Question

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

Answer 1

Considering refraction at first surface

$X,P,Y$ First image in formed at image distance $V_1$

$\frac{{\mu }_1}{-u}+\frac{{\mu }_2}{v_1}=\frac{{\mu }_2-{\mu }_1}{R_1}......\left(1\right)$

For second surface $X{P}_{2}Y$ object distance is $V_1$ image distance is $V$

$\frac{-{\mu }_2}{v_1}+\frac{{\mu }_1}{v}=\frac{{\mu }_1-{\mu }_2}{R_2}=\frac{{\mu }_2-{\mu }_1}{-R_2}.......\left(2\right)$

Adding $\left(1\right)$ & $\left(2\right)$

$\frac{1}{v}-\frac{1}{u}=\frac{({\mu }_2-{\mu }_1)}{{\mu }_1}[\frac{1}{R_1}-\frac{1}{R_2}]$

But $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$ $;\frac{{\mu }_2}{{\mu }_1}=\mu$

$\frac{1}{f}=(\mu -1)[\frac{1}{R_1}-\frac{1}{R_2}]$