Explain how the image is formed by thin convex lens and derive `-1/u+1/v=(n_(21)-1)[(1)/(R_1)+(1)/(R_2)]`

Explanation

Figure shows the geometry of image formation by a double convex lens. The image formation can be seen in terms of two steps.
The first refracting surface forms the image `I_1`, of the object O figure (b).

The image `I_1`, acts as a virtual object for the second surface that forms the image at I figure (c).

Thus, the image formed by thin lens is shown in figure (a).

For refraction at interface ABC,
`therefore (n_1)/(OB)+(n_2)/(BI_1)=(n_2-n_1)/(BC_1)` ... (1)
A similar procedure applied to the interface ADC gives,
`therefore-(n_2)/(DI_1)+(n_2)/(DI)=(n_2-n_1)/(DC_1)` ... (2) (Refractive index of medium on right side of ADC is n, while on its left it is `n_2`).
For a thin lens, `BI_1`, = `DI_1`. Adding equations (1) and (2),
`(n_1)/(OB)+(n_1)/(DI)=(n_2-n_1)[(1)/(BC_1)+(1)/(DC_2)]`
Suppose the object is at infinity i.e. OB `rarr infty` and DI = f.
`therefore (n_1)/(infty)+(n_2)/(f)=(n_2-n_1)[(1)/(BC_1)+(1)/(DC_2)]`
`therefore (n_1)/(f)=(n_2-n_1)[(1)/(R_1)-(1)/(R_2)]`
`[ because BC_1=R_1" and "DC_2=-R_2]`
Now dividing by `n_1` on both sides `(1)/(infty)=0`
`therefore 1/f=(n_2/n_1-1)[(1)/(R_1)-(1)/(R_2)]`
`therefore 1/f=(n_(21)-1)[(1)/(R_1)-(1)/(R_2)](because n_(21)=(n_2)/(n_1))` ... (3)
equation (3) is known as the lens maker.s formula.
It is useful to design lenses of desired focal length using surfaces of suitable radii of curvature.
Note that the formula is true for a concave lens also. In that case R, is negative, R, positive and therefore f is negative.