Derivation of Lens Formula - Definition, FAQs

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.

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.

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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.

• 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

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

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

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