(a) Deduce the expression, by drawing a suitable ray diagram, for the refractive index of a triangular glass prisms in terms of the angle of minimum deviation (D) and the angle of prism (A).
Draw a plot showing the variation of the angle of deviation with the angle of incidence.
(b) Calculate the value of the angle of incidence when a ray of light incident on one face of an equilateral glass prism produces the emergent ray, which just grazes along the adjacent face. Refractive index of prism is `sqrt2`

Explanation

(a) Plot of angle of deviation vs Angle of Incidence
From Snell's Law `mu = (sin i)/(sin r)`
Also `r = r^(1) " rArr delta m + A = 2i`
`rArr i = (A + delta m)/(2)`
`A = 2r " " rArr r = (A)/(2)`
`mu = (sin i)/(sin r) " " rArr sqrt2 = (sin i)/(sin 30^(@)) " " rArr sin i = (sqrt2)/(2) = (1)/(sqrt2)`
`i = 45^(@)`
(b) `(sin r_(2))/(sin 90^(@)) = (1)/(mu) and r_(2) = c`
`sin c = (1)/(mu) = (1)/(sqrt2) = sin 45^(@)`
`:. r_(1) + r_(2) = A`
`r_(1) + r_(2) = A " " rArr r_(1) = 15^(@)`
`(sin i)/(sin r_(1)) = mu = sqrt2 " " (sin i)/(sin 15^(@)) = sqrt2 " " rArr sin i = sqrt2 sin r`
`sin i = sqrt2 {sin (45^(@) - 30^(@))} = sqrt2 {sin 45^(@) cos 30^(@) - cos 45^(@) sin 30^(@)}`
`= sqrt2 {(1)/(sqrt2) xx (sqrt3)/(2) - (1)/(sqrt2) xx (1)/(2)} = ((sqrt3)/(2) - (1)/(2)) = (1.732 -1)/(2) = (732)/(2) = sin^(-1) (0.366) = 21.5^(@)`