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

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Central Idea: The ray will get transmitted through face $A C$ if $i_{A C} < i_{C}$

Consider the ray diagram is shown below.

A ray of light incident on face $A B$ at an angle $θ$

$r_{1} =$ Angle of refraction on face $A B$

$r_{2} =$ Angle of incidence at face $A C$

For transmission of light through face $A C$

$i_{A C} < i_{C}$

or $A - r_{1} < i_{C}$

or $\sin (A - r_{1}) < \sin i_{C}$

or $\sin (A - r_{1}) < \frac{1}{μ}$

$A - r_{1} < \sin^{- 1} (\frac{1}{μ})$

or $\sin r_{1} > \sin [A - \sin^{- 1} (\frac{1}{μ})]$

Now, applying Snell's law at the face $A B$

$1 \times \sin θ = μ \sin r_{1}$

or $\sin r_{1} = \frac{\sin θ}{μ}$

$\Rightarrow \frac{\sin θ}{μ} > \sin [A - \sin^{- 1} (\frac{1}{μ})]$

or $θ > \sin^{- 1} [μ \sin \{A - \sin^{- 1} (\frac{1}{μ})\}]$

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