Question

The angle of incidence for a ray of light at a refracting surface of a prism is 45^o. The angle of prism is 60^o. If the ray suffers minimum deviation through the prism, the angle of minimum deviation and refractive index of the material of the prism respectively, are

30^o, $square root of 2$

45^o ; $square root of 2$

30^o; $fraction numerator 1 over denominator square root of 2 end fraction$

45^o; $fraction numerator 1 over denominator square root of 2 end fraction$

Solution

30^o, $square root of 2$

Consider a ray of light PQ incident on the surface AB and moves along RS, after passing through the prism ABC.

Given that, incident ray suffers minimum deviation. therefore, the ray inside the prism must be parallel to the base BC of the prism.
From the geometry of the prism and the ray diagram, it is evident that
Angle of incidence, i = 45^o
Angle of refraction, r = r' = 30^o
Angle of emergence, e = 45^o
Therefore, minimum deviation suffered by the ray is,

straight delta subscript min space equals space left parenthesis straight i plus straight e right parenthesis space minus space left parenthesis straight r plus straight r apostrophe right parenthesis space space space space space space space space equals space 90 to the power of straight o space minus space 60 to the power of straight o space space space space space space space space space space equals space 30 to the power of straight o

Also, using the formula,

straight mu space equals space fraction numerator sin space open parentheses begin display style fraction numerator straight A plus straight delta subscript straight m over denominator 2 end fraction end style close parentheses over denominator sin space straight A divided by 2 end fraction

where,

straight mu

is the refractive index of the material of the prism.
A = Angle of prism = 60^o
Therefore,

straight mu space equals space fraction numerator sin space open parentheses begin display style fraction numerator 60 to the power of straight o plus 30 to the power of straight o over denominator 2 end fraction end style close parentheses over denominator sin space begin display style 60 to the power of straight o over 2 end style end fraction space space space equals space fraction numerator sin space 45 to the power of straight o over denominator sin space 30 to the power of straight o end fraction space space equals space fraction numerator begin display style bevelled fraction numerator 1 over denominator square root of 2 end fraction end style over denominator 1 divided by 2 end fraction space space equals fraction numerator 2 over denominator square root of 2 end fraction space space equals space square root of 2

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