Angle of minimum deviation , Prism Formula

Refraction through a prism

Let ABC is a Prism having Prism angle A and refracting index of material of prism is μ .

The incident ray first gets refracted at L . Snell’s law of refraction at L yields,

$ \displaystyle \frac{sin i_1}{sin r_1} = n $

$ \displaystyle sin r_1 = \frac{sin i_1}{n} $

$ \displaystyle r_1 = sin^{-1} (\frac{sin i_1}{n}) $

Then the refracted ray again gets refracted at M.

Applying Snell’s law at M , we obtain

$ \displaystyle \frac{sin r_2}{sin i_2} = \frac{1}{n} $

$ \displaystyle sin i_2 = n(sin r_2) $

$ \displaystyle i_2 = sin^{-1}(n sin r_2) $

In Quadrilateral ALOM , ∠L + ∠M = 180

Hence , ∠O + ∠A = 180 ..(i)

In Triangle OLM ,

∠O + r₁ + r₂ = 180 …(ii)

From (i) & (ii)

⇒  (r₁ + r₂) = A ………..(iii)

Angle of Deviation

The angle of deviation can be given as

δ = ∠PLM + ∠PML

⇒ δ = (i₁ − r₁) + (i₂ − r₂)

⇒ δ = (i₁ + i₂) − (r₁ + r₂)

Putting r₁ + r₂ = A    we obtain

δ = i₁ + i₂ − A

As we know that  Deviation ,  δ = i₁ + i₂ – A

From (i ~ δ) graph we conclude that deviation δ is minimum when, ray of light passes symmetrically through the prism , i₁ = i₂ = i (say)

⇒  δ_m = 2i − A

⇒ δ_m + A = 2 i

$ \displaystyle i = \frac{\delta_m +A}{2}$   …(i)

Since i₁ = i₂

⇒ r₁ = r₂ = r (say)

r₁ + r₂ = A

2r = A

⇒ r = A/2   …(ii)

Putting i & r in terms of A and δ_m in Snell’s Law

$ \displaystyle n = \frac{sini}{sin r} $

we obtain an expression for minimum deviation δ_m as

$ \displaystyle n = \frac{sin(\delta_m + A)/2}{sin A/2} $

Note : If the angle of prism is very small, sin A ≈ A

⇒ δ_m = (n −1)A

This equation is valid only when the refracting angle of the prism is very small.

Since prisms generally are made of optically denser medium w.r.t. air, can the phenomenon of total internal reflection take place over here ?

The phenomenon of total internal internal reflection can also be observed in case of prisms. This can be understood by the following cases.

When the angle of incidence for a prism is 90 degree then the ray just grazes along the surface of the prism . This is known as grazing incidence and in this case the angle of refraction at the first refracting surface is equal to the critical angle of the material of the prism.

Similarly, when the angle of emergence is equal to 90 degree (i.e. ray is incident at the critical angle at the second refracting face) then the case is of grazing emergence.

Both, in case of grazing emergence or grazing incidence, the angle of deviation for the prism is maximum.

Recall: The wavelength is inverse the proportional to the refractive index.

Also Read :

Reflection of Light at a Plane Surface Reflection at Spherical Surface & Mirror Formula Lateral Magnification Refraction of Light , Laws of Refraction , Relation between real and apparent depth Refraction through Number of media Total internal reflection Refraction through a prism Refraction at Curved surface Lens maker’s formula Combinations of Lenses Simple Magnifier Compound Microscope Telescope

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