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 N_{1}O and N_{2}O be the two normals at points Q and R respectively. In the figure, ∠i be the angle of incidence, ∠r_{1,} and ∠r_{2} 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, δ_{1}+ r_{1} = i⇒ δ _{1}= (i − r_{1}) …..(i)Also, δ _{2} + r_{2} = e⇒ δ _{2} = (e − r_{2}) ……(ii)From ΔDQR, δ = δ _{1} + δ_{2} ……(1)Put the value of δ _{1} and δ_{2} from (i), (ii) in (1), δ = (i − r _{1}) + (e − r_{2})⇒ δ = i − r _{1} + e − r_{2}⇒ δ = (i + e) – (r _{1} + r_{2}) ……(2)From quad. AQOR, A + O = 180° ……(iii) From ΔOQR, O + r _{1} + r_{2} = 180° ……(iv)From (iii) and (iv), we have A + O = O + r _{1} + r_{2}⇒ A = r _{1} + r_{2} ……(3)Put the value of (r _{1} + r_{2}) 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 r _{1} = i/r_{1} ⇒ i = μr_{1}and μ = sin e/sin r _{2} = e/r_{2} and ⇒ e = μr_{2}Put these values of i and e in equation (4), ⇒ A + δ = μr _{1} + μr_{2}⇒ A + δ = μ (r _{1} + r_{2})⇒ A + δ = μA [∴ r _{1} + r_{2} = 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 ∠ r _{1} = ∠ r_{2} = ∠ 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. |