Find the value of the angle of emergence from the prism. Refractive index of the glass is √3 .

We are asked to find the angle of emergence from a prism whose refractive index is √3.

For a thin prism the relation between the refractive index $n$, the prism apex angle $A$ and the angle of minimum deviation $D\_\{min\}$ is

$n\; =\; \backslash frac\{\backslash sin\backslash !\backslash left(\backslash frac\{A\; +\; D\_\{min\}\}\{2\}\backslash right)\}\{\backslash sin\backslash !\backslash left(\backslash frac\{A\}\{2\}\backslash right)\}$

When the prism is used such that the angle of incidence equals the angle of emergence (the condition for minimum deviation), the angle of emergence is equal to the angle of deviation $D\_\{min\}$.

Given $n\; =\; \backslash sqrt\{3\}$, we substitute into the formula and solve for $D\_\{min\}$ assuming a common apex angle of the prism, $A\; =\; 60^\{\backslash circ\}$ (as shown in the figure).

First compute the denominator:

$\backslash sin\backslash !\backslash left(\backslash frac\{A\}\{2\}\backslash right)\; =\; \backslash sin\backslash !\backslash left(\backslash frac\{60^\{\backslash circ\}\}\{2\}\backslash right)\; =\; \backslash sin\; 30^\{\backslash circ\}\; =\; 0.5$

Set up the equation:

$\backslash sqrt\{3\}\; =\; \backslash frac\{\backslash sin\backslash !\backslash left(\backslash frac\{60^\{\backslash circ\}\; +\; D\_\{min\}\}\{2\}\backslash right)\}\{0.5\}$

Multiply both sides by 0.5:

$\backslash sin\backslash !\backslash left(\backslash frac\{60^\{\backslash circ\}\; +\; D\_\{min\}\}\{2\}\backslash right)\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$

We know that

$\backslash sin\; 60^\{\backslash circ\}\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$

Hence

$\backslash frac\{60^\{\backslash circ\}\; +\; D\_\{min\}\}\{2\}\; =\; 60^\{\backslash circ\}$

Solving for $D\_\{min\}$ gives

$60^\{\backslash circ\}\; +\; D\_\{min\}\; =\; 120^\{\backslash circ\}$

$D\_\{min\}\; =\; 60^\{\backslash circ\}$

Because the angle of emergence equals the angle of minimum deviation, the angle of emergence from the prism is also 60°.