Derivation of Lens Formula - Definition, FAQs

Derivation of Lens Formula - Definition, FAQs

Vishal kumarUpdated on 06 Sep 2025, 04:35 PM IST

Lenses are pieces of curved glass or some transparent material, and they bend light to create images. Lenses are commonly found in cameras, eyeglasses, microscopes and telescopes. The formation of an image by lenses is understood with the help of the lens formula, a mathematical expression that relates the object distance (u), image distance (v), and focal length (f) of a lens. In this article, we shall derive the lens formula step by step using simple ray diagrams and basic geometry, all for the benefit of a simple-to-understand concept for students and for application to numerical problems.

This Story also Contains

  1. What Is A Lens?
  2. What is the Lens Formula?
  3. Lens Formula Derivation
Derivation of Lens Formula - Definition, FAQs
Derivation of Lens Formula

What Is A Lens?

The lens is a transparent object that may be made of glass or plastic, and it bends (or refracts) light rays to create images. These are lenses with curved surfaces that primarily use the principle of refraction. Lenses are found in eyeglasses, cameras, microscopes, and magnifying glasses.

Lenses are based on two basic types:

  • Convex lens (converging lens): Thicker in the middle, bending light rays to bring them together, and can create real or virtual images.
  • Concave lens (diverging lens): Thinner in the middle, diverging light rays, and always forming a virtual image.

The lens is divided into two types depending on how the light rays behave when they travel through the lens. These are further divided into the following types.

types of lens

What is the Lens Formula?

The lens equation or lens formula is an equation that links the focal length, image distance, and object distance.

$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$

where,

v is the distance of the image from the optical centre of the lens, u is the distance of the object from the optical centre of the lens and f is the focal length of the lens.

Also read-

Lens Formula Derivation

Let AB represent an object at a distance greater than the focal length f of the convex lens that is located at right angles to the primary axis. The image A' B' is generated between O and $F_1$ on the same side as the item, and it is virtual and erect.

lens formula derivation

  • Focul length = $OF_1$ = f
  • Object distance = OA =u
  • Image distance = $OA_1$ = v

$\triangle O A B$ and $\triangle$ O A' B' are similar

\begin{array}{l}\because \angle B A O=\angle B^1 A^1 O=90^{\circ}, \text { vertex is common for both the triangles } \\ \text { so } \angle A O B=\angle A^1 O B^1, \therefore \angle A B O=\angle A^1 B^1 O\end{array}

$\frac{A' B'}{A B}=\frac{O A'}{O A}---(1)$

$\triangle O C F_1$ and $\triangle F_1 A' B' $ are similar

$\frac{A' B'}{O C}=\frac{A' F_1}{O F_1}$

But from the ray diagram, we see that OC = AB

$\begin{aligned} & \frac{A' B'}{A B}=\frac{A' F_1}{O F_1}=\frac{O F_1-O A'}{O F_1} \\ & \frac{A' B'}{A B}=\frac{O F_1-O A'}{O F_1}---(2)\end{aligned}$

From equation (1) and equation (2), we get

$\begin{aligned} & \frac{O A'}{O A}=\frac{O F_1-O A'}{O F_1} \\ & \frac{-v}{-u}=\frac{-f--v}{-f} \\ & \frac{v}{u}=\frac{-f+v}{-f} \\ & -v f=-u f+u v\end{aligned}$

Dividing throughout by uvf

$-\frac{1}{u}=-\frac{1}{v}+\frac{1}{f}$

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

Read more-

Frequently Asked Questions (FAQs)

Q: Concave lenses come in a variety of shapes and sizes.
A:

Concave lenses can be found in a variety of real-world applications.

Telescopes and binoculars

Nearsightedness can be corrected with eyeglasses.

Cameras.

Flashlights.

Q: What is the lens formula?
A:

  1/f=1/v−1/u

Q: What type of lens is the human eye?
A:

The human eye has a convex (biconvex) lens.

Q: Which lens is used to correct myopia?
A:

A concave lens is used to correct myopia.

Q: Who discovered the lens maker formula?
A:

 The lens maker formula was discovered by Rene Descartes.

Q: When is a Concave Lens' Image Virtual?
A:

Only when the object and image are on the same side of the lens is the picture generated by a concave lens virtual.

Q: When Do Convex Lenses Act Like Combined Lenses?
A:

The combined lens works as a convex lens if the focal length of the second lens is greater than the focal length of the first lens.

Q: What is the answer to the lens formula ?
A:

The lens formula is the relationship between the object's distance u, the image's distance v, and the lens's focal length f. With the right sign conventions, this law can be applied to both concave and convex lenses.