(i) Consider a thin lens made up of a medium of refractive index `n_2` is placed in a medium of refractive index `n_1` (ii) Let `R_1` and `R_2` be the radii of curvature of two spherical surfaces (1) and (2) respectively and P be the pole as shown in figure.
(iii). Consider a point object O on the principal axis. The ray which falls very close to P, after refraction at the surface (1) forms image at I'. Before it does so, it is again refracted by the surface (2). (iv) Therefore the final image is formed at I.The general equation for the refraction at a spherical surface is given from Equation
`(n_2)/(v) - (n_1)/(u) = ((n_2 - n_1))/(R_1)`
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) " " ...(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) " " ...(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 rearranging,
` 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)) " " ....(3)`
(v). If the object is at infinity, the image is formed at the focus of the lens. Thus, for ` u = oo , v = f` Then the equation becomes.
` 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)) " " ....(4)`
(vi) 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)) " " ...(5)`
(vii) The above equation is called the lens maker's formula, because it tells the lens manufactures what curvature is needed to make a lens of desired focal length with a material of particular refractive index. (viii) This formula holds good also for a concave lens. By comparing the equations (3) and (4) we can write,
`1/v + 1/u = 1/f " " ...(6)`
(ix) This equation is known as lens equation which relates the object distance u and image distance v with the focal length f of the lens. This formula holds good for a any type of lens.
