A plot between the angle of deviation (δ) and angle of incidence (i) for triangular prism is shown below. Explain any given value of ‘δ’ corresponds to two values of angle of incidence? State the significance of point P on the graph. Use this information to derive an expression for refractive index of the material of the prism.

In general, any given value of 8, except for i = e corresponds to two values i and e. This, in fact, is expected from the symmetry of i and e as δ = i + e – A, i.e., δ remains the same if i and e are interchanged.

Fig.(a)
Point P is the point of minimum deviation. This is related to the fact that the path of the ray as shown in Fig (b) can be traced back resulting in the same angle of deviation. At the minimum deviation D_m, the refracted ray inside the prism becomes parallel to the base.
For δ = D_m,
i = e
and

straight r subscript 1 space equals space straight r subscript 2 space equals space straight r

2 straight r space equals space straight A

straight r space equals space straight A divided by 2

In the same way

straight D subscript straight m space equals space 2 straight i space minus space straight A

straight i space equals space fraction numerator straight A plus straight D subscript straight m over denominator 2 end fraction

∴ The refractive index of the prism is

box enclose straight mu space equals fraction numerator sin space left parenthesis straight A plus straight D subscript straight m right parenthesis over denominator sinA divided by 2 end fraction. end enclose

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