Deviation Of Light By A Prism

A prism is a homogeneous, transparent medium bounded by two plane surfaces inclined at an angle A with each other. These surfaces are called the refracting surfaces, and the angle between them is called the angle of prism (A).

Let us consider the refraction of monochromatic light through a prism. Now, let i and e represent the angle of incidence and angle of emergence, respectively. r1 and r2 are two angles of refraction. If 𝜇 is the refractive index of the material of the prism, then: 

                                    𝜇 = sin i / sin r1

                                    𝜇 = sin e / sin r2

The angle between the incident ray and the emergent ray is known as the angle of deviation 𝝳. For refraction, through a prism, it is found that:

i+e  = A+𝞭 and r1 + r2 = A

If sin r2 ≥1/𝜇, then the ray will be totally reflected at the second refracting surface.

For A = 60° and 𝜇 = 1.5, if i < 28°, the total internal reflection will occur at face ac, and then 𝞭 will be meaningless.

Minimum angle of deviation(𝝳m)

When the angle of incidence i in a prism is increased, the value of angle of deviation 𝝳 first decrease and then expand. The minimum value of deviation is called the angle of minimum deviation. This situation arises when i = e, at the point of minimum deviation, the light ray passes symmetrically through the prism and r1 = r2.

Refractive index

In the position of minimum deviation, as i = e, hence i = (A+𝝳m)/2 and because r1 = r2 = r, hence r = A/2. Therefore , the refractive index of the material of the prism is: 

     𝜇 = sin i /sin r

        = sin [(A+𝝳m)/2] /sin[A/2]

Relation between refractive index and the angle of minimum deviation 

Let the angle of minimum deviation be 𝛿m. For minimum deviation, i = i’ and r = r’. 

We have,

                      𝛿m =  i+i’-A

                          = 2i-A

                        i  = (A+𝛿m)/2 ………(i)

Also    r+r’ = A

               r = A/2 …………..(ii)

The refractive index is 

                          𝜇 = sin i / sin r ………..(iii)

Using (i) and (ii),

                  𝜇 = (sin(A+𝛿m)/2 ) .1/ sin A/2

If the angle of prism a is small, 𝛿m is also small. Equation then becomes 

                            𝜇 = ((A+𝛿m)/2 ).1/(A/2)

                            𝛿m = (𝜇-1)A.

The deviation is therefore independent of the angle of incidence and depends on the refractive angle of the prism and the refractive index for monochromatic light. For small angle A, 𝝳 will also be small and 𝝳m = (𝜇-1)A.                        

Important facts regarding prism

1. The refracted ray inside a prism bends towards the base.
2. In minimum deviation position, the refracted ray inside a prism is parallel to the base, and the ray passes symmetrically through the prism.
3. For minimum deviation by prism, the angle of incidence must be 90°.
4. In equilateral prism (A = 60°), when the ray suffers minimum deviation, r1 = r2 = 30°.
5. If the angle of prism A is greater than twice the critical angle (i.e., 2C) of glass of which the prism is made, then there will be no emergent ray. The angle a is then called the limiting angle of the prism.

Dispersion of light 

When a ray of white light passes through a prism, it gets split into rays of constituent colours or wavelengths. This phenomenon of separation of white light into its component colours is called the dispersion of light.

Dispersion occurs because the refractive index is not strictly constant for a particular material but rather varies slightly with the wavelength of light.

Angular dispersion (θ)

It is defined as the difference in deviations suffered by the two extreme colours, i.e., red and violet.

Dispersive power (𝞈)

The ratio of angular dispersion between two colours to the deviation of mean ray (i.e., the ray of yellow colour) produced by a prism is called the dispersive power of the material of the prism for those colours.

1. For small A, the dispersive power depends on the material of the prism only but not on the angle of the prism.
2. 𝞈 can be greater than or equal to zero but cannot be less than zero. It is zero for vacuum, nearly equal to zero for air, and greater than zero for all other refracting media.

Conclusion 

A prism is a homogenous, transparent medium bounded by two plane surfaces inclined at an angle A with each other. These surfaces are called the refracting surfaces, and the angle between them is called the refracting angle or the angle of prism A. The angle between the incident ray and the emergent ray is called the angle of deviation.