Question

(i) Plot a graph to show variation of the angle of deviation as a function of angle of incidence for light passing through a prism. Derive an expression for refractive index of the prism in terms of angle of minimum deviation and angle of prism.

(ii) What is dispersion of light? What is its cause?

(iii) A ray of light incident normally on one face of a right isosceles prism is totally reflected, as shown in fig. What must be the minimum value of refractive index of glass? Give relevant calculations.

Solution

i)

If the angle of incidence is increased gradually, then the angle of deviation first decreases, attains a minimum value ( $straight delta subscript straight m$ ) and then again starts increasing.

When angle of deviation is minimum, the prism is at a minimum deviation position.

When space straight delta space equals space straight delta subscript straight m comma space straight e space equals space straight i space and space straight r subscript 2 space equals space straight r subscript 1 space equals space straight r space space space space space space space space space space space... space left parenthesis straight i right parenthesis because space straight r subscript 1 space plus space straight r subscript 2 space equals space straight A From space left parenthesis straight i right parenthesis comma space we space get straight r space plus space straight r space equals space straight A straight r space space equals space straight A over 2 Also comma space we space have straight A space plus space straight delta space equals space straight i space plus space straight e Substituting space straight delta space equals space straight delta subscript straight m space and space straight e space equals space straight i comma straight A space plus space straight delta subscript straight m space equals space straight i space plus space straight i space straight i space equals space fraction numerator open parentheses straight A space plus space straight delta subscript straight m close parentheses over denominator 2 end fraction because space straight mu space equals space fraction numerator sin space straight i over denominator sin space straight r end fraction Therefore comma space straight mu space equals space fraction numerator sin space open parentheses fraction numerator straight A space plus space straight delta subscript straight m over denominator 2 end fraction close parentheses over denominator sin space open parentheses begin display style straight A over 2 end style close parentheses end fraction

ii)
The splitting of light into its component colors is known as dispersion of light.
When a narrow beam of light is incident on a glass prism, the emergent light splits into seven colours called VIBGYOR.
Reason for dispersion: The colors in the spectrum have different wavelengths. The wavelength of violet light is smaller than that of red light. the refractive index of a material in terms of the wavelength of the light is given by the Cauchy's expression.

straight mu space equals space straight a space plus space straight b over straight lambda squared space plus space straight c over straight lambda to the power of 4

; where a, b and c are constants.
Thus the refractive index is different for different colours and hence, dispersion occurs.
iii)

The light which is incident normally on one face of a right-angled isosceles triangle prism is totally reflected.

straight mu space equals space fraction numerator 1 over denominator sin space straight c end fraction space semicolon space where space straight c space is space the space critical space angle space equals space 45 to the power of straight o Therefore comma space straight mu space equals space fraction numerator 1 over denominator sin space 45 to the power of straight o end fraction straight mu space equals space square root of 2 space equals space 1.41 space

Therefore, minimum value of the refractive index of the glass is 1.41

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