Lens Maker's Formula

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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)

Note:

The lens maker's formula can be derived for a concave lens in the same way. The ray diagram is as follows:

Note:

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.