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  • Show that for a material with refractive index mu geq sqrt{2} light incident at any angle shall be guided along a length perpendicular to the incident face.

Show that for a material with refractive index \mu \geq \sqrt{2} light incident at any angle shall be guided along a length perpendicular to the incident face.

Answers (1)

Let the ray incident on face AB at angle i, after refraction, it travels along PQ and then interacts with face AC which is perpendicular to the incident face.

Any ray is guided along AC if the angle the ray makes with the face AC \left ( \phi \right ) is greater than the critical angle. From figure

\phi +r=90^{o}, therefore \sin\; \phi =\cos\; r\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; ......(i)

If \phi is the critical angle it means, \sin\; \phi \geq \frac{1}{\mu }\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; ......(ii)

From (i) and (ii), \cos \; r \geq \frac{1}{\mu ^{2}}\; or\; 1-\cos ^{2}r\leq 1-\frac{1}{\mu ^{2}}

i.e., \Rightarrow \sin ^{2}r\leq 1-\frac{1}{\mu ^{2}}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; .....(iii)

Applying Snell's law on face AB,

1. \sin \; i=\mu \sin\; r

Or 1. \sin^{2} \; i=\mu^{2} \sin^{2}\; r\Rightarrow \sin^{2}\; r=\frac{1}{\mu ^{2}} \sin^{2}\; i\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; .....(iv)

From (i) and (ii), \frac{1}{\mu ^{2}}\sin^{2}i\leq 1-\frac{1}{u^{2}}

Or \sin^{2}i\leq \mu ^{2}-1

When i=\frac{\pi }{2}, then we have the smallest angle \phi.

If it is greater than the critical angle, then all other angles of incidence shall be more than the critical angle

Thus, 1\leq \mu ^{2}-1\; or \; \mu ^{2}\geq 2

\Rightarrow \mu \geq \sqrt{2}. This is the required result.

Posted by

infoexpert23

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