## Spherical Lens Formula

### Introduction

In this section, we will be looking at a mathematical relation that connects the focal length, the object distance, and the image distance of a lens. This can be of great help in solving mathematical problems to find out the position of the image and the object.

### Explanation

**New Cartesian Sign Convention:**

While approaching the refraction of light by spherical lenses mathematically, a set of sign conventions is followed, called the **New Cartesian Sign Convention**.

According to this convention, the optical center of a spherical lens is taken as the **origin**, and the principal axis is taken as the **x-axis** of the cartesian coordinate system.

It is used to solve the numerical problems on reflection by spherical lenses.

According to the New Cartesian Sign Convention, the distances that should be taken to be positive and negative are shown below.

The focal length of a convex lens is always **positive** as it lies towards the right, and the focal length of a concave lens is always **negative** as it lies towards the left.

**Lens Formula:**

The mathematical formula that relates the focal length of a lens (f), the object distance (u), and the image distance (v) is called the lens formula. It is given by,

The lens formula can be used for both the convex and the concave lens.

**Questions and Solutions:**

**A concave lens has a focal length of 30 cm. At what distance should the object be placed from the lens so that it forms an image at 20 cm from the lens? What can you conclude about the nature and the position of the image from the result?**

**Solution:**

Given that,

The type of lens = concave lens

Focal length (f) = – 30 cm

Image distance (v) = – 20 cm

Object distance (u) = ?

Position of the image = ?

The lens formula is given by,

From this, we can conclude that the image is located on the **same side **of the lens as the object at a distance of 60 cm from the optical center.

**A convex lens has a focal length of 30 cm. An object is located at a distance of 45 cm from the lens. Find the position of the image so formed.**

**Solution:**

Given that,

The type of lens = convex lens

Focal length (f) = 30 cm

Image distance (v) = ?

Object distance (u) = – 45 cm

Position of the image = ?

The lens formula is given by,

Thus, it can be concluded that the image is located at a distance of 90 cm towards the right of the lens.

**Summary**

- While approaching the refraction of light by spherical lenses mathematically, a set

of sign conventions is followed, called the**New Cartesian Sign Convention.** - According to this convention, the optical center of a spherical lens is taken as the
**origin**and the principal axis is taken as the**x-axis**of the cartesian coordinate system and the signs are chosen according to the cartesian coordinate system. - The formula that links the above three quantities is called the lens formula, which is

given by,

1/f = 1/v -1/u - The lens formula is applicable to both the lenses.

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