The thickness at the centre of a plano-convex | vedclass

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(D) Given: Thickness $t = 3 \, mm = 0.3 \, cm$,Diameter $D = 6 \, cm$,so radius $r = 3 \, cm$. Speed of light in lens $v = 2 	imes 10^{8} \, m/s$. Sp

Vedclass · Accepted Answer

$30$

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(D) Given: Thickness $t = 3 \, mm = 0.3 \, cm$,Diameter $D = 6 \, cm$,so radius $r = 3 \, cm$. Speed of light in lens $v = 2 	imes 10^{8} \, m/s$. Speed of light in vacuum $c = 3 	imes 10^{8} \, m/s$.Refractive index $\mu = \frac{c}{v} = \frac{3 	imes 10^{8}}{2 	imes 10^{8}} = 1.5$.From the geometry of the lens,$R^{2} = r^{2} + (R - t)^{2}$,where $R$ is the radius of curvature.$R^{2} = r^{2} + R^{2} + t^{2} - 2Rt$.Neglecting $t^{2}$ as $t$ is very small,we get $2Rt = r^{2}$,so $R = \frac{r^{2}}{2t}$.$R = \frac{3^{2}}{2 	imes 0.3} = \frac{9}{0.6} = 15 \, cm$.Using the lens maker's formula: $\frac{1}{f} = (\mu - 1) \left( \frac{1}{R} - \frac{1}{\infty} ight) = \frac{\mu - 1}{R}$.$f = \frac{R}{\mu - 1} = \frac{15}{1.5 - 1} = \frac{15}{0.5} = 30 \, cm$.

The thickness at the centre of a plano-convex lens is $3 \, mm$ and the diameter is $6 \, cm$. If the speed of light in the material of the lens is $2 \times 10^{8} \, m/s$,the focal length of the lens is ............ $cm$.

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