Question

Derive expression for lens maker’s formula.

(OR) 

Prove 1/f = (n-1) (1/R₁ - 1/R₂)

Answer 1

Procedure : 

• Imagine a point object ‘O’ placed on the principal axis of the thin lens 
• Let this lens be placed in a medium of refractive index n_a and let refractive index of lens be n_b. 
• Consider a ray, from ‘O’ which is incident on the convex surface of the lens with radius of curvature R₁ at A. 
• The incident ray refracts at A. 
• It forms image at Q, if there were no concave surface. From figure Object distance PO = – u;

Image distance PQ = v = x 

Radius of curvature R = R₁ 

n₁ = n_a and n₂ = n_b .

But the ray that has refracted at A suffers another refraction at B on the concave surface with radius of curvature (R₂). 

At B the ray is refracted and reaches I. 

The image Q of the object due to the convex surface. So I is the image of Q for concave surface. 

Object distance u = PQ = + x 

Image distance PI = v 

Radius of curvature R = – R₂ 

The refraction of the concave surface of lens is medium -1 and surrounding is medium –2. 

∴ n₁ = n_b and n₂ = n_a

Substituting values in \(\frac {n_2}{v} - \frac {n_1}{u}= \frac {(n_2-n_1)}{R} = \frac {n_a}{v}- \frac {n_b}{x} = \frac {(n_b-n_a)}{R_2}\)...(2)

Adding (1) and (2) and dividing both sides by n_a we have

This is called lens maker's formula.