Derivation of Prism Formula: Introduction, Step-by-Step Guide and Applications

A prism changes the direction of light due to refraction and this change can be measured using the prism formula. The derivation of prism formula helps explain the relationship between the refractive index of the prism material, the prism angle and the angle of deviation.

Have you ever thought why a light ray bends while passing through a prism or why different colours spread out when white light enters it? The answer lies in the way light refracts inside the prism. This article explains the mathematical derivation of prism formula, its related terms and the mathematical derivation of the angle of deviation.

Table of Contents

• Introduction to Prism
• Understanding the Prism Formula and Its Terms
• Mathematical Derivation of Prism Formula
• Mathematical Derivation of Angle of Deviation
• Frequently Asked Questions about Derivation of Prism Formula

Introduction to Prism

A prism is a transparent optical object made of glass or another transparent material. It has flat and polished surfaces that refract light.

Interestingly, a prism does much more than simply bend light. It can also split white light into different colours, producing a spectrum.

Some common uses of prisms include:

• Binoculars
• Cameras
• Telescopes
• Spectrometers
• Optical instruments

Understanding the Prism Formula and Its Terms

The prism formula relates the refractive index of a prism to its angle of deviation and prism angle. At minimum deviation, the refractive index of the prism is given by:

$\mu=\frac{\sin\left(\frac{A+\delta_m}{2}\right)} {\sin\left(\frac{A}{2}\right)}$

Do you know? This formula is widely used in optics to calculate how much a light ray bends while passing through a prism.

Important terms used in derivation of prism formula

Before moving ahead, first try to understand a few important terms which are related to the prism formula.

Term	Symbol	Meaning
Angle of Incidence	$(i_1)$	Angle between the incident ray and the normal at the first surface
Angle of Refraction	$(r_1)$	Angle formed by the refracted ray at the first surface
Angle of Emergence	$(i_2)$	Angle at which the light ray leaves the prism
Second Angle of Refraction	$(r_2)$	Refraction angle at the second surface
Angle of Prism	(A)	Angle between the two refracting surfaces of the prism
Angle of Deviation	$(\delta)$	Total angle through which the light ray bends
Minimum Deviation	$(\delta_m)$	Smallest possible angle of deviation produced by the prism
Refractive Index	$(\mu) or (n)$	Measure of how much the prism material bends light

These quantities are used throughout the derivation of the prism formula.

Mathematical Derivation of Prism Formula

The prism formula is derived by using Snell’s law of refraction.

So, according to Snell's Law,

$\mu = \frac{\sin i}{\sin r}$

Where Μ is the refractive index of the prism material, i is the angle of incidence and r is the angle of refraction.

Step 1: Expression for Angle of Deviation

Whenever a light ray passes through a prism, the total angle of deviation (δ) is,

$\delta = (i_1-r_1)+(i_2-r_2)$

Or, we can also write, 

$\delta=i_1+i_2-(r_1+r_2)$

Step 2: Relation Between Prism Angle and Refraction Angles

From the geometry of the prism, we know that

$A=r_1+r_2$

where A is the angle of the prism.

After substituting into the deviation equation, we get,

$\delta=i_1+i_2-A$

Therefore,

$i_1+i_2=A+\delta$

Step 3: Condition of Minimum Deviation

At minimum deviation,

$i_1=i_2=i$

and

$r_1=r_2=r$

Since

$A=r_1+r_2$

From this we get, 

A=r+r=2r

Hence,

$r=\frac{A}{2}$

Also, we know that

i+i=A+δm 

$2i=A+\delta_m$

$2i=\frac{A+\delta_m}{2}$

Where \ delta_m is the minimum angle of deviation.

Step 4: Apply Snell’s Law

By using Snell’s Law,

$\mu=\frac{\sin i}{\sin r}$

Substituting the values of i and r, we get

$\mu=\frac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

Here, μ is the refractive index of the prism material, A is the angle of the prism and \delta_m is the minimum angle of deviation.

This is the standard prism formula used to determine the refractive index of a prism material.

Mathematical Derivation of Angle of Deviation

When a ray of light passes through a prism, it bends at both refracting surfaces. The total bending experienced by the light ray is known as the angle of deviation (δ).

Let consider

•  $i_1$ = Angle of incidence
•  $r_1$ = Angle of refraction at the first surface
•  $r_2$ = Angle of refraction at the second surface
• e = Angle of emergence
• A = Angle of prism
• δ = Angle of deviation

Step 1: Express the Total Deviation

The total deviation is equal to the sum of deviations at both surfaces of the prism. So mathematically we can write, 

$\delta=\delta_1+\delta_2$

$\delta=(i_1-r_1)+(e-r_2)$

$\delta=(i_1+e)-(r_1+r_2)$

Step 2: Find the Relation Between A,  $r_1, and r_2$

From the geometry of the prism,

$r_1+r_2=A$

This is a very important relation used in prism calculations.

Step 3: Substitute the Value of  $r_1+r_2$

Substituting,

$r_1+r_2=A$

into the deviation equation, 

$\delta=(i_1+e)-A$

This is the general formula for the angle of deviation produced by a prism.

Angle of Deviation for a Thin Prism

Interestingly! The calculations become much simpler for a thin prism. For a thin prism, the prism angle is very small. Therefore,

$\sin\theta \approx \theta$

Using Snell's law,

$n=\frac{\sin i}{\sin r_1}$

For small angles,

$n\approx\frac{i}{r_1}$

Therefore,

$i\approx nr_1$

Similarly, 

$n=\frac{\sin e}{\sin r_2}$

$e\approx nr_2$

Adding both equations, we get

$i+e=n(r_1+r_2)$

Since

$r_1+r_2=A$

we get

i+e=nA

From the angle of deviation formula,

$i+e=A+\delta$

Substituting,

nA=A+\delta

Therefore,

${\delta=A(n-1)}$

This is known as the prism formula for a thin prism.

For a given prism, both A and n remain constant. Therefore, the angle of deviation depends on the properties of the prism itself.

In this article, we learned how the derivation of prism formula is obtained using Snell's law and the geometry of a prism. We also studied the important terms related to the prism formula, the concept of angle of deviation, and the derivation for a thin prism. These concepts help explain how light bends inside a prism and form the basis of many optical instruments used in science and technology.