Derive lens maker's formula for a thin convex lens.

Explanation

Basic assumptions in derivtion of Lens - maker's formula:
(i) Aperture of lens should be small
(ii) Lenses should be thin
(iii) Object should be point sized and placed on principal axis.

Suppose we have a thin lens of material of refractive index `n_(2)`, placed in a medium of refractive index `n_(1)`, let o be a point object placed on principle axis then for refraction at surface ABC we get image at `I_(1)`
`therefore (n_(2))/(V^(1)), (n_(1))/(v) = (n_(2)-n_(1))/(R^(1))......(1)` But the refracted ray before goes to meet at `I_(1)` falls on surface ADC and refracts at `I_(2)`
finally, hence `I_(1)` works as a virtual object `2^(nd)` refracting surface
`therefore (n_(1))/(V)-(n_(2))/(V^(1)) = (n_(1)-n_(2))/(R_(2)).....(2)`
Equation (1) + (2)
`(n_(1))/(V)-(n_(1))/(u) = (n_(2)-n_(1)) ((1)/(R_(1))-(1)/(R_(2)))`
`therefore (1)/(V)-(1)/(u) = (n_(21)-1) ((1)/(R_(1))-(1)/(R_(2)))........(3)`
`"If"u = oo , v =f`
`(1)/(f) = (n_(21)-1) ((1)/(R_(1))-(1)/(R_(2)))........(4)`
Which is lens maker's formula.