Derive th lens maker's formula.

Explanation

Let `r_(1)` and `r_(2)` be radii of curvature of the thin lens.
Let O be the point object at a distance u from the pole of the first curved surface of the lens. Real image is formed at `I^(1)`, at a distance `v^(1)` from the pole P. This image is formed in the denser medium.
We know that from the refraction formula,
`(n_(1))/(-u) + (n_(2))/(v) = (n_(2) - n_(1))/(r)`
i.e., `(n_(1))/(-u) + (n_(2))/(v^(1)) = (n_(2) - n_(1))/(r_(1))` ....(1)
Where `n_(1)` is the refractive index of rarer medium and `n_(2)` that of the denser medium `(n_(2) gt n_(1))`. For the second surface, real image at `I^(1)` will serve as a virtual object and the distance is taken as -ve. The object space is the lens medium for refraction through the second curved surface. Final image is formed in air I and at a distance of .v. from Q.
`(n_(1))/(-u) + (n_(2))/(v) = (n_(2) - n_(1))/(-r_(2))` ...(2)
Adding (1) and (2) we get
`(n_(1))/(-u) + (n_(1))/(v) = (n_(2) - n_(1))((1)/(r_(1)) - (1)/(r_(2)))`
Or `(1)/(-u) + (1)/(v) = (n_(2) - n_(1))/(n_(1)) ((1)/(r_(1)) - (1)/(r_(2)))` ......(3)
When `u = oo, v = f`
The term on the L.H.S can replaced by 1/f where f is the focal length of lens
i.e., `(1)/(f) = [(n_(2) - n_(1))/(n_(1))]((1)/(r_(1)) - (1)/(r_(2))) = (n_(21) - 1)((1)/(r_(1)) - (1)/(r_(2)))`
`(1)/(f) = (n_(21) -1)((1)/(r_(1)) - (1)/(r_(2)))` .......(4)
The equation (4) is called the lens maker.s formula.