Question

The focal length of a plano-convex lens is equal to its radius of curvature. The value of the refractive index of its material is-

Answer 1

Hint: In case of lens usually there are two types used. Concave lens and convex lens. All will serve different purposes. Properties of different lenses are different. In the case of concave lenses they always form virtual images. While convex lenses form both virtual and real images. Inverted image in the sense it is real. Plano convex lenses have almost the same properties as convex lenses.

Formula used:
$\dfrac{1}{f} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$

Complete solution Step-by-Step:
Biconvex lens means the two surfaces of the lens are convex. There is a plano convex lens which means one side of the lens is planar and the other side of lens is convex.
Lens are the things which will allow the light to pass through them with convergence or divergence. Depending on whether the lens will converge or diverge the rays they are divided into convex and concave lenses. Their capacity of convergence or divergence is given by power of the lens.
Power of the lens is the inverse of focal length. We have a lens maker formula in order to determine the power and thus focal length of the lens.
$\dfrac{1}{f} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$
Where ‘f’ is the focal length and ‘${R_1}$’ is the radius of the first surface of the lens and ‘${R_2}$’ is the radius of second surface of the lens and ‘$\mu $’ is the refractive index of the lens.
For plano-convex lenses the ‘${R_1}$’ will be zero and ‘${R_2}$’ will be equal to ‘-R’.
So lens maker equation for plano convex lens will be
$\eqalign{
  & \dfrac{1}{f} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right) \cr
  & \Rightarrow \dfrac{1}{f} = \left( {\mu - 1} \right)\left( {\dfrac{1}{R}} \right) \cr
  & \Rightarrow \dfrac{1}{R} = \left( {\mu - 1} \right)\left( {\dfrac{1}{R}} \right) \cr
  & \therefore \mu = 2 \cr} $

Hence refractive index of the lens will be ‘2’

Note:
The lens refractive index can be varied in such a way that a convex lens can behave as the concave lens without any change in shape. This is when we place the lens in the medium of refractive index greater than the lens refractive index, then the convex lens would change into concave while the concave lens would change into convex.