Refraction through Prism

aRefraction through a Prism

Let us consider a prism XYZ with refraction angle A. A ray of light PQ is incident on face XY at an angle i. This is refracted along QR in the prism and finally emerges out along RS.Here r1 is angle of refraction in the first face, r2 is the angle of incidence in the second face and e is the angle of emergence at the second face. Angle 𝛿 between direction of incident ray and emergent ray is called angle of deviation. So ∠STL=𝛿 is angle of deviation. MN and ON are normal drawn on face XY and XZ, respectively. The angle of deviation produced at the first face 𝛿1=∠TQR =∠TQN-∠RQN =i-r1 And that produced at the second face a

𝛿2=	∠TRQ
	=∠TRN-∠QRN
	=e-r2

Since the deviations are in the same direction, the net deviation produced by the prism isa

𝛿=∠RTL	=𝛿1+𝛿2
	=i-r1+e-r2
	=i+e-(r1+r2)

In △QTR,∠TQR=i-r1∠TRQ=e-r 2∠QRT=180-𝛿So,in △QRT, we have,a

∠Q+∠R+∠T=180

or, i-r1+e-r2+180-𝛿=180

or, 𝛿=(i+e)-(r1+r2)……………………….(i)

In △QRN,r1+r2+∠RNQ=180∘……………….. (ii) In quadrilateral QXRN,∠QXR +∠XRN+∠RNQ+∠NQX=360∘or, A+90∘+∠RNQ+90∘=360∘or, A+∠RNQ=180∘……………………… (iii)Comparing eqn (ii) and (iii),A=r1+r2 ..........(iv)Putting this value in eqn (i), 𝛿=i+e-A 𝛿+A=i+e Thus, the sum of angle of incidence and angle of emergence in a prism is equal to sum of angle of deviation and angle of prism. Minimum DeviationAngle of deviation is expresses as𝛿+A=i+e ............(i)This suggest that angle of deviation varies with angle of incidence. When we plot a graph of variation of angle of deviation with the angle of incidence, it is observed that it decreases at first reaches a minimum value and increases. The lowest value of deviation is known as angle ofminimum deviation 𝛿m.

At the condition of minimum deviation,i) angle of incidence is equal to angle of emergence i.e. i = eii) angle of refraction at first face is equal to angle of incidence at second face. i.e. r1= r2we know,A=r1+r2=r+r=2ror, r=A

2 .....................(ii)Applying condition of minimum deviation in equation (i)a

or,	𝛿m=i+e-A
or	𝛿m=i+i-A
	2i=𝛿m+A
or	i=𝛿m+A 2…………..(iii)

From Snell's law,a

𝜇=sini sinr

𝜇=sin(A+𝛿m 2) sinA 2

This is expression minimum deviation of a prism. Deviation in small angle plismABC is a prism with small angle A of about 10∘ -12∘.PQ is a ray incident on face AB at an angle of incidence i. It is refracted along QR and finally emerges out along RS. Here, e is angle of emergence, r1 is angle of refraction at first face, r2 is angle of incidence on second face and 𝛿 is angle of deviation.The angle of deviation is given by,𝛿=(i+e)-(r1+r2)……………………….(i)

At face AB,𝜇=Sini

Sinr1Since the angle of incidence is small, the angle of refraction is also small. For small angle i and r 1sini≈i, and sinr1≈r1So,𝜇=i

r1 or, i=𝜇r1Similarly, at face AC,e=𝜇r2Substituting value of i and e in 𝜇, we geta

𝛿	=𝜇r1+𝜇r2-(r1+r2 )
	=𝜇(r1+r2)-(r1+r2)
	=(r1+r2)(𝜇-1)
	=A(𝜇-1)
∴𝛿=A(𝜇	-1)

This is the expression for angle of deviation of a ray of light in a small angle of prism. It indicates that, deviation in a small angled prism is independent of angle of incidence.Grazing Incidence and Grazing EmergenceWhen a ray light on a face of a prism with an angle of incidence 90∘, the ray lies on the surface and is refracted through the prism. This refraction of the prism is called the grazing incidence. Since the maximum value of the angle of incidence is 90∘, the angle of deviation of prism is maximum.

When the emergent ray from the prism makes an angle of 90∘ with normal, the ray lies on the surface. This condition is known as grazing emergence.Limiting angle of PrismThe largest value for the angle of prism for which the incident ray travel along the refracting faces on the prism is called limiting angle of prism.The limiting angle of prism is given by,a

A=C+C

A=2C

Hence, the limiting angle of prism is twice the critical angle.