Image Formation Formula For Curved Surfaces

1. Let us consider a curved surface separating two media of refractive indices n1 and n2.

2. Let n1 be the rarer medium and n2 be the denser medium.

3. An object is placed on the principle axis at a point O in the rarer medium.

4. Consider two rays coming from the object O.

5. The first ray travels along the principal axis meet the pole and the refracted ray passes through the pole undeviating .

6. The second  ray making an angle α with the principal axis strikes the curved surface at the interface at A.

7. Let the angle of incidence be θ1.  The refracted ray passes through the denser medium and meets the principal axis at point I. Let the angle of refraction be θ2.

8. The two refracted rays meet at point I and there the image is formed.

9. Let the angle made by second refracted ray with principal axis be γ.

10. Let the angle between the normal and the principal axis be β.

11. Let the object distance (u) be PO and the Image distance(v) be PI. Let PC be the radius of curvature (R).

                  From  Δ ACO, θ1= α + β

                 From  ΔACI,  β = θ2 + γ

                        β – γ = θ2

   We know that according to Snell’s law,

        n1 sin θ1 = n2 sin θ2

  Substituting the values of θ1  and θ2

       n1 sin (α+β) = n2 sin(β-γ)

 12.  The angles α,β and γ is considered as very small if the rays are very close to the principal axis . Since the rays are parallel to the principal axis, they are called paraxial rays and this approximation is called paraxial approximation.

                  Therefore sin(α+β) = α+β and sin(β-γ) = β-γ

n1 (α+β) =  n2 (β-γ)

n1 α + n1 β = n2 β - n2 γ

And Tanα = AN/NO

Tanβ = AN/NC

Tanγ =AN/NI

Since all angles are very small,

    α = AN/NO

β = AN/NC

γ = AN/NI

substituting the angle values, we get

n1 AN/NO + n1 AN/NC = n2 AN/NC – n2 AN/NI

here N becomes P(POLE)

n1/PO + n1/PC = n2/PC – n2/PI

n1/PO + n2/PI = (n2 – n1)/PC

Sign Convention:

All the distance are measured from the pole

Distance measured in the direction of the incident ray are taken as positive and in opposite direction is taken as negative.

The height measured above the axis is taken as positive and below the axis is taken as negative.

Here PO= object distance = -u

PI= image distance = v

PC= radius of curvature= R

 By substituting the values, we get  

                      n2/v – n1/u = (n2 – n1)/R , for curved surfaces. 

Note: For plane surface, radius of curvature (R) approaches infinity then 1/R becomes 0. Therefore,

For plane surfaces,

n2/v – n1/u = 0

                                   n2/v  =  n1/u