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  4. A ray of light is incident at 60° on one face of a prism of angle 30° and the emergent ray makes 30° with the incident ray. The refractive index of the prism is

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Q. A ray of light is incident at $60^{\circ}$ on one face of a prism of angle $30^{\circ}$ and the emergent ray makes $30^\circ$ with the incident ray. The refractive index of the prism is

KEAMKEAM 2011Ray Optics and Optical Instruments

Solution:

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Here, $i = 60^{\circ}, A = 30^{\circ}, \delta= 30^{\circ}$
As $i+ e = A + \delta $
$ e = A + \delta - i = 30^{\circ} + 30^{\circ} - 60^{\circ} = 0^{\circ} $
Hence emergent ray is normal to the surface.
image
$e= 0^{\circ} $
$ \Rightarrow r_{2} = 0^{\circ} $
As $r_{1} +r_{2} = A $
$ \therefore r_{1} = A -r_{2} = 30^{\circ} - 0^{\circ} = 30^{\circ} $
$ \mu =\frac{ sin\, i}{sin \,r_{1}} = \frac{sin\, 60^{\circ}}{sin \,30^{\circ}} $
$= \frac{\sqrt{3}}{2}\times \frac{2}{1} = \sqrt{3} $
$= 1.732$