Refraction at plane surfaces

A large square container with thin transparent vertical walls and filled with water $\left(\text{refractive index }\frac43\right)$ is kept on a horizontal table. A student holds a thin straight wire vertically inside the water 12 cm from one of its corners, as shown schematically in the figure. Looking at the wire from this corner, another student sees two images of the wire, located symmetrically on each side of the line of sight as shown. The separation (in cm) between these images is ______.

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 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 $t\times10^{-9}$ s, where $t$ is ______.

[Speed of light $c = 3\times10^8$ m/s]

Three glass cylinders of equal height $H = 30$ cm and same refractive index $n = 1.5$ are placed on a horizontal surface as shown in figure. Cylinder I has a flat top, cylinder II has a convex top and cylinder III has a concave top. The radii of curvature of the two curved tops are same ($R = 3$ m). If $H_1$, $H_2$, and $H_3$ are the apparent depths of a point $X$ on the bottom of the three cylinders, respectively, the correct statement(s) is/are:

A monochromatic light is travelling in a medium of refractive index $n = 1.6$. It enters a stack of glass layers from the bottom side at an angle $\theta = 30^\circ$. The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $n_m = n - m\Delta n$, where $n_m$ is the refractive index of the $m^{th}$ slab and $\Delta n = 0.1$ (see the figure). The ray is refracted out parallel to the interface between the $(m-1)^{th}$ and $m^{th}$ slabs from the right side of the stack. What is the value of $m$?

A parallel beam of light is incident from air at an angle $\alpha$ on the side PQ of a right angled triangular prism of refractive index $n=\sqrt2$. Light undergoes total internal reflection in the prism at the face PR when $\alpha$ has a minimum value of 45°. The angle $\theta$ of the prism is

A transparent slab of thickness $d$ has a refractive index $n(z)$ that increases with $z$. Here $z$ is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $n_1$ and $n_2\,(>n_1)$, as shown in the figure. A ray of light is incident with angle $\theta_i$ from medium 1 and emerges in medium 2 with refraction angle $\theta_f$ with a lateral displacement $l$.

Which of the following statement(s) is(are) true?

A monochromatic beam of light is incident at 60° on one face of an equilateral prism of refractive index $n$ and emerges from the opposite face making an angle $\theta(n)$ with the normal (see the figure). For $n=\sqrt3$ the value of $\theta$ is 60° and $\dfrac{d\theta}{dn}=m$. The value of $m$ is

For two structures namely $S_1$ with $n_1=\sqrt{45}/4$ and $n_2=3/2$, and $S_2$ with $n_1=8/5$ and $n_2=7/5$ and taking the refractive index of water to be 4/3 and that of air to be 1, the correct option(s) is(are)

A right angled prism of refractive index $\mu_1$ is placed in a rectangular block of refractive index $\mu_2$, which is surrounded by a medium of refractive index $\mu_3$, as shown in the figure. A ray of light '$e$' enters the rectangular block at normal incidence. Depending upon the relationships between $\mu_1$, $\mu_2$ and $\mu_3$, it takes one of the four possible paths '$ef$', '$eg$', '$eh$' or '$ei$'.

Match the paths in List I with conditions of refractive indices in List II and select the correct answer using the codes given below the lists:

For light incident from air on a meta-material, the appropriate ray diagram is

Choose the correct statement.

Water (with refractive index $=\frac43$) in a tank is 18 cm deep. Oil of refractive index $\frac74$ lies on water making a convex surface of radius of curvature '$R=6$ cm' as shown. Consider oil to act as a thin lens. An object 'S' is placed 24 cm above water surface. The location of its image is at '$x$' cm above the bottom of the tank. Then '$x$' is

A ray OP of monochromatic light is incident on the face AB of prism ABCD near vertex B at an incident angle of 60° (see figure). If the refractive index of the material of the prism is $\sqrt3$, which of the following is (are) correct?

Two transparent media of refractive indices $\mu_1$ and $\mu_3$ have a solid lens shaped transparent material of refractive index $\mu_2$ between them as shown in figures in Column II. A ray traversing these media is also shown in the figures. In Column I different relationships between $\mu_1$, $\mu_2$ and $\mu_3$ are given. Match them to the ray diagrams shown in Column II.

A ball is dropped from a height of 20 m above the surface of water in a lake. The refractive index of water is 4/3. A fish inside the lake, in the line of fall of the ball, is looking at the ball. At an instant, when the ball is 12.8 m above the water surface, the fish sees the speed of ball as [Take $g=10\ \mathrm{m/s^2}$.]

A light beam is traveling from Region I to Region IV (Refer Figure). The refractive index in Regions I, II, III and IV are $n_0$, $\frac{n_0}2$, $\frac{n_0}6$ and $\frac{n_0}8$, respectively. The angle of incidence $\theta$ for which the beam just misses entering Region IV is

A ray of light traveling in water is incident on its surface open to air. The angle of incidence is $\theta$, which is less than the critical angle. Then there will be

Speed of light is