please give me the derivation of refraction at the spherical surfaces ?

 

 

 

Object at O, centre of curvature at C, virtual image at B. Pole at P. A is point of incidence. AP ’ is dropped. Refractive index of rarer medium n ₁ and denser medium n ₂ .

In parallaxial approximation A is very near to P thus PP’ ? 0

(Lens maker’s formula using concave and convex surfaces)

Let a ray of light from the object O incident upon a concave surface of radius R ₁ and come out of the convex surface radius R ₂ . Let the rarer medium have a r.i. n ₁ and denser medium n ₂ .

Since the lens is thin thus P’P ? 0

Let

OP’= -u

O’P’=-v’

Radius = -R ₁

Applying the Gaussian formula for concave surface,