Question

Starting with an expression for refraction at a single spherical surface, obtain Lens Maker's Formula.

Answer 1

LOL' and LO'L are the two curved surfaces forming a thin convex lens of radii R₁ and R₂ of refractive index μ. A is any object

After refraction through surface LOL', B is the virtual image. For surface LO'L ', B behaves as object and after refraction e is the image.

∴ For thin lens A is an object and C is its image 

For curved surface LOL', A = object; B = Image hence applying the formula 

                                              `mu/v-1/v = (mu-1)/R`

We have                                  `mu/v-1/v = (mu-1)/R_1`           ...... (1) 

∴                                                   `g mua = 1/(amug)`

Hence                                  `(1/mu)/(V') - 1/V =(1/mu-1)/R_2`           .....(2)

Multiplying eqn. (2) by μ, we have 

                                             `1/V-mu/(V') = (1-mu)/(R_2)`            ....(3)

Adding eqn. (2) and (3) 

                                               `1/V-1/mu =(mu-1)/(R_1) +(1-mu)/R_2`

⇒                                                     `1/(f') = (mu-1)(1/(R_1)-1/(R_2))`

where f' : is the second focal length of the lens.