Refraction Through Prism

Table of Contents

What is Prism?

A prism is a portion of a transparent medium bounded by two plane faces inclined to each other at a suitable angle. Angle ‘A’ is called the angle of Prism.

Refraction Through Prism:

Let ABC be the principal section of the Prism. Let PQ be the incident ray, QR be the refracted ray and RS be the emergent ray. Let N₁O and N₂O be the two normals at points Q and R respectively. In the figure, ∠i be the angle of incidence, ∠r_1, and ∠r₂ be the angles of refraction and ∠e be the angle of emergence. When the emergent ray is produced backward, it meets the incident ray at D. Then ∠EDR is called the angle of deviation (δ).

Calculation of Angle of Deviation:

From the figure, δ₁+ r₁ = i
⇒ δ₁= (i − r₁) …..(i)

Also, δ₂ + r₂ = e
⇒ δ₂ = (e − r₂) ……(ii)

From ΔDQR,
δ = δ₁ + δ₂ ……(1)

Put the value of δ₁ and δ₂ from (i), (ii) in (1),
δ = (i − r₁) + (e − r₂)
⇒ δ = i − r₁ + e − r₂
⇒ δ = (i + e) – (r₁ + r₂) ……(2)

From quad. AQOR,
A + O = 180° ……(iii)

From ΔOQR,
O + r₁ + r₂ = 180° ……(iv)

From (iii) and (iv), we have
A + O = O + r₁ + r₂
⇒ A = r₁ + r₂ ……(3)

Put the value of (r₁ + r₂) from (3) in (2),
δ = (i + e) – A
A + δ = i + e

If μ be the refractive index of the material of Prism and angles are small.
μ = sin i/sin r₁ = i/r₁ ⇒ i = μr₁
and μ = sin e/sin r₂ = e/r₂ and ⇒ e = μr₂

Put these values of i and e in equation (4),
⇒ A + δ = μr₁ + μr₂
⇒ A + δ = μ (r₁ + r₂)
⇒ A + δ = μA [∴ r₁ + r₂ = A]
⇒ δ = μA – A
⇒ δ = (μ – 1) A ……(5)

This is the required expression for the angle of deviation in the case of Prism.

Derivation of Prism Formula:

The variation of the angle of deviation with the angle of incidence is shown in the figure. From the graph, when i increases, δ decreases first, reaches a minimum, and then δ increases.

From the graph, we conclude that there is one and only one angle of incidence, for which deviation produced by the prism is minimum.

In the minimum deviation position,
δ = δ_m
∠ i = ∠ e
∠ r₁ = ∠ r₂ = ∠ r……(6)

Put (6) in (3),
A = r + r
⇒ A = 2r
⇒ r = A/2 ……(7)

Put (6) in (4),
A + δ_m = i + 1
⇒ A + δ_m = 2i
⇒ i = (A + δ_m)/2 ……(8)

According to Snell’s Law,
μ = sin i/ sin r ……(9)

Put (7) and (8) in (9), and we have
μ = sin [(A + δ_m)/2] / sin (A/2) ……(10)

Equation (10) is the required expression for the Prism formula.

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