Question

Establish the refraction formula for a thin lens:
$\frac{1}{f}=(\mu -1)(\frac{1}{R_1}-\frac{1}{R_2})$.

Answer 1

Formula for refraction through spherical surface (lens maker's formula).
$\frac{\mu }{v}-\frac{1}{u}=\frac{\mu -1}{R}$ ........(1)
Path of ray, directions and distances are shown in Fig. Let radii of curvature of lens $L$ are $R_1$ and $R_2$. For the surface $AB$ of the radius of curvature $R_1$ of lens $L$, the object is $O$ and $I^{{}^{\prime }}$ is the image.
Therefore on putting values of distance in equation (1) as per Fig.,
$\frac{\mu }{v^{{}^{\prime }}}-\frac{1}{u}=\frac{\mu -1}{R_1}$ ........(2)
[Here $\mu$ is the refractive index of glass with respect to air]
For the surface $A^{{}^{\prime }}{B}^{{}^{\prime }}$ of radius of curvature $R_2$, $I^{{}^{\prime }}$ is the object and $I$ is the image. Substituting values in the formula of refraction,
$\frac{\left(1/\mu \right)}{v}-\frac{1}{v^{{}^{\prime }}}=\frac{\left(1/\mu \right)-1}{R_2}$
[Here $\frac{1}{\mu }$ is the refractive index of air with respect to glass]
Multiplying each term by $\mu$,
$\frac{1}{v}-\frac{\mu }{v^{{}^{\prime }}}=\frac{1-\mu }{R_2}$
or $\frac{1}{v}-\frac{\mu }{v^{{}^{\prime }}}=\frac{-(\mu -1)}{R_2}$ ......(3)
Adding equations (2) and (3),
$\frac{1}{v}-\frac{1}{u}=(\mu -1)[\frac{1}{R_1}-\frac{1}{R_2}]$ .......(4)
If the focal length of lens $L$ is $f$ then substituting $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$ is equation (4),
$\frac{1}{f}=(\mu -1)[\frac{1}{R_1}-\frac{1}{R_2}]$