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Q.

A narrow parallel beam of light is incident on a transparent sphere of refractive index $\frac{4}{n} .$ If the beam finally gets focused at a point situated at a distance = 2 $\times$ (radius of sphere) from the centre of the sphere, then find n.

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answer is 3.

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Detailed Solution

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from first reaction:

$\frac{μ_{2}}{v} - \frac{μ_{1}}{u} = \frac{μ_{2} - μ_{1}}{R},$

$\frac{n}{v} - \frac{1}{\infty} = \frac{μ - 1}{R} \Rightarrow \frac{μ}{v} = \frac{μ - 1}{R}$

$v = \frac{μR}{μ - 1}$

For $2^{n d}$ refraction:

$u' = \frac{μR}{μ - 1} - 2 R = \frac{(2 - μ) R}{(μ - 1)}, and v = R$

For second surface

$\frac{μ_{2}}{v} - \frac{μ_{1}}{u} = \frac{μ_{2} - μ_{1}}{R}$

$\frac{1}{R} - \frac{μ}{u'} = \frac{1 - μ}{- R}$

$\Rightarrow \frac{1}{R} - \frac{μ (μ - 1)}{(2 - μ) R} = \frac{μ - 1}{R}$

$\Rightarrow μ = \frac{4}{3} = \frac{4}{n}$

$\Rightarrow n = 3$

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