Derive th lens maker's formula.

Explanation

Consider a thin convex lens placed in a medium of refractive index `n_1`. Let `n_2` be the refractive index of the material of the lens.
Let O be the a point luminous object placed on the principle axis at a distance u from its optic centre as shown in fig. A ray OP incident along the principle axis proceeds undeviated. Another paraxial ray incident on the lens along OM is refracted along MN and emerges along NI. The emergent ray meet at I on the principle axis.I is the real image of object O.
The formation of image I can be considered in two stages.
1. Refraction at surface APB: In the absence of the second surface AQB, the refracted rays MN and PQ intersect at `I^1` in the medium of refractive index `n_1`. `I_1` is real image of object O formed due to refraction at the surface APB. For this refraction from rarer to denser medium. Object distance =OP=u. Image distance `=PI^1=v^1`
`therefore n_2/(v.)=n_1/u=(n_2-n_1)/R_1` ....(1)
Refraction at a surface AQB: Rays MN and PQ incident on the surface AQB undergo refraction and emerge out of the lens. The emergent rays intersect at I on the principal axis `I^1` acts as virtual object for the refraction at AQB and its real image is I. The refraction is from denser to rarer medium. For the refraction
Object distance = `QI^1=v^1`
Image distance =QI=v.
i.e., `n_1/v-n_2/(v.)=-((n_2-n_2)/R_2)` ...(2)
Adding equations (1) and (2)
`n_1/v-n_1/u=(n_2-n_1)/R_1-(n_2-n_1)/R_2=(n_2-n_1){1/R_1-1/R_2}`
Dividing throughout by `n_1`
`1/v-1/u=(n_2-n_1)/n_1{1/R_1-1/R_2} =(n_2/n_1-1){1/R_1-1/R_2}`
i.e., `1/v-1/u=(n_21-1) {1/R_1-1/R_2}` where `n_(21)=n_2/n_1`
If `u=oo` then the incident beam on the lens is a parallel beam. The final image will be formed at the principal focus . In this case, v=f.
`therefore 1/f=(n_21-1){1/R_1-1/R_2}`
This relation is known as lens maker.s formula. If the refractive index of the material of the lens is n and it is placed in air then
`1/f=(n-1){1/R_1-1/R_2}`