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Condition for grazing emergence

Condition for grazing emergence :- For grazing emergence, i₂ = 90°, we can determine the angle of incidence i₁ for such grazing emergence.

$\displaystyle For\text{ }{{i}_{2}}={{90}^{\circ }}\Rightarrow {{r}_{2}}=C$

As for a prism,

$\displaystyle {{r}_{1}}+{{r}_{2}}=A$

$\displaystyle \Rightarrow {{r}_{1}}=A-{{r}_{2}}=A-C$

Applying Snell’s Law at interface AB, we get

$\displaystyle \frac{\sin {{i}_{1}}}{\sin {{r}_{1}}}=\mu$

$\displaystyle \Rightarrow \sin {{i}_{1}}=\mu \sin {{r}_{1}}=\mu \sin \left( A-C \right)$

$\displaystyle \Rightarrow \sin {{i}_{1}}=\mu \left[ \sin A\cos C-\cos A\sin C \right]$

$\displaystyle \Rightarrow \sin {{i}_{1}}=\mu \left[ \left( \sin A \right)\sqrt{1-{{\sin }^{2}}C}-\left( \cos A \right)\left( \sin C \right) \right]$

$\displaystyle \Rightarrow \sin {{i}_{1}}=\mu \left[ \left( \sin A \right)\sqrt{1-\frac{1}{{{\mu }^{2}}}}-\left( \cos A \right)\left( \frac{1}{\mu } \right) \right]$

$\displaystyle \Rightarrow \sin {{i}_{1}}=\mu \left[ \left( \sin A \right)\frac{\sqrt{{{\mu }^{2}}-1}}{\mu }-\left( \cos A \right)\left( \frac{1}{\mu } \right) \right]$

$\displaystyle \Rightarrow \sin {{i}_{1}}=\left[ \sin A\sqrt{{{\mu }^{2}}-1}-\cos A \right]$

$\displaystyle \Rightarrow {{i}_{1}}={{\sin }^{-1}}\left[ \sin A\sqrt{{{\mu }^{2}}-1}-\cos A \right]$

Note :- The light ray will emerge out of the prism only if the angle of incidence i₁ is greater than the above value.(This is because as A = i₁ + i₂ , so if i₁ increases than i₂ decreases and hence i₂ < 90°)

Condition for grazing emergence

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