In an experiment for determination of the foc | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given: $u = 10 \pm 0.1 	ext{ cm}$,$v = 20 \pm 0.2 	ext{ cm}$.Using the lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$.For a convex lens

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(A) Given: $u = 10 \pm 0.1 	ext{ cm}$,$v = 20 \pm 0.2 	ext{ cm}$.Using the lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$.For a convex lens,$u$ is negative,so $\frac{1}{f} = \frac{1}{v} - \frac{1}{-u} = \frac{1}{v} + \frac{1}{u}$.Calculating $f$: $\frac{1}{f} = \frac{1}{20} + \frac{1}{10} = \frac{1+2}{20} = \frac{3}{20} \implies f = \frac{20}{3} 	ext{ cm}$.Differentiating the lens formula: $-\frac{df}{f^2} = -\frac{dv}{v^2} - \frac{du}{u^2} \implies \frac{df}{f^2} = \frac{dv}{v^2} + \frac{du}{u^2}$.Substituting the values: $\frac{df}{f^2} = \frac{0.2}{(20)^2} + \frac{0.1}{(10)^2} = \frac{0.2}{400} + \frac{0.1}{100} = \frac{0.2 + 0.4}{400} = \frac{0.6}{400}$.Relative error $\frac{df}{f} = f^2 	imes \frac{0.6}{400} = \left(\frac{20}{3}ight)^2 	imes \frac{0.6}{400} = \frac{400}{9} 	imes \frac{0.6}{400} = \frac{0.6}{9} = \frac{6}{90} = \frac{1}{15}$.Wait,re-evaluating the derivative: $\frac{df}{f^2} = \frac{dv}{v^2} + \frac{du}{u^2} \implies df = f^2 \left( \frac{dv}{v^2} + \frac{du}{u^2} ight)$.Percentage error = $\frac{df}{f} 	imes 100 = f \left( \frac{dv}{v^2} + \frac{du}{u^2} ight) 	imes 100 = \frac{20}{3} \left( \frac{0.2}{400} + \frac{0.1}{100} ight) 	imes 100 = \frac{20}{3} \left( \frac{0.2 + 0.4}{400} ight) 	imes 100 = \frac{20}{3} 	imes \frac{0.6}{400} 	imes 100 = \frac{20 	imes 0.6}{3 	imes 4} = \frac{12}{12} = 1 \%$.Therefore,$n = 1$.

Similar Questions

The temperatures of two bodies measured by a thermometer are $t_1 = 20^{\circ}C \pm 0.5^{\circ}C$ and $t_2 = 50^{\circ}C \pm 0.5^{\circ}C$. The temperature difference and the error therein are:

The random error in the arithmetic mean of $100$ observations is $x$; then the random error in the arithmetic mean of $400$ observations would be

The density of a cube is measured by measuring its mass and length of its sides. The percentage error in the measurement of mass and length are $5 \%$ and $6 \%$ respectively. The percentage error in the measurement of density is (in $\%$)

$A$ force '$F$' is applied on a square plate of side '$L$'. If the percentage error in determining '$L$' is $3\%$ and that in '$F$' is $2\%$, then the percentage error in determining the pressure is: (in $\%$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module