frac{1 / 4}{(30)^2}\right] \times 100 \%$

$=5\left[\frac{1}{3600}+\frac{1}{900}\right] \times 100 \%$

$=5\left[\frac{5}{36}\right] \%=\frac{25}{36} \% \approx 0.69 \%$

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