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 at a distance u from the pole of the first curved surface of the lens . Real image is formed at I. at a distance v. from the pole `P_(1)` 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.)=(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 - ve . The object space is the lens me - dium for refractive through the second curved surface . final image is formed in air at I and at a distance of .v. from ` P_(2)`
` (n_1)/(-u)+(n_2)/(v)=(n_2-n_1)/(r )`
` i.e., (n_2)/(-v)+(n_1)/(v)=(n_(2)-n_(1))/(-r_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)))`
when `u= oo , v= f`
when `u=f , v=oo`
` therefore ` The term 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 (4) is called the lens maker.s formula
Note :(1) using the equation (4) , it can be shown that
`""_(1) N_(2) = 1 -[(r_1r_2)/(f(r_1-r_(2)))]` where ` ""_(1) N_(2) =(n_(2))/(n_(1))`
(2) for radius of curvature the letter .R. may be used .