Question

Write the formula for refraction of light on a spherical (concave or convex) surface. Establish lens formula $\dfrac{1}{f} = \left( {n - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$ with its help. Symbols used have their usual meaning.

Answer 1

Hint : The focal length of a lens depends upon the refractive index of the material of the lens and the radii of curvatures of the two surfaces. Then on using the formula for refraction at a single spherical surface we can find the lens maker’s formula.

Formula Used: The formulae used in the solution are given here.
$\dfrac{1}{f} = \left( {n - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$ where $f$ is the focal length of the mirror, $n$ is the refractive index, ${R_1}$ is the radius of curvature of the lens surface closer to the light source, ${R_2}$ is the radius of curvature of the lens surface farther from the light source.

Complete step by step answer
Refraction is the change in direction of light when it passes from one medium to another. The working of a lens is based on the refraction of light when they pass through it.
The refractive index of a lens is the relative measure, i.e. is the ratio of velocities of light in two mediums. It is dependent upon the relative refractive index of the surrounding medium where it is measured. More the refractive index, the more the light beam bends away from normal while travelling from high density medium to low density medium.
A spherical mirror has the shape of a section from the surface of a hollow sphere. If the inside surface of the mirror is polished, it is a concave mirror. If the outside is polished, it is a convex mirror.
Let us consider the thin lens shown in the image above with 2 refracting surfaces having the radii of curvatures ${R_1}$ and ${R_2}$ respectively. Let the refractive indices of the surrounding medium and the lens material be ${n_1}$ and ${n_2}$ respectively.
The complete derivation of the lens maker formula is described below. Using the formula for refraction at a single spherical surface we can say that,
For the first surface,
$\dfrac{{{n_2}}}{{{v_1}}} - \dfrac{{{n_1}}}{u} = \dfrac{{{n_2} - {n_1}}}{{{R_1}}}$ where radii of curvature is ${R_1}$ , and ${v_1}$ is the image distance from the first refracting surface, $u$ is the object distance.
For the second surface,
$\dfrac{{{n_1}}}{v} - \dfrac{{{n_2}}}{{{v_1}}} = \dfrac{{{n_1} - {n_2}}}{{{R_2}}}$ where radii of curvature is ${R_2}$ , and ${v_1}$ is the image distance from the first refracting surface, which is now acting as the object and $v$ is the final image distance.
Adding the two equations,
$\dfrac{{{n_1}}}{v} - \dfrac{{{n_1}}}{u} = \left( {{n_2} - {n_1}} \right)\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right]$
$\Rightarrow \dfrac{1}{v} - \dfrac{1}{u} = \left( {\dfrac{{{n_2}}}{{{n_1}}} - 1} \right)\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right]$
Now, let $u = \infty$ and $v = f$ ,
$\Rightarrow \dfrac{1}{f} = \left( {\dfrac{{{n_2}}}{{{n_1}}} - 1} \right)\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right]$
But, it is also known that,
$\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ , as evident from above.
$\therefore$ It can be said that, $\dfrac{1}{f} = \left( {n - 1} \right)\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right]$ holds true, when $n = \dfrac{{{n_2}}}{{{n_1}}}$ . This is known as the Lens Makers’ Formula.

Note
The Lens Makers’ Formula has a few limitations. The lens should not be thick so that the space between the 2 refracting surfaces can be small. The medium used on both sides of the lens should always be the same.