Lens Makers Formula

Description:

For various optical instruments, lenses of different focal lengths are used.

Focal length of a lens depends upon radii of curvatures of two surfaces and refractive index of the lens material.

Lens maker’s formula is used by lens manufacturers for making lenses of desired focal lengths.

Consider a thin glass lens having centers of curvatures C₁ and C₂ with radii of curvatures R₁ and R₂ respectively as shown in figure −

When a point object ’P’ is kept on principle axis which is formed by joining the centers of curvature, one ray passes along the principle axis without deviation.

Other ray PQ is refracted along QR due to first surface at a point ‘Q’, if there was no second surface then the refracted ray would have reaches the point I’, but due to second surface, the ray undergoes another refraction at point R and reaches at point I.

For refraction at surface-1 −

When ray moves from rarer to denser medium, we have −

μ₂/v - μ₁/u = μ₂ - μ₁/R

From figure, we get −

v = OI′ and u = OP and R = OC₁

Putting values −

μ₂/+OI′ - μ₁/-OP = μ₂ - μ₁/-OC₁

∴ μ₂/+OI′ + μ₁/OP = μ₂ - μ₁/OC₁...........(1)

For refraction at surface-2 −

When ray moves from denser to rarer medium, we have −

μ₁/v - μ₂/u = μ₁ - μ₂/R

From figure, we get −

v = OI and u = OI′ and R = OC₂

Putting values −

μ₂/+OI - μ₁/+OI′ = μ₁ - μ₂/-OC₂

∴ μ₂/OI - μ₁/OI′ = μ₂ - μ₁/OC₂.......(2)

Adding equation 1 and equation 2, we get −

μ₁/OI + μ₁/OP = (μ₂ - μ₁)(1/OC₁ + 1/OC₂)

Now, from figure considering values for lens −

OI = +v and OP = -u

Putting values for lens −

μ₁/v + μ₁/-u = (μ₂ - μ₁)(1/R₁ + 1/-R₂)

μ₁(1/v - 1/u) = (μ₂ - μ₁)(1/R₁ - 1/R₂)

Therefore,

1/f = (μ₂/μ₁ - 1)(1/R₁ - 1/R₂)

Writing in terms of power −

P = (μ₂₁ - 1)(1/R₁ - 1/R₂)