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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 ligh

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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 distance from the first surface is the diameter,$v = 2r$.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$$\mu = 2\mu - 2$$\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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