A transparent cube of side d, made of a material of refractive index \mu _{2} , is immersed in a liquid of refractive index \mu _{1}(\mu _{1}<\mu _{2}). A ray is incident on the face AB at an angle \theta (shown in the figure). Total internal reflection takes place at point E on the face BC.

Then \theta must satisfy:

 

  • Option 1)

    \Theta <sin^{-1}\frac{\mu _{1}}{\mu _{2}}

     

     

  • Option 2)

    \Theta >sin^{-1}\sqrt{\frac{\mu ^{2}_{2}}{\mu ^{2}_{1}}-1}

  • Option 3)

    \Theta <sin^{-1}\sqrt{\frac{\mu ^{2}_{2}}{\mu ^{2}_{1}}-1}

  • Option 4)

    \Theta >sin^{-1}\frac{\mu _{1}}{\mu _{2}}

 

Answers (1)

\mu_{1}\sin \Theta =\mu_{2}\sin r_{1}

90-r_{1}>c

sin(90-r_{1})>sin c

cosr_1>\frac{\mu_1}{\mu_2}

\sqrt{1-\left ( \frac{\mu _{1}\sin \Theta }{\mu _{2}} \right )^{2}}> \frac{\mu _{1}}{\mu _{2}}

1-\frac{\mu _{1}^{2}}{\mu ^{2}_{2}}\sin ^{2}\Theta > \frac{\mu _{1}^{2}}{\mu ^{2}_{2}}

\frac{\mu ^{2}_{2}-\mu _{1}^{2}}{\mu ^{2}_{1}}> \sin ^{2}\Theta

\Theta < \sin ^{-1}\sqrt{\frac{\mu_2 ^{^{2}}}{\mu ^{2}_{1}}-1}

 


Option 1)

\Theta <sin^{-1}\frac{\mu _{1}}{\mu _{2}}

 

 

Option 2)

\Theta >sin^{-1}\sqrt{\frac{\mu ^{2}_{2}}{\mu ^{2}_{1}}-1}

Option 3)

\Theta <sin^{-1}\sqrt{\frac{\mu ^{2}_{2}}{\mu ^{2}_{1}}-1}

Option 4)

\Theta >sin^{-1}\frac{\mu _{1}}{\mu _{2}}

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