Derive th lens maker's formula.

Explanation

Lens maker.s formula

Let `r_1` and `r_2` be the 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., at a distance from the pole `P_1`. This image is formed in the denser medium.
We know that from the refraction formula,
`n_1/(-u)+u_2/v =(n_2-n_1)/r`
i.e., `n_1/(-u)+n_2/(v.)=(n_2-n_1)/r_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 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 at I and at a distance of .d from `P_2`
`n_1/(-u)+v_2/v = (n_2-n_1)/r`
i.e., `n_2/(v.)+v_1/v = (n_2-n_1)/(-r_2)` ....(ii)
Adding (i) and (ii), we get
`n_1/(-u)+n_1/v=(n_2-n_1)(1/r_1+1/r_2)`...(iii)
or `1/(-u)+1/v=(n_2-n_1)/n_1 (1/r_1-1/r_2)`
when `u=oo`, v=f
when u=f, v=`oo`
`therefore` The terms on the L.H.S. can be 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)`
The equation (iv) is called the lens maker.s formula .