The principal section of a glass prism is an isosceles triangle $ABC$ with $AB = AC$. The face $AC$ is silvered. $A$ ray of light is incident normally on the face $AB$ and after two reflections, it emerges from the base $BC$ perpendicular to the base. The angle $BAC$ of the prism is: (in $^{\circ}$)

The ratio of the angle of minimum deviation of a prism in air to that when it is dipped in water is (given $_a\mu_g = 3/2$ and $_a\mu_w = 4/3$).

$A$ light ray is incident on a prism at an angle of $45^o$ and produces minimum deviation. If the refractive index of the prism is $\sqrt{2}$,then the prism angle is .......... $^o$.

White light is passed through a prism of angle $5^{\circ}$. If the refractive indices for red and blue colors are $1.641$ and $1.659$ respectively,the angular dispersion between them will be..... (in $^{\circ}$)

The angle of minimum deviation for a prism of refractive index $1.5$ is equal to the angle of the prism. The angle of the prism is . . . . . . . $(\sin 48^{\circ} 36^{\prime} = 0.75)$

Two equilateral-triangular prisms $P_1$ and $P_2$ are kept with their sides parallel to each other,in vacuum,as shown in the figure. $A$ light ray enters prism $P_1$ at an angle of incidence $\theta$ such that the outgoing ray undergoes minimum deviation in prism $P_2$. If the respective refractive indices of $P_1$ and $P_2$ are $\sqrt{\frac{3}{2}}$ and $\sqrt{3}$,then $\theta = \sin^{-1}\left[\sqrt{\frac{3}{2}} \sin \left(\frac{\pi}{\beta}\right)\right]$,where the value of $\beta$ is: