A paraxial beam is incident on a glass (n = 1 | vedclass

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(D) Step $1$: Refraction at the spherical surface.Using the refraction formula at a spherical surface: $\frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2 -

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$8$

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(D) Step $1$: Refraction at the spherical surface.Using the refraction formula at a spherical surface: $\frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$.Here,$n_1 = 1$ (air),$n_2 = 1.5$ (glass),$u = -\infty$,and $R = +6\, cm$.$\frac{1.5}{v_1} - \frac{1}{-\infty} = \frac{1.5 - 1}{6} \implies \frac{1.5}{v_1} = \frac{0.5}{6}$.$v_1 = \frac{1.5 	imes 6}{0.5} = 18\, cm$.This means the rays would converge at $18\, cm$ from the pole of the spherical surface.Step $2$: Refraction at the plane surface.The rays are traveling from glass $(n_2 = 1.5)$ to air $(n_1 = 1)$.The distance of the virtual object from the plane surface is $d = v_1 - R = 18 - 6 = 12\, cm$.Using the apparent depth formula for refraction from a denser to a rarer medium: $v = d 	imes \frac{n_{air}}{n_{glass}}$.$v = 12 	imes \frac{1}{1.5} = 12 	imes \frac{2}{3} = 8\, cm$.Thus,the final distance of the point of convergence $F$ from the plane surface is $8\, cm$.

$A$ paraxial beam is incident on a glass $(n = 1.5)$ hemisphere of radius $R = 6\, cm$ in air as shown. The distance of the point of convergence $F$ from the plane surface of the hemisphere is......$cm$.

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