Question

For the given incident ray as shown in the figure, the condition of total internal reflection of this ray the minimum refractive index of the prism will be:

A. \[\dfrac{{\sqrt 3 + 1}}{2}\]
B. \[\dfrac{{\sqrt 2 + 1}}{2}\]
C. \[\sqrt {\dfrac{3}{2}} \]
D. \[\sqrt {\dfrac{7}{6}} \]

Answer 1

Hint: The given problem can be resolved by the mathematical relation between the refractive index of any medium with the critical angle of reflection. Moreover, the mathematical relation under the Snell’ s law can also be used to obtain the desired result for the solution.


Complete step-by-step solution
Given: The figure showing the direction of the incident ray.
The condition for the critical reflection is given as,
\[\sin {\theta _c} = \dfrac{1}{\mu }.........................\left( 1 \right)\]
Here, \[\mu \] is the refractive index of the medium and \[{\theta _c}\] is the critical angle.
Applying the trigonometric relation as,
\[\cos {\theta _c} = \sqrt {1 - {{\sin }^2}{\theta _c}} \]
Substituting the value of equation 1 in above equation as,
\[\begin{array}{l}
\cos {\theta _c} = \sqrt {1 - {{\sin }^2}{\theta _c}} \\
\cos {\theta _c} = \sqrt {1 - \dfrac{1}{{{\mu ^2}}}} \\
\mu \cos {\theta _c} = \sqrt {{\mu ^2} - 1} ...........................\left( 2 \right)
\end{array}\]

Now, applying the Snell’s law for the first incident ray as,

\[{\mu _a}\sin {\theta _i} = \mu \sin {\theta _r}\]
Here, \[{\mu _a}\] is the refractive index of air and its value is 1. And \[{\theta _i}\], \[{\theta _r}\] are the incident angle and refracted angle respectively and their values are \[45\;^\circ \]and \[\left( {90 - {\theta _c}} \right)\].
Substituting the values as,
\[\begin{array}{l}
{\mu _a}\sin {\theta _i} = \mu \sin {\theta _r}\\
\left( 1 \right) \times \sin 45\;^\circ = \mu \sin \left( {90 - {\theta _c}} \right)\\
\dfrac{1}{{\sqrt 2 }} = \mu \cos {\theta _c}
\end{array}\]
Substituting the values of equation 2 in above equation as,
\[\begin{array}{l}
\dfrac{1}{{\sqrt 2 }} = \mu \cos {\theta _c}\\
\dfrac{1}{{\sqrt 2 }} = \sqrt {{\mu ^2} - 1} \\
\mu = \sqrt {\dfrac{3}{2}}
\end{array}\]
Therefore, the minimum refractive index of prism is \[\sqrt {\dfrac{3}{2}} \] and option C is correct.


Note: To resolve the given condition the appropriate analysis of the direction of the light rays is to be made, along with the exact mathematical relation of the critical Angle. Moreover, the concepts of refractive index and Snell’s law are to be remembered to achieve the complete analysis of the given condition.