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

**be the principle focus and**

*F**be the focal length.*

**f**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 μ_{1} to medium of refractive index μ_{2} through a curved surface. Then,

**μ _{2}/v – μ_{1}/u = (μ_{2}-μ_{1})/R**. Click here to know derivation of this formula.

**When light ray enters lens**, object distance = -OB

Image distance = BI_{1}

Radius = R_{1}

Hence, μ_{2}/BI_{1} – μ_{1}/-OB = (μ_{2}-μ_{1})/R_{1}

⇒ μ_{2}/BI_{1} + μ_{1}/OB = (μ_{2}-μ_{1})/R_{1} —(1)

**When light ray exits lens**, Object distance = DI_{1}

Image distance = DI

Radius = -R_{2}

Hence, μ_{1}/DI – μ_{2}/DI_{1} = -(μ_{1}-μ_{2})/R_{2}

⇒ μ_{1}/DI – μ_{2}/DI_{1} = (μ_{2}-μ_{1})/R_{2} —(2)

For thin lens point B is very close to point D. Thus, DI_{1} = BI_{1}

⇒ μ_{1}/DI – μ_{2}/BI_{1} = -(μ_{1}-μ_{2})/R_{2} —(3)

(1) + (3) ⇒ μ_{1}(1/OB + 1/DI) = (μ_{2}-μ_{1})(1/R_{1} – 1/R_{2})

But, OB = -u, DI = v

Hence, **μ _{1}(-1/u + 1/v) = (μ_{2}-μ_{1})(1/R_{1} – 1/R_{2})** —(4)

When u=∞, v=f

⇒ μ_{1}(0 + 1/v) = (μ_{2}-μ_{1})(1/R_{1} – 1/R_{2})

⇒ **μ _{1}/f = (μ_{2}-μ_{1})(1/R_{1} – 1/R_{2})** —(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**

## Picked For You

- Concave Mirror- Principal Focus, Image Formation, Uses
- Convex Mirror- Principal Focus, Image Formation, Uses
- Uses of Convex Mirror and It’s Applications in Daily Life

**To solve and ask more and more questions download Kunduz app for free.**