On the hypotenuse of a right-angled prism $(30^{\circ}-60^{\circ}-90^{\circ})$ with a refractive index of $1.50$,a drop of liquid is placed as shown in the figure. Light is allowed to fall normally on the short face of the prism. In order for the ray of light to undergo total internal reflection at the hypotenuse,what is the maximum value of the refractive index of the liquid?

$A$ glass prism of refractive index $1.5$ is immersed in water (refractive index $4/3$) as shown in the figure. $A$ light beam incident normally on face $AB$ undergoes total internal reflection at face $BC$.

The critical angle for a denser-rarer interface is $45^{\circ}$. The speed of light in the rarer medium is $3 \times 10^8 \, m/s$. The speed of light in the denser medium is:

Assertion $(A)$: Optical fibres are widely used in communication networks. Reason $(R)$: Optical fibres are small in size, light in weight, flexible, and there is no scope for interference in them.

$A$ planar structure of length $L$ and width $W$ is made of two different optical media of refractive indices $n_1=1.5$ and $n_2=1.44$ as shown in the figure. If $L \gg W$,a ray entering from end $AB$ will emerge from end $CD$ only if the total internal reflection condition is met inside the structure. For $L = 9.6 \ m$,if the incident angle $\theta$ is varied,the maximum time taken by a ray to exit the plane $CD$ is $t \times 10^{-9} \ s$,where $t$ is. . . . . . . [Speed of light $c = 3 \times 10^8 \ m/s$]

$A$ ray of light refracts from medium $1$ into a thin layer of medium $2$, crosses the layer, and is incident at the critical angle on the interface between medium $2$ and $3$ as shown in the figure. If the angle of incidence of the ray is $\theta$, the value of $\theta$ is