Light is incident normally on the short face of a $30^{\circ} - 60^{\circ} - 90^{\circ}$ prism as shown in the figure. A liquid is poured on the hypotenuse of the prism. If the refractive index of the prism is $\sqrt{3}$ , find the maximum value of refractive index of the liquid above which TIR dosen't take place.

1.3

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1.5

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1.4

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1.2

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Solution

The correct option is B $1.5$
Light will get totally reflected only when it travels from denser to rarer medium.

Hence, the liquid will be relatively rarer medium as compared to the prism.

The maximum value of referactive index of liquid will be corresponding to the critical angle.

$sin i_{c} = \frac{μ_{r}}{μ_{d}}$

From geometery,
$∠ i = 60^{\circ}$
$sin 60^{\circ} \geq sin i_{c} = \frac{μ_{r}}{μ_{d}}$
$μ_{r} \leq μ_{d} sin 60^{\circ}$
$μ_{r} \leq \sqrt{3} \times \frac{\sqrt{3}}{2}$
$μ_{r} \leq 1.5$
Hence, the maximum value of refractive index of liquid is $1.5$

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