Derivation Of Lens Formula

Introduction

The study of optics is one of the fundamental branches of physics that deals with the behaviour of light and its interaction with matter. One of the most significant applications of optics is the formation of images using lenses. The derivation of lens formula is an important concept in optics that helps us understand the behaviour of light rays as they pass through a lens. Let’s look at it in detail.

Refraction of Light

Before we dive into the derivation of lens formula, it is essential to understand the concept of refraction. Refraction is the bending of light when it passes through a medium of varying optical density. This phenomenon occurs because the speed of light changes as it moves from one medium to another.

The bending of light can be explained using Snell’s Law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant when light passes from one medium to another. This constant is known as the refractive index of the medium.

Lenses and their Types

A lens is a piece of glass or any other transparent material that can refract light. Lenses are used to bend light in a way that allows us to focus on an image. There are two types of lenses, convex and concave.

• Convex lenses are thicker at the centre and thinner at the edges. When parallel light rays pass through a convex lens, they converge at a point called the focal point.
• Concave lenses are thinner at the centre and thicker at the edges. When parallel light rays pass through a concave lens, they diverge and do not converge at a point.

Lens Formula

Derivation of Lens Formula

A lens is a transparent object with two curved surfaces, which are either both convex or one is convex and the other is concave. A lens can form images by refracting light rays that pass through it. The lens formula relates the distance of the object from the lens, the distance of the image from the lens, and the focal length of the lens. The focal length is defined as the distance between the lens and the point where parallel light rays converge after passing through the lens.

The derivation of lens formula is based on the principle of refraction. Let us consider a thin convex lens with an object placed at a distance ‘u’ from the lens. The light rays from the object pass through the lens and converge to form an image at a distance ‘v’ from the lens. According to the law of refraction, the product of the refractive index of the lens and the sine of the angle of incidence is equal to the product of the refractive index of the surrounding medium and the sine of the angle of refraction.

Using the above law, we can derive the lens formula as follows:

(1/u) + (1/v) = (1/f)

Where,

u = distance of the object from the lens

v = distance of the image from the lens

f = focal length of the lens

Sign Convention

The lens formula uses a sign convention to denote the direction of light rays. The direction of the light ray is positive when it is in the direction of the incident ray and negative when it is in the opposite direction. Similarly, the distance of the object from the lens is positive when it is on the same side as the incident light ray and negative when it is on the opposite side. The distance of the image from the lens is positive when it is on the opposite side of the lens from the incident light ray and negative when it is on the same side as the incident light ray.

Applications of Lens Formula

The lens formula has many practical applications in optics. Such as:

• Designing lenses for cameras, telescopes, microscopes, and other optical instruments.
• Correcting vision problems such as myopia (nearsightedness), hyperopia (farsightedness), and astigmatism using corrective lenses.
• Developing eyeglasses, contact lenses, and intraocular lenses for eye surgery.

The lens formula is also used to calculate the magnification of an image, which is the ratio of the size of the image to the size of the object. The magnification can be calculated using the following formula:

m = (-v/u)

Where,

m = magnification

u = distance of the object from the lens

v = distance of the image from the lens

Recommended Articles:

Derivation of Escape Velocity: Introduction, Formula, And Earth
Hall Effect: Introduction, Theory, Principle, Principle, And Applications
Heat Equation: Introduction, Assumptions, Derivation, And Applications
Derivation of Kinetic Energy
Derivation Of Law Of Conservation Of Momentum