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(B) For refraction at a spherical surface,the formula is given by: $\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}$.Here,the light travel

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$2$

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(B) For refraction at a spherical surface,the formula is given by: $\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}$.Here,the light travels from air $(\mu_1 = 1)$ to the sphere $(\mu_2 = \mu)$.The object is at infinity,so $u = -\infty$.The image is formed at the farther surface,so the image distance $v = 2r$ (the diameter of the sphere).The radius of curvature of the first surface is $R = +r$.Substituting these values into the formula:$\frac{\mu}{2r} - \frac{1}{-\infty} = \frac{\mu - 1}{r}$.Since $\frac{1}{\infty} = 0$,we get:$\frac{\mu}{2r} = \frac{\mu - 1}{r}$.Multiplying both sides by $r$:$\frac{\mu}{2} = \mu - 1$.Rearranging the terms:$1 = \mu - \frac{\mu}{2} = \frac{\mu}{2}$.Therefore,$\mu = 2$.

$A$ narrow parallel beam of light is incident paraxially on a solid transparent sphere of radius $r$ kept in air. What should be the refractive index if the beam is to be focused at the farther surface of the sphere?

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