(a) Deduce the expression, by drawing a suitable ray diagram, for the refractive index of a triangular glass prism in terms of the angle of minimum deviation (D) and the angle of prism (A). Draw a plot showing the variation of the

#School

(a) Deduce the expression, by drawing a suitable ray diagram, for the refractive
index of a triangular glass prism 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 the prism is $\sqrt{2}$ .

Answers (1)

a )

at a minimum angle of deviation
$i= e\, ,r_{1}= r_{2}$
$\therefore D_{m}= i+e-A$
             $= 2i-A$
$i= \frac{A+D_{m}}{2}$
By snell's law
$n_{1}\sin i= n_{2}\sin r$
$n_{1}\sin\left ( \frac{A+D_{m}}{2} \right )= n_{2}\sin \left ( \frac{A}{2} \right )$
$\frac{n_{2}}{n_{1}}= \frac{\sin \left ( \frac{A+D_{m}}{2} \right )}{\sin \left ( \frac{A}{2} \right )}$

b)
   $\frac{\sin r_{2}}{\sin 90}= \frac{1}{\sqrt{2}}$
$r_{2}= 45^{\circ}$
$r_{1}+r_{2}= A$
$r_{1}+45= 60^{\circ}$
$r_{1}= 15^{\circ}$
$\frac{\sin i}{\sin r_{1}}= \sqrt{2}$
$\sin i= \sqrt{2}\sin 15^{\circ}$
$i= \sin^{-1}\left ( \sqrt{2}\sin 15 \right )= 21\cdot 47^{\circ}$