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Asked by Prashant DIGHE | 30 Jul, 2019, 08:05: AM

Expert Answer

Angle of deviation δ = i+e-A ...................(1)

where i is angle of incidence, e is angle of emergence and A is refracting angle of prism as shown in figure.

we are given that δ = 23° and A = 60°, hence from eqn.(1), we get e+i = 83° ....................(2)

In eqn.(1), angle of incidence i and angle of emergence e can be replaced with one another.

If i is angle of incidence, e is angle of emergence. Similarly if e is angle of incidence then i is angle of emergence.

If two angles of incidence differing by 23°, e - i = 23° ....................(3)

By solving eqn.(2) and eqn.(3), we get i = 30° and e = 53°

Let r₁ be the angle of refraction at surface PQ and r₂ be the angle of incidence at surface PR

Let μ be refractive index of material of prism

Hence we have, sin(i)/sin(r₁) = μ or sin(r₁) = 0.5/μ , sin(e) / sin(r₂) = μ or sin(r₂) = 0.8/μ .....................(4)

In Prism, we have the relation, r₁ + r₂ = A , hence sin( r₁ + r₂ ) = √3/2 .........................(5)

if we expand LHS of eqn.(5) and use the substitutions given by eqn.(4), we get

begin mathsize 12px style fraction numerator 0.5 over denominator mu end fraction square root of 1 space minus space fraction numerator 0.64 over denominator mu squared end fraction end root space plus space square root of 1 space minus space fraction numerator 0.25 over denominator mu squared end fraction end root space cross times space fraction numerator 0.8 over denominator mu end fraction space equals space fraction numerator square root of 3 over denominator 2 end fraction square root of mu squared space minus space 0.64 end root space space plus space 1.6 space square root of mu squared space minus space 0.25 end root space equals space square root of 3 space mu squared space.............. left parenthesis 6 right parenthesis end style

Refractive index μ can be obtained by solving eqn.(6), but easier method is to substituite the different values

given in list of options for answer and choose the correct value that satisfies the eqn.(6).

I am finding that μ =

begin mathsize 12px style fraction numerator square root of 43 over denominator 5 end fraction end style

satisfies eqn.(6), hence refractive index μ =

begin mathsize 12px style fraction numerator square root of 43 over denominator 5 end fraction end style

Answered by Thiyagarajan K | 31 Jul, 2019, 11:16: AM

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