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 

                                              ${\displaystyle \frac{\mu }{v}-\frac{1}{v}=\frac{\mu -1}{R}}$

We have                                  ${\displaystyle \frac{\mu }{v}-\frac{1}{v}=\frac{\mu -1}{R_1}}$           ...... (1) 

∴                                                   ${\displaystyle g\mu a=\frac{1}{a\mu g}}$

Hence                                  ${\displaystyle \frac{\frac{1}{\mu }}{V\prime }-\frac{1}{V}=\frac{\frac{1}{\mu }-1}{R_2}}$           .....(2)

Multiplying eqn. (2) by μ, we have 

                                             ${\displaystyle \frac{1}{V}-\frac{\mu }{V\prime }=\frac{1-\mu }{R_2}}$            ....(3)

Adding eqn. (2) and (3) 

                                               ${\displaystyle \frac{1}{V}-\frac{1}{\mu }=\frac{\mu -1}{R_1}+\frac{1-\mu }{R_2}}$

⇒                                                     ${\displaystyle \frac{1}{f\prime }=(\mu -1)(\frac{1}{R_1}-\frac{1}{R_2})}$

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