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LENS MAKERS EQUATION DERIVATION

Lens Maker's Formula

It is a relation that connects focal length of a lens to radii of curvature of the two surfaces of the lens and refractive index of the material of the lens.

The following assumptions are made for the derivation:

The lens is thin, so that distances measured from the poles of its surfaces can be taken as equal to the distances from the optical centre of the lens.
The aperture of the lens is small.
Point object is considered.
Incident and refracted rays make small angles.



Consider a convex lens (or concave lens) of absolute refractive index m2 to be placed in a rarer medium of absolute refractive index m1.

Considering the refraction of a point object on the surface XP1Y, the image is formed at I1 who is at a distance of V1.

CI1= P1I1 = V1 (as the lens is thin)

CC1 = P1C1 = R1

CO = P1O = u

It follows from the refraction due to convex spherical surface XP1Y


The refracted ray from A suffers a second refraction on the surface XP2Y and emerges along BI. Therefore I is the final real image of O.

Here the object distance is


(Note P1P2 is very small)


(Final image distance)

Let R2 be radius of curvature of second surface of the lens.

 It follows from refraction due to concave spherical surface from denser to rarer medium that



Adding (1) & (2)







The lens maker's formula indicates that a convex lens can behave like a diverging one if m1 > m2 i.e., if the lens is placed in a medium whose m is greater than the m of lens. Similarly a concave lens can be made convergent.


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