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Wave Optics

Question
CBSEENPH12039386

(i) A ray of light incident on face AB of an equilateral glass prism shows the minimum deviation of 30°. Calculate the speed of light through the prism.


(ii) Find the angle of incidence at face AB so that the emergent ray grazes along the face AC.

Solution


At the minimum deviation, the refracted ray inside the prism becomes parallel to its base.
Angle of minimum deviation is given as Dm = 30°
Since, the prism is equilateral, So, A = 60° 
Refractive index of the prism
straight mu space equals space fraction numerator Sin space open parentheses begin display style fraction numerator straight A space plus space straight D subscript straight m over denominator 2 end fraction end style close parentheses over denominator Sin space begin display style straight A over 2 end style end fraction
space equals space fraction numerator Sin space begin display style 90 over 2 end style over denominator Sin space begin display style 60 over 2 end style end fraction
space equals space fraction numerator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator 1 divided by 2 end fraction space equals space square root of 2

We know that μ = v1/v2
Hence the speed of light in prism would be 1/√2 times the speed of light in air i.e = 3 x108 /1.414 = 2.121 x108 m/s
(ii) 

From Snell's law, we know that fraction numerator sin space straight i over denominator sin space straight r end fraction space equals space straight mu subscript 12
For the emergent ray to graze at the face AC, the angle of refraction should be 90
So, applying snell's law at face AC, we get
fraction numerator sin space straight i subscript AC over denominator sin space straight r subscript AC end fraction space equals space straight mu subscript 21 space rightwards double arrow space fraction numerator sin begin display style space end style begin display style begin display style straight i end style subscript AC end style over denominator sin begin display style space end style begin display style begin display style 90 end style to the power of straight o end style end fraction space equals space fraction numerator 1 over denominator square root of 2 end fraction space or space straight i subscript AC space equals space 45 to the power of straight o

From figure, we can see that angle of refraction at face AB is 15
So applying Snell's law we get
Sin iAB = sin rAB x μ12
or iAB = sin-1 (√sin 15°)

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