A narrow parallel beam of light is incident paraxially on a solid transparent sphere of radius R . What should be the refractive index if the beam is to be focused at the surface of the sphere,

For refraction at a spherical surface,

$$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$$

For the first surface of the sphere,  $$n_1=1,\qquad n_2=n$$

The incident beam is parallel to the principal axis, so  $$u=-\infty$$

For the rays to be focused at the surface of the sphere, the image is formed at the second surface. Since the distance between the two surfaces is $$2R$$,  $$v=2R$$

Substituting in the refraction formula,

$$\frac{n}{2R}-\frac{1}{-\infty}=\frac{n-1}{R}$$

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

$$\frac{n}{2}=n-1$$

$$n=2n-2$$

$$n=2$$

Hence, the correct option is B.