Obtain lens maker's formula and medium its signification. Lens maker's formula and lens equation:

Explanation

Let us consider a thin lens made up of a medium refractive index `n_(2)` is placed in a medium of refractive index `n_(1)`. Let `R_(1) and R_(2)` be the radii of curvature of two spherical surfaces (1) and (2) respectively and P be the pole. Consider a point object O on the principa axis. The ray which falls very close to P, after at the surface (I) Forms image at I.. Before it dose so. it is again refracted by the surface (2). Therefore the final image is formed at I. The general equation for the refraction at a spherical surface is given by
`(n_(2))/(v)-(n_(1))/(u)=((n_(2)-n_(1)))/(R)`
For the refracting surface (1), the light goes from `n_(1) to n_(2)`.
`(n_(2))/(v)-(n_(1))/(u)=((n_(2)-n_(1)))/(R_(1))`
For the refracting surface (2), the light goes from medium `n_(2) to n_(1)`.
`(n_(1))/(v)-(n_(2))/(v.)=((n_(1)-n_(2)))/(R_(2))`
Adding the above two equations (1) and (2)
`(n_(1))/(v)-(n_(1))/(u)=(n_(2)-n_(1))((1)/(R_(1))-(1)/(R_(2)))`
Further simplifying and rerranging.
`(1)/(v)-(1)/(u)=((n_(2)-n_(1))/(n_(1)))((1)/(R_(1))-(1)/(R_(2)))`
`(1)/(v)-(1)/(u)-(n_(2)/(n_(1))-1)((1)/(R_(1))-(1)/(R_(2)))`
If the object is at infinity, the image is formed at the focus of the lens. Thus, for `u = oo, v = f`.
`(1)/(f)-(1)/(oo)=(n_(2)/(n_(1))-1)((1)/(R_(1))-(1)/(R_2))`
`(1)/(f)=(n_(2)/(n_(1))-1)((1)/(R_(1))-(1)/(R_(2)))`
If the refractive index of the lens is `n_(2)` and it is placed in air, then `n_(2) = n and n_(1) = 1`. So the equation (4) becomes,
`(1)/(f)=(n-1)((1)/(R_(1))-(1)/(R_(2)))`
The above equation is called the lens maker.s because it tells the lens manufactures what curvature is needed to make a lens of desired foal lenght with a material of particular refractive index. This formula holds good also for a concave lens. By comparing the equation (3) and (4) we can write,
`(1)/(v) - (1)/(u) = (1)/(f)`
This euqtion is known as lens equation which relates the object distance u and image distance v with focal f of the lens. This formula holds good for a any type of lens.