$A$ prism has a refractive index of $1.5$. If a light ray undergoes total internal reflection at the interface with a liquid,what is the maximum possible refractive index of the liquid? (Refer to the provided diagram for geometry).

Show that for a material with refractive index $\mu \geqslant \sqrt{2}$,light incident at any angle shall be guided along a length perpendicular to the incident face.

At the interface between two materials having refractive indices $n_1$ and $n_2$,the critical angle for reflection of an electromagnetic wave is $\theta_{1C}$. The $n_2$ material is replaced by another material having refractive index $n_3$,such that the critical angle at the interface between $n_1$ and $n_3$ materials is $\theta_{2C}$. If $n_3 > n_2 > n_1$,$\frac{n_2}{n_3} = \frac{2}{5}$,and $\sin \theta_{2C} - \sin \theta_{1C} = \frac{1}{2}$,then $\theta_{1C}$ is:

$A$ ray of light is incident on the hypotenuse of a right-angled prism after travelling parallel to the base inside the prism. If $\mu$ is the refractive index of the material of the prism, the maximum value of the base angle for which light is totally reflected from the hypotenuse is

$A$ transparent cube of side $d$,made of a material of refractive index $\mu_2$,is immersed in a liquid of refractive index $\mu_1$ (where $\mu_1 < \mu_2$). $A$ ray is incident on the face $AB$ at an angle $\theta$ (as shown in the figure). Total internal reflection takes place at point $E$ on the face $BC$. The angle $\theta$ must satisfy:

The wavelengths of light in two liquids $x$ and $y$ are $3500 \, \mathring{A}$ and $7000 \, \mathring{A}$ respectively. The critical angle of $x$ with respect to $y$ is .......... $^o$.