Derivation of Prism Formula: Refraction through Prism

Prism Formula is based on the concept of bending of light when it passes through a prism. Prism formula is derived using Snell’s Law. Snell’s Law is used to describe the relationship between the incident angle and the angle of refraction. This law takes into account the refractive indices of the two mediums. Prism is a transparent and polished flat optical element that reflects light. These are formed from any transparent material with the wavelengths that they are designed for. The Prism formula in physics is given by,

\(\mu ={{sin\frac{A+\delta _{m}}{2}}}/{sin\frac{A}{2}}\)

Key Takeaways: Prism, Snell’s Law, Refraction, Refractive Index, Dispersion, Incident Ray, Refracted Ray, Angle of Prism, Angle of Deviation, Light

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What is Prism?

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Prism is a transparent, 3-dimensional object with two flat polished surfaces inclined at an acute angle to each other. These surfaces reflect light. 

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Derivation of Prism Formula

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The derivation of prism formula is as follows – 

By Snell’s law, we know that,

 \(\mu = \frac{\sin i}{\sin r}\)

δ = i₁ - r₁ + i₂ - r₂

δ = i₁ + i₂ - (r₁ + r₂) ….(i)

∠ALO + ∠AMO = 2rt∠s 

∠LAM + ∠LOM = 2rt∠s….(ii) (sum of 4 angles of a quadrilateral = 4 rt∠s) 

∠r₁ + ∠r₂ + ∠LOM = 2rt∠s…...(iii)

∠LAM = ∠r₁ + ∠r₂ ….(comparing eq(ii) and eq(iii))

A = ∠r₁ + ∠r₂ 

Substituting the value of A in eq(i)

δ = i₁ + i₂ - A

i₁ + i₂ = A + δ 

∠i₁ = ∠i₂

∠r₁ = ∠r₂ = ∠r

∠ALM = ∠LMA = 90° - ∠r 

Thus, AL = LM and LM II BC

∠A = ∠r₁ + ∠r₂ 

A = 2r….(Since ∠r₁ = ∠r₂ = ∠r)

r = A/2

i₁ + i₂ = A + δ

i₁ + i₁ = A + δm

2i₁ = A + δm

i₁ = A + δm/2 

Now, according to Snell’s Law,

μ = sini/sinr

The video below explains this:

Prism Formula Detailed Video Explanation:

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Types of Prism 

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Prisms are classified into various categories based on their properties and use. These prism types are as mentioned below:

• Dispersive Prism: They split the light into a spectrum of different colours. For example, the Triangular prism.
• Reflective Prism: They rotate, invert and displace the beam of light from the source. For example, Pentaprism, retroreflector.
• Deflecting Prism: They deflect the beam of light at a fixed angle. For example, Wedge prisms.
• Polarising Prism: They break the light beam by variation in polarisation. For example, Nicol prism.
• Beam-splitting Prism: This type of prism breaks a beam into more than two beams. For example, the dichroic prism.

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Refraction through Prism

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Light bends when it travels from one medium to another. This phenomenon is called Refraction through prism. 

• White light is formed of seven different colours - Violet, Indigo, Blue, Green, Yellow, Orange, and Red.
• Whenever light enters a prism, it bends due to the difference in the refractive index of the two mediums.
• The various constituents of the white light get deviated at various angles depending upon their wavelengths. 
• Violet colour shows maximum deviation because it has the lowest wavelength and red colour deviates the least since it has the highest wavelength.
• In this way, the prism refracts seven colours of light at different angles. 
• This phenomenon of separation of white light into its constituent colours is called Dispersion.

Refraction through prism

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Key Definitions

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Here are some basic definitions and formulae related to the derivation of prism formula and its concept:

• Incident Ray: The ray which falls on the surface separating two different mediums is called Incident Ray.
• Refracted Ray: The ray which enters the second medium with a change in direction is called Refracted Ray.
• Angle of Incidence: The angle formed between the incident ray and the normal is the angle of incidence. It is denoted by i.
• Angle of Refraction: The angle formed between refracted ray and normal is the angle of refraction. It is denoted by r.

Refraction through a Prism

• Angle of Prism: The angle formed between the two lateral faces of a prism is called Prism Angle. It is denoted by A.
• Angle of Deviation: The angle formed by extending the incident ray forward and emergent ray backwards is called the Angle of Deviation. It is denoted by δ.

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Previous Year Questions

1. An isosceles prism of angle 120 degree has a refractive index of 1.44 … (JEE Advanced 1995)
2. Monochromatic light is incident on a glass prism of angle A … (JEE Mains 2015)
3. An object approaches a convergent lens from the left of the lens … (KCET 2020)
4. The refracting angle of prism is A and refractive index … (KCET 2020)
5. A ray of light passes through an equilateral prism … (KEAM)

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Things to Remember

• The prism formula is derived from the principle of Snell’s Law.
• Snell’s law states the relationship between the angle of incidence and the angle of refraction.
• The bending of light takes place when it travels from one medium to another. This phenomenon is called Refraction.
• The angle of a Prism is the angle formed between two lateral faces of a prism.

• The Prism formula is given by, \(\mu ={{sin\frac{A+\delta _{m}}{2}}}/{sin\frac{A}{2}}\).

Prism Experiment