Monochromatic light is incident on a glass prism of angle A. If the refractive index of the material of the prism is\mu, a ray, incident at an angle \Theta, on the face AB would get transmitted through the face AC of the prism provided :

  • Option 1)

    \Theta > \sin ^{-1}\left [ \mu \sin \left ( A-\sin ^{-1} \left ( \frac{1}{\mu } \right )\right ) \right ]

  • Option 2)

    \Theta < \sin ^{-1}\left [ \mu \sin \left ( A-\sin ^{-1} \left ( \frac{1}{\mu } \right )\right ) \right ]

  • Option 3)

    \Theta >\cos ^{-1}\left [ \mu \sin \left ( A+\sin ^{-1} \left ( \frac{1}{\mu } \right )\right ) \right ]

  • Option 4)

    \Theta < \cos ^{-1}\left [ \mu \sin \left ( A+\sin ^{-1} \left ( \frac{1}{\mu } \right )\right ) \right ]

 

Answers (1)

As we learnt in

Relation between angle of incidence and angle of refaction -

\mu _{1}\sin i= \mu _{2}\sin r
 

- wherein

\mu _{1}= refractive index of medium of incidence.

\mu _{2}=  refractive index of medium where rays is refracted.

i= angle of incidence.

r= angle of refraction.

 

 

Where r2 = C     \angle N_{2}Rc=90^{o}

where c = critical angle

sinc=\frac{1}{\nu}=sinr_{2}

Applying Snell's law at R

\mu sin r_{2}=1\ sin90^{o}                        (i)

Snell's law at Q 

   1+sin\theta=\mu sinr r_{1}                    (ii)

A=r_{1}+r_{2}\ \Rightarrow\ \; r_{1}=A-r_{2}

sin \theta=\mu sin(A-r_{2})\ \; \Rightarrow\ \; sin\theta=\mu sin A cos r_{2}-cosA

cos\ r_{2}=\sqrt{1-sin^{2}r_{2}}=\sqrt{1-\frac{1}{\mu^{2}}}                                (iv)                    (iii)

\therefore    From equation (iii) and (iv) 

    sin\theta=\mu sinA \sqrt{1-\frac{1}{\mu^{2}}}-cos A

theta=sin^{-1}\left[\mu sin(A-sin^{-1})\left(\frac{1}{\mu} \right ) \right ]

Correct option is 1.

 

 


Option 1)

\Theta > \sin ^{-1}\left [ \mu \sin \left ( A-\sin ^{-1} \left ( \frac{1}{\mu } \right )\right ) \right ]

This is the correct option.

Option 2)

\Theta < \sin ^{-1}\left [ \mu \sin \left ( A-\sin ^{-1} \left ( \frac{1}{\mu } \right )\right ) \right ]

This is an incorrect option.

Option 3)

\Theta >\cos ^{-1}\left [ \mu \sin \left ( A+\sin ^{-1} \left ( \frac{1}{\mu } \right )\right ) \right ]

This is an incorrect option.

Option 4)

\Theta < \cos ^{-1}\left [ \mu \sin \left ( A+\sin ^{-1} \left ( \frac{1}{\mu } \right )\right ) \right ]

This is an incorrect option.

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