Write the basic assumptions used in the derivation of lens- maker's formula and hence derive this expression.

Explanation

Basic assuptions in the derivation of lens-maker.s formula are :
(a) (i) The lens is a thin and homogenous one.
(ii) Aperture of the lens is small as compared to its radii of curvature.
(iii) Object is supposed to be situated on left side of lens so that incident rays travel from left to right.
Consider a point object O situated on the principal axis of a biconvex lens, whose two surfaces have radii of curvature `R_(1) and R_(2),` respectively. Due to refraction at 1st surface of lens an image I. is formed for the object O. If `OC=u, CU.=v.`, then using the refraction formula at a single spherical surface, we have
`(n_(2))/(v.)-(n_(1))/(u)=(n_(2)-n_(1))/(R_(1))`

The image I. behaves as a virtual object for refraction at the second surface of the lens and the final real image is formed at I. Thus, for second surface applying refraction formula, we have
`(n_(1))/(v)-(n_(2))/(v.)=(n_(1)-n_(2))/(R_(2))=(n_(2)-n_(1))/((-R_(2)))`
Adding (i) and (ii), we have
`(n_(1))/(v)-(n_(1))/(u)=(n_(2)-n_(1))((1)/(R_(1))-(1)/(R_(2)))`
` or (1)/(v)-(1)/(u)=((n_(2))/(n_(1))-1)((1)/(R_(1))-(1)/(R_(2)))=(n_(21)-1)((1)/(R_(1))-(1)/(R_(2)))`
If `u=oo`, then by ddefinition `v=f` and , hence,
`(1)/(f)-(1)/(oo)=(n_(21)-1)((1)/(R_(1))-(1)/(R_(1))-(1)/(R_(2)))`
`rArr (1)/(f)=(n_(21)-1)((1)/(R_(1))-(1)/(R_(2)))`