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 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))" "` .......(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 .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))" "` .......(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`
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)))" " ` ......(4)
Note: (1) Using the equation (4), it can be shown that
`" "_(1)n_(2)=1-[(r_(1)r_(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.