Refraction At Spherical Surface: Derivation and Formula

Ray Optics and Optical Instruments 2025 ( Physics Ray Optics and Optical Instruments )

Jaya Sharma
Updated on Sep 4, 2025 02:34 IST

By Jaya Sharma, Assistant Manager - Content

In this lesson, we will be learning about refraction at spherical surfaces. We will discuss how light bends as it passes from one medium to another through a curved boundary.

refraction at spherical surfaces

This explains the behaviour of mirrors, lenses and other optical instruments such as microscopes, telescopes and cameras. Once you have learnt about spherical surfaces, start working on the NCERT exercise of Ray Optics and Optical Instruments.

Table of content
  • Understanding Refraction at Spherical Surface
  • Sign Convention for Refraction at Spherical Surface
  • Refraction at Spherical Surface Formula
  • Expression for Refraction at a Single Spherical Surface
  • Key Terms Related to Refraction at Spherical Surface
  • Notes on Class 12th Physics
  • NCERT Solutions Related to 12th Physics Chapter
  • About the Reviewer
View More
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Understanding Refraction at Spherical Surface

A spherical surface is a portion of a sphere, like surface of a lens, which may be convex or concave. Refraction at spherical surface is the bending of light when it passes from one transparent medium to another through a curved boundary. 

The basic principle behind refraction at spherical surface is the different refractive index and change in speed. All these factors cause the light to bend and are governed by the Snell's Law. This type of refraction mostly occurs in lenses and other optical instruments. 

For your understanding, we are including a simulation that explains refraction at spherical surface:

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Sign Convention for Refraction at Spherical Surface

The sign convention for refraction at spherical surfaces is a set of rules that determines the signs of different quantities involved in the refraction of light through spherical surfaces. These are also known as the New Cartesian Sign convention. Let us take a look at each one:

sign conventions
  1. Origin: Every distance is measured from optical center (pole) of spherical surface. Pole (P) is the point where principal axis intersects spherical surface.
  2. Incident Light direction: We will always consider that incident light travels from left to right.
  3. Distances in the direction of incident light: All distances that are measured in the same direction as incident light are taken as positive.
  4. Distance opposite to direction of incident light: Distances that are measured in direction which is opposite to the incident light are considered negative.
  5. Heights measured upward: All heights that are measured upwards and perpendicular to principal axis are considered as positive.
  6. Heights measured downwards: All heights measured downwards and perpendicular to principal axis are considered as negative.
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Refraction at Spherical Surface Formula

The formula for refraction at spherical surface is as follows: n 2 v - n 1 u = n 2 - n 1 R Let us take a look at each term used in the formula:

  • n 1 : Refractive index of medium lying on the side of object (incident light)
  • n 2 : Refractive index of medium lying on the side of image (refracted light)
  • u: Distance of object from pole of spherical surface
  • v: Distance of image from pole of spherical surface
  • R: Radius of curvature of spherical surface

This refraction at spherical surface formula is used for determining the position and nature of the image which is formed by refraction at a spherical surface.

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Expression for Refraction at a Single Spherical Surface

Let us take a look at the derivation of the expression of refraction at a single spherical surface:

Refraction at a Single Surface
  • μ 1 : Refractive index of medium that contains the object (on the left side)
  • μ 2 : Refractive index of medium that will contain the image (on the right side)
  • A: Point of incidence on spherical surface
  • u: The distance of the object from the pole N (it will be positive if the object is on the left)
  • l: The distance of the image form pole N (it will be positive if the image is on the right)
  • R: Radius of curvature of spherical surface (it will be positive if center of curvature S is on right)
  • i: Angle of incidence
  • r: Angle of refraction
  • J: Object point
  • I: Image point

Let us take a look at the step-by-step derivation of refraction at spherical surface.

First, we will apply Snell’s law:

μ 1 sin i = μ 2 sinr

For smaller angles, sin i ≈ i and sin r ≈ r,

so: μ 1 i = μ 2 r

As we look at the triangle JAQ, we notice that α = A Q J Q = h - u

Here, u is accompanied by a negative sign since it is measured opposite to the direction of light.

In △IAQ, β = A Q I Q = h l

For a spherical surface: γ = A Q S Q = h R

Now, let us relate i, r and γ.

As it is visible from the image:

i = α+γ

r = β−γ

Let us substitute these in Snell's law, which will result in:

μ 1 ( α + γ ) = μ 2 ( β γ )

Let us now express α, β, and γ as h.

μ 1 ( h - u + h R ) = μ 2 ( h l - h R )

We will now simplify and solve for l by cancelling h from both the sides: μ 1 ( 1 - u + 1 R ) = μ 2 ( 1 l - 1 R ) μ 1 ( 1 R - 1 u ) = μ 2 ( 1 l - 1 R ) We will now rearrange the above equation to obtain the final expression: μ 2 l - μ 1 u = μ 2 - μ 1 R

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Key Terms Related to Refraction at Spherical Surface

The following table gives a glance at important terminologies used in this topic:

Term

Definition

Pole

The geometric center of a spherical surface .

Center of Curvature

The center of sphere which includes the spherical surface.

Radius of Curvature

The radius of the sphere.

Object Distance

Object’s distance from the pole.

Image Distance

Image’s distance from the pole.

Real Image

Image formed by convergence of refracted rays.

Virtual Image

Image formed by apparent divergence of refracted rays.

Refracted Ray

The light ray that bends and travels through the second medium after refraction.

Refractive Index

A dimensionless measure denoted by μ that indicates how much does a medium slow down light

Spherical Surface A curved surface which is a part of a sphere that has a constant radius of curvature.
Refraction The phenomenon in which the light bends as it passes from one medium to another medium with varying refractive indices.
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About the Reviewer

This lesson has been reviewed by subject matter expert Raghvendra Kumar, who has years of experience in teaching. He has qualified for the AIEEE and GATE exams. To maintain the quality of our content, we have referred to the NCERT as well as standard Physics books.

Q&A Icon
Commonly asked questions
Q:  

What is the formula for refraction at spherical surface from rarer to denser medium?

A: 

Whenever a light ray travels from a rarer medium to a denser medium through a spherical surface, relationship between object distance (u), image distance (v), radius of curvature (R) and refractive index ( n 1 and n 2 ) is given by refraction formula for spherical surfaces as follows.

n 2 v - n 1 u = n 2 - n 1 R

 

  • n 1 is the refractive index of medium that contains the object (on the left side)
  • n 2 is refractive index of medium that will contain the image (on the right side)
Q:  

Which among denser or rarer media has a higher refractive index?

 

A: 

When these two media are compared, then, it is evident that rarer medium has lower refracted index as compared to denser medium. Denser media like glass and water have higher refractive index whereas rarer medium like air and vacuum have lower refractive index. Due to this, the light will bend towards the normal when it travels from rarer to denser medium. On the other hand, light will bend away from normal when it travels from denser to a rarer medium.

Q:  

How can you tell whether a medium has low or high index of refraction?

A: 

You can identify whether a medium has higher or lower refractive index through the three ways. The first step is to observe the direction of bending. In this case, if the light is bending towards the normal, the second medium has higher refractive index. If the light bends away from normal, first medium has higher index of refraction. 

In the second method, you can use Snell's law. If the second angle is smaller than the first one, second medium has higher refractive index. In case the first angle is smaller than second one, first medium has higher index of refraction. The third method is critical angle method where, if the light undergoes total internal reflection at boundary, the first medium has higher refractive index.

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