Question

A narrow pencil of parallel light is incident normally on a- solid transparent sphere of radius r. What should be the refractive index if the pencil is to he focused (a) at the surface of the sphere, (b) at the centre of the sphere.

Answer 1

Given :

We should consider two cases:

For refraction at first surface A:

u= -∞

μ1=1

μ2=?

A) when focused on surface V=2r

R=r

μ2/V- μ1/u= μ2-μ1/R

μ2/2r = μ2-1/r

μ2= 2μ2-2

μ2= 2

B) When focused at centre  

u= r1

R=r

μ2/V - μ1/u= μ2-μ1/R

μ2/R = μ2-1/r

μ2= μ2-1

Which is not possible

hence it cannot focus at centre

Answer 2

The refractive index if the pencil is to focus at the surface of the sphere.

Explanation:

Given data in the question  :

There are two situations which we should consider:

For first surface A refraction:

u = -∞

$\mu 1$ = 1

$\mu 2$ = ?

(A) When focused at the surface of the sphere :

          V = 2 r

          R = r

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

        $\frac{\mu_{2}}{2 r}=\frac{\mu_{2}-1}{r}$

         $\mu_{2}=2 \mu_{2}-2$

$2 \mu_{2}-\mu_{2}=2$        

          $\mu_{2}=2$

(B) Once focused at the centre  of the sphere :

          u =  $r_1$

          R = r

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

         $\frac{\mu_{2}}{R}=\frac{\mu_{2}-1}{R}$

       $\mu_{2} R=R\left(\mu_{2}-1\right)$      

          $\mu_{2}=\left(\mu_{2}-1\right)$

Which could not be achieved

So, it cannot concentrate in the center .