Lens Maker Formula- Definition, Derivation, Numericals

Lenses help people see who have weak eyesight. However, have you ever wondered, how manufacturers make lens with so perfect focal length. They use lens maker formula to determine the focal length and radius of curvature of the lens needed. Different optical instruments use lenses of different focal lengths. The focal length of a lens depends on the refractive index of the lens and the radii of curvature. Before knowing about lens maker formula, you can read about lens formula and its derivation.

What is Lens Maker Formula?

Lens maker formula gives relation between refractive index, focal length, and radius of curvature for any lens.

Lens maker formula is given by:-

f = (μ₂/μ₁ – 1)(1/R₁ – 1/R₂)

μ₂ is refractive index of material of lens.

μ₁ is refractive index of material outside lens.

R₁ is radius of curvature of first side of lens.

R₂ is radius of curvature of second side of lens.

Lens Maker Formula Derivation

When light ray travels from medium of refractive index μ₁ to medium of refractive index μ₂ through a curved surface. Then,

μ₂/v – μ₁/u = (μ₂-μ₁)/R. Click here to know derivation of this formula.

When light ray enters lens, object distance = -OB

Image distance = BI₁

Radius = R₁

Hence, μ₂/BI₁ – μ₁/-OB = (μ₂-μ₁)/R₁

⇒ μ₂/BI₁ + μ₁/OB = (μ₂-μ₁)/R₁ —(1)

When light ray exits lens, Object distance = DI₁

Image distance = DI

Radius = -R₂

Hence, μ₁/DI – μ₂/DI₁ = -(μ₁-μ₂)/R₂

⇒ μ₁/DI – μ₂/DI₁ = (μ₂-μ₁)/R₂ —(2)

For thin lens point B is very close to point D. Thus, DI₁ = BI₁

⇒ μ₁/DI – μ₂/BI₁ = -(μ₁-μ₂)/R₂ —(3)

(1) + (3) ⇒ μ₁(1/OB + 1/DI) = (μ₂-μ₁)(1/R₁ – 1/R₂)

But, OB = -u, DI = v

Hence, μ₁(-1/u + 1/v) = (μ₂-μ₁)(1/R₁ – 1/R₂) —(4)

When u=∞, v=f

⇒ μ₁(0 + 1/v) = (μ₂-μ₁)(1/R₁ – 1/R₂)

⇒ μ₁/f = (μ₂-μ₁)(1/R₁ – 1/R₂) —(5)

Hence, f = (μ₂/μ₁ – 1)(1/R₁ – 1/R₂). This is lens maker formula.

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