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Lens Maker Formula- Definition, Derivation, Numericals

Lens maker formula gives relation between refractive index, focal length, and radius of curvature for any lens. Read more about it below.

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Posted by Mahak Jain, 23/9/2021
Lens Maker Formula- Definition, Derivation, Numericals

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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 = (μ21)/R. Click here to know derivation of this formula.

lens formula

When light ray enters lens, object distance = -OB

Image distance = BI1

Radius = R1

Refraction at first surface of lens formula
Refraction at first surface of lens

Hence, μ2/BI1 – μ1/-OB = (μ21)/R1

⇒ μ2/BI1 + μ1/OB = (μ21)/R1(1)

When light ray exits lens, Object distance = DI1

Image distance = DI

Radius = -R2

Hence, μ1/DI – μ2/DI1 = -(μ12)/R2

⇒ μ1/DI – μ2/DI1 = (μ21)/R2(2)

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

⇒ μ1/DI – μ2/BI1 = -(μ12)/R2(3)

(1) + (3) ⇒ μ1(1/OB + 1/DI) = (μ21)(1/R1 – 1/R2)

But, OB = -u, DI = v

Hence, μ1(-1/u + 1/v) = (μ21)(1/R1 – 1/R2)(4)

When u=∞, v=f

⇒ μ1(0 + 1/v) = (μ21)(1/R1 – 1/R2)

μ1/f = (μ21)(1/R1 – 1/R2)(5)

Hence, f = (μ21 – 1)(1/R1 – 1/R2). This is lens maker formula.


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