Q.A cylindrical jar of height $h$ is filled to the brim with a transparent liquid of refractive index $\mu$. A small dot is marked at the centre of the bottom surface of the jar. An opaque circular disc is to be floated flat on the top surface of the liquid, centred directly above the dot (symmetric about the vertical centre-line). Find the minimum diameter of this disc so that, viewed from above, the dot cannot be seen from any direction.

The key here is to first check whether the ray refracts into air or reflects back into the medium. The refractive index given is $1.414$, which is $\sqrt{2}$ — a familiar value. When light travels from a denser medium to a rarer one (air, $n=1$), total internal reflection occurs if the angle of incidence exceeds the critical angle. The critical angle $i_c$ is found from Snell’s law at the limiting case where the angle of refraction is $90^\circ$.

Let’s work through it.

1. Find the critical angle. Snell’s law: $n_1 \sin i_c = n_2 \sin 90^\circ$. Here $n_1 = 1.414$, $n_2 = 1$. So

Hence $i_c = 45^\circ$.

2. Compare the given angle of incidence with the critical angle.

   The incident angle is $48^\circ$, which is greater than $45^\circ$. Therefore, the ray does not enter air — it undergoes total internal reflection at the boundary.

3. Determine the deviation in total internal reflection.

   For reflection, the angle of reflection equals the angle of incidence. The ray is turned back into the same medium. The deviation $\delta$ is the angle through which the ray's direction is bent from its original path. …

Concept and Intuition

The angle of contact (or contact angle) is defined as the angle between the tangent to the liquid surface and the solid surface, measured inside the liquid. It is determined by the balance of three interfacial tensions: solid–liquid ($\gamma_{SL}$), solid–vapor ($\gamma_{SV}$), and liquid–vapor ($\gamma_{LV}$). This balance is given by Young’s equation:

Here, $\theta$ is the contact angle. Crucially, $\gamma_{SL}$, $\gamma_{SV}$, and $\gamma_{LV}$ are intrinsic properties of the materials involved — they depend only on the chemical nature of the solid, the liquid, and the vapor (or air), not on the shape or orientation of the solid.

A common pitfall is to think that tilting the rod changes the “effective” contact angle, but the contact angle is measured locally at the three-phase line (where solid, liquid, and vapor meet). That local geometry is unchanged because the solid’s surface chemistry is the same. The rod’s orientation only changes the macroscopic shape of the meniscus, not the microscopic balance of forces at the contact line.

Step-by-Step Reasoning

1. Identify the governing principle The contact angle $\theta$ is given by Young’s equation:

The right-hand side depends only on the three interfacial tensions. These tensions are fixed for a given solid, liquid, and surrounding vapor.

2. Check if orientation changes the interfacial tensions

   When the rod is placed vertically, the solid surface is vertical. When placed horizontally, the solid surface is horizontal (or at a different orientation relative to gravity). However, the chemical composition of the rod’s surface does not change — it is the same material. Therefore, $\gamma_{SV}$, $\gamma_{SL}$, and $\gamma_{LV}$ remain exactly the same.

3. Apply Young’s equation …

Inside the rod the ray travels at $60^\circ$ to the axis. The cylindrical side (rod–air interface) is parallel to the axis, so its normal is perpendicular to the axis. The angle of incidence at that wall, measured from the normal, is:

$\theta = 90^\circ - 60^\circ = 30^\circ$ …

Concept & Intuition

When two waves of amplitudes $A_1$ and $A_2$ superimpose, the resultant intensity depends on the phase difference between them. The maximum intensity occurs when the waves are in phase (constructive interference), giving amplitude $A_1 + A_2$. The minimum intensity occurs when they are exactly out of phase (destructive interference), giving amplitude $|A_1 - A_2|$. Since intensity is proportional to the square of the amplitude, the ratio of maximum to minimum intensity is $\left(\frac{A_1+A_2}{A_1-A_2}\right)^2$. The problem gives this ratio as $9:4$, so we can set up an equation and solve for the ratio of amplitudes.

Step-by-step solution

1. Write the intensity ratio in terms of amplitudes Maximum intensity $I_{\text{max}} \propto (A_1 + A_2)^2$ Minimum intensity $I_{\text{min}} \propto (A_1 - A_2)^2$ Hence

2. Set the given ratio The problem states $I_{\text{max}} : I_{\text{min}} = 9 : 4$, so

3. Take the square root (positive root, since amplitudes are positive)

4. Cross-multiply and solve for $A_2/A_1$ $$$$

Concept & Intuition

Total internal reflection (TIR) is a phenomenon where light is completely reflected back into the original medium, with no transmission into the second medium. This happens only when light tries to leave a denser medium (like glass or water) into a rarer one (like air). If the angle of incidence is too shallow (less than the critical angle), some light escapes by refraction. But if the angle is steep enough (greater than the critical angle), the refracted ray would have to bend so much that it never actually leaves — instead, all the energy bounces back. So two conditions are necessary: the direction must be from denser to rarer, and the angle must be larger than a specific threshold.

Step-by-step reasoning

1. Identify the medium condition

   TIR cannot occur when light goes from a rarer to a denser medium (e.g., air to glass) because the refracted ray always bends toward the normal, so some light always enters the denser medium. The ray must travel from a denser to a rarer medium.

   → Statement (b) is correct; statement (a) is incorrect.

2. Identify the angle condition

   The critical angle is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is exactly 90°. If the angle of incidence is less than the critical angle, refraction occurs (some light escapes). If it is greater than the critical angle, no refraction is possible — total internal reflection occurs. …