Derivation - Lens Maker's Formula | Refraction by a lens - Param Himalaya
Derivation : Consider a lens made of a material of absolute refractive index $n_2$. This lens is placed in a medium of absolute refractive index $n_1$ ($n_1 < n_2$). The lens is bounded by two spherical refracting surfaces $XP_1Y$ and $XP_2Y$. $C_1$ and $C_2$ be their centres of curvature and $R_1$ and $R_2$ be their radii of curvature respectively. $C$ is the optical centre of the lens.
STEP 1. Refraction at Surface $XP_1Y$ : Let O be a point object lying in the rarer medium on the principal axis of the refracting surface $XP_1Y$. The incident ray OA after refraction at A bends towards the normal $AC_1$ and meets the principal axis at $I_1$ if the second surface $XP_2Y$ were not present. So, $I_1$ is the real image of the object O.
Since object lies in the rarer medium, so we have
$-\frac{n_1}{u} + \frac{n_2}{v_1} = \frac{n_2 - n_1}{R_1} \qquad ... (i)$
STEP 2. Refraction at Surface $XP_2Y$ :} In fact, the ray AB refracted by the first surface $XP_1Y$ is refracted at B by the second surface $XP_2Y$ and it finally meets the principal axis at I. The point $I_1$ acts as a virtual object for the spherical surface $XP_2Y$ (Figure 53). Now it is the situation, when object is placed in the denser medium and image is formed in the rarer medium.
$-\frac{n_2}{v_1} + \frac{n_1}{v} = \frac{n_1 - n_2}{-R_2} = \frac{n_2 - n_1}{R_2} \qquad ... (ii)$
STEP 3. Adding equations (i) and (ii), we get
$-\frac{n_1}{u} + \frac{n_1}{v} = (n_2 - n_1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
$-\frac{1}{u} + \frac{1}{v} = \left(\frac{n_2 - n_1}{n_1}\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
$-\frac{1}{u} + \frac{1}{v} = \left(\frac{n_2}{n_1} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
But $\frac{n_2}{n_1} = n$, i.e. relative refractive index of the lens with respect to the rarer medium (say air).
i.e.
$-\frac{1}{u} + \frac{1}{v} = (n - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
If the object is at infinity, the image is formed at the principal focus of the lens. That is, if $u = -\infty$, $v = f$. Hence eqn. (iii) becomes
$-\frac{1}{\infty} + \frac{1}{f} = (n - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
$0 + \frac{1}{f} = (n - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$
$\frac{1}{f} = (n - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \qquad ... (iii)$
Eqn. (iv) is known as Lens maker's formula.}
$\frac{1}{f} = (n - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \qquad ... (iv)$
Comparing eqns. (iii) and (iv), we get,
$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$
which is known as Lens formula.