Arrives at lens equation from lens maker's formula .

Explanation

(i) Consider a thin lens made up of a medium of refractive index `n_2` is placed in a medium of refractive index `n_1`. Let `R_1` and `R_2` be the radii of curvature of two spherical surfaces (1) and (2) respectively and P be the pole.
(ii) Consider a point object O on the principal axis. The ray which falk very close to P, after refraction at the surface (1) forms image at 1
(iii) Before it does so, it is again refracted by the surface (2). Therefore the final image is formed at 1.
(iv) The general equation of the refraction at a spherical surface is given from equation ,
`(n_2)/(v) - (n_1)/(mu) = ((n_2 - n_1))/(R)`
(v) For the refracting surface (1), the light goes from `n_1` to `n_2`
`(n_2)/(v) - (n_1)/(u) = ((n_2- n_1))/(R_1) " " .....(1)`
(vi) For the refracting surface (2), the light goes from medium `n_2` to `n_1`
` (n_1)/(v) - (n_2)/(v') = ((n_1 - n_2))/(R_2) " " ....(2)`
(vii) Adding the above two equations (1) and (2)
` (n_1)/(v) - (n_1)/(u) = (n_2 - n_1) ((1)/(R_1) - (1)/(R_2) )`
Further simplifying and rearranging ,
` 1/v - 1/u = ((n_2 - n_1))/(n_1) ((1)/(R_1) - (1)/(R_2))`
` 1/v - 1/u = ((n_2)/(n_1) - 1) ((1)/(R_1) - (1)/(R_2)) " " ...(3)`
(viii) If the object is at infinity, the image is formed at the focus of the lens. Thus, for ` u = oo , v =f ` .Then the equation becomes.
` 1/f - 1/oo = (n_2/n_1 - 1) ((1)/(R_1) - (1)/(R_2))`
` 1/f = (n_2/n_1 - 1) ((1)/(R_1) - (1)/(R_2)) " " ....(4)`
(ix) If the refractive index of the lens is `n_2` and it is placed in air, then` n_2` = n and `n_1` = 1. So the equation (4) becomes,
` 1/f = (n-1) ((1)/(R_1) - (1)/(R_2)) " " ...(5)`
The above equation is called the lens maker's formula, because it tells the lens manufactures what curvature is needed to make a lens of desired focal length with a material of particular refractive index.