Derivation of Prism Formula

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What is Prism?

In Physics, a prism is defined as a transparent, polished flat optical element on which light reflects. In a prism, one surface must be angled because objects with two parallel polished surfaces are not considered as prisms. Prisms are made up of materials that are transparent to the wavelengths for which they are designed. The materials generally used to make a prism are glass, plastic, and fluorite. The traditional shape of a prism is somewhat triangular.

Types of Prism

Dispersive Prisms -  Dispersive prisms are used for breaking up the light into its constituent colours because the refractive index is dependent on the frequency. The white light that enters into this prism is a mixture of various wavelengths and frequencies. After passing through the prism, each light frequency bent in a different direction. Some examples of dispersive prisms are Triangular prism, Abbe prism, Pellin-Broca prism, Amici prism, Compound prism, and Grism.

Reflective Prisms - Reflective prisms are used for reflecting light. To flip, invert, rotate, deviate or displace the light beam, a reflective prism is used. This type of prisms is used in binoculars or single-lens reflex cameras for the purpose of getting an erect image. If it is not used in these devices, then the image will be upside down for the users. Some common examples of reflective prisms are Porro prism, Porro-Abbe prism, Amici roof prism, Pentaprism, Abbe-Koenig prism, Schmidt-Pechan prism, Bauernfeind prism, Dove prism, and Retroreflector.

Polarizing Prisms - Polarizing prisms are used for splitting a beam of light into components of varying polarization. Polarizing prisms are made up of materials that are known as birefringent crystalline. Some common examples of the polarizing prism are Nicol prism, Wollaston prism, Nomarski prism, Rochon prism, Senarmont prism, Glan-Foucault prism, Glan-Taylor prism, and Glan-Thompson prism.

Deflecting Prisms - Deflecting prisms are used for deflecting the beam of light at a fixed angle. Beam steering can be done by using a pair of these prisms. On rotating the prisms, the beam gets deflected into any desired angle within the limit if a conical “Field of Regards”. Some examples of deflecting prisms are Rhomboid prism and Deck prisms.

Beam-Splitting Prisms -  Prisms that are used to splitting a single beam into two or more beams are known as beam-splitting beams. Some of the common examples of this prism are Beam splitter cube and Dichroic prism.

Derive the Prism Formula

\[\mu = \frac{sin i}{sin r}\]    (by Snell’s law)

\[\delta = i_{1} - r_{1} + i_{2} - r_{2}\] . . . . (Eq. 1)

\[\delta = i_{1} + i_{2} - (r_{1} + r_{2})\]

\[\angle ALO + \angle AMO = 2rl \angle s\]  (From Quadrilateral and ㄥALO = ㄥAMO = 90°)

ㄥLAM + ㄥLOM = 2rt ∠s\] (sum of four ∠s of a quadrilateral = 4rt ∠s)  (Eq.2)

\[\angle r_{1} + \angle r_{2}\] + ㄥLOM = 2rt ∠s (Eq.3)

ㄥLAM = \[\angle r_{1} + \angle r_{2}\] ( Comparing Eq.2 and Eq.3)

\[A = \angle r_{1} + \angle r_{2}\]

\[\delta = i_{1} + i_{2} - A\] ( Substituting A in Eq.1)

\[i_{1} + i_{2} = A + \delta \angle i_{1} = \angle i_{2}\]

\[\angle r_{1} = \angle r_{2} = \angle r\]

ㄥALM = ㄥLMA = 90° - ∠r\]

Thus, AL = LM and LM॥ BC

ㄥA = \[\angle r_{1} + \angle r_{2}\]

A = 2r ( Since, \[\angle r_{1} = \angle r_{2} = \angle r\] )

\[r = \frac{A}{2} i_{1} + i_{2} = A + \delta\]

\[i_{1} + i_{2} = A + \delta m 2 i_{1} = A + \delta m\]

\[i_{1} = \frac{(A + \delta m)}{2}\]

∴  \[\mu = [ sin [\frac{(A + \delta m)}{2}]]/[Sin(A/2)]\]

The prism formula and the answer to the common question ‘How to prove prism formula?’ are discussed above.

The above derivation can also be termed as the refractive index of prism formula derivation. Students can write this solution when they are asked to derive an expression for the refractive index of prism in terms of angle of prism and angle of minimum deviation.

Derive an Expression for Deviation Through a Thin Prism

A prism whose refracting angle is small ( within 10°) is known as a thin prism. In this type of prisms, by default, the angle of incidence and the angle of refraction is also small.

Consider a thin prism of angle A and refractive index n. Let i₁ be the angle of incidence, r₁ be the angle of refraction at the first face, and in the second face, r₂ be the angle of incidence, and i₂ be the angle of emergence. Let us consider d as the angle of deviation.

d = (i₁ + i₂) - A……….. (1)

A = r₁ + r₂       ………… (2)

For a thin prism when the angle of incidence i₁ is small  the angles r₁, r₂ , and i₂ are also small (For small values of θ, sinθ θ in radians)

∴ n = sin i₁/ sin r₁ i₁/r₁  or    i₁ = nr₁

Also, n = sin i₂/ sin r₂ i₂/r₂ or   i₂ = nr₂

From Eq.1

d = (nr₁ + nr₂) - A = n(r₁ + r₂) - A = nA - A ( ∵ A = r₁ + r₂)

∴  d = A(n - 1)

Using this above formula, students can derive an expression for deviation through a thin prism.

FAQ (Frequently Asked Questions)

1. What is the Difference Between a Pyramid and a Prism?

Answer: A pyramid and a prism look the same, but they have some dissimilarities as well. The first being the base of both the shapes. Prisms have two congruent or identical bases, whereas the pyramid has only one base. The shape of the pyramid and prism base depends on the overall three-dimensional shape of the object if the three- dimensional shape varies, the object’s bases also vary from each other. For example, the base can have a square, rectangle, triangle, hexagon, pentagon, or octagon shape. The base of the pyramid and the prism can never be circular or oval-shaped.

2. What are the Characteristics of a Prism?

Answer: The characteristics of a prism are as follows.

  • A prism is a three-dimensional object having two identical surfaces facing each other. The identical surfaces facing each other are termed as bases.

  • The shape of the base of a prism can be a triangle, square, rectangle, or any other polygon.

  • Some other faces of the prism are parallelograms or rectangles.

  • The prism base can never be circular or oval-shaped.

  • The angle of deviation of prism depends on the angle of incidence and the angle of refraction.