Physics
Study Material

# Lens Formula with Derivation and Magnification for Thin Lens

This article have two methods of derivation of lens formula. One is similar triangle method and other is refraction through curved surface. First one is used in 10th class and second one is used in 12th class. Scroll down to see both separately.

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Posted by Mahak Jain, 20/9/2021

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Lenses and mirrors have become an integral part of human life. Many of us might have been sightless without lenses on our eyes. Mirrors are also used in daily life. Read more about uses of concave mirrors and convex mirrors. To use these lenses and mirrors we need to have a formula to calculate image and object distances. In this article we are going to study lens formula, magnification and power of lenses and their combination. The article contains two derivations of same formula. Firstly, using similar triangle properties and second using refraction through curved surface.

## What is Lens Formula?

In optics, the relationship between the distance of an image from the pole (v), the distance of an object from the pole (u), and the focal length (f) of the lens is given by lens formula. Lens formula is applicable for both convex and concave lenses.

Note: The following formula is applicable only for thin lenses. Almost all the lenses used in daily life are thin lenses.

The formula is as follows:

1/f = 1/v – 1/u

## Lens Formula Derivation

#### Method 1- Using Similar Triangles

Let us consider a convex lens with an optical center (pole) O. Let us place an object AB perpendicular to the principal axis at a distance u from O. Image formed is A’B’  as shown in the figure. Let F  be the principle focus and f be the focal length.

In △ABO and △A’B’O

∠AOB = ∠A’OB’ and ∠ABO = ∠A’B’O

Therefore, △ABO and △A’B’O are similar.

Thus, A’B’/AB = OB’/OB —(1)

Similarly, △A’B’F and △OCF are similar

Thus, A’B’/OC = FB’/OF

But, OC = AB

Hence, A’B’/AB = FB’/OF —(2)

Equating equations (1) and (2), we get

OB’/OB = FB’/OF

but FB’ = OB’ – OF

OB’ / OB = (OB’-OF) / OF

Substituting u, v and f with sign convention.

OB = -u, OB’ = v, OF = f

v / -u = (v-f) / f

1/f = 1/v – 1/u

This above equation is called lens formula.

#### Method 2- Using Concept of Refraction at Curved Surface

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.

When light ray enters lens, object distance = -OB

Image distance = BI1

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

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)

Comparing equation (4) and (5), we get

1/f = 1/v – 1/u. This is lens formula.

## Magnification of Lens

Magnification is the number of times image is larger than object.

Thus, m = Image height/Object height

In the above figure 1, △ABO and △A’B’O are similar

So, AB/u = A’B’/v

⇒ A’B’/AB = v/u

Here, AB is object height and A’B’ is image height.

Hence, m = v/u