3 questions from 2026
A container of height \( 2\,\text{m} \), length \( 2\,\text{m} \) and breadth \( 1\,\text{m} \) is made of insulating vertical walls and two large area horizontal metal plates (\( M_1 \) and \( M_2 \)) which extend far beyond the vertical walls in all directions. The container is partitioned into two equal chambers with a thin insulating vertical wall. The partition wall contains a small hole of cross-sectional area \( \sqrt{10}\,\text{cm}^2 \) near its bottom edge. Initially the hole is closed and the left chamber of the container is completely filled with a liquid of dielectric constant \( \varepsilon_r = 15 \) and the right chamber is empty (\( \varepsilon_r = 1 \)). At time \( t = 0 \), the hole is opened and the liquid flows from the left chamber to the right chamber. In both the chambers, the space above the liquid has \( \varepsilon_r = 1 \) and is maintained at atmospheric pressure. The schematic of the container at a time \( t > 0 \) is shown in the figure. [Given: acceleration due to gravity is \( 10\,\text{m s}^{-2} \).]
The height (in \( \text{m} \)) of the liquid in left chamber at \( t = 500\,\text{s} \) is ________.
A container of height \( 2\,\text{m} \), length \( 2\,\text{m} \) and breadth \( 1\,\text{m} \) is made of insulating vertical walls and two large area horizontal metal plates (\( M_1 \) and \( M_2 \)) which extend far beyond the vertical walls in all directions. The container is partitioned into two equal chambers with a thin insulating vertical wall. The partition wall contains a small hole of cross-sectional area \( \sqrt{10}\,\text{cm}^2 \) near its bottom edge. Initially the hole is closed and the left chamber of the container is completely filled with a liquid of dielectric constant \( \varepsilon_r = 15 \) and the right chamber is empty (\( \varepsilon_r = 1 \)). At time \( t = 0 \), the hole is opened and the liquid flows from the left chamber to the right chamber. In both the chambers, the space above the liquid has \( \varepsilon_r = 1 \) and is maintained at atmospheric pressure. The schematic of the container at a time \( t > 0 \) is shown in the figure. [Given: acceleration due to gravity is \( 10\,\text{m s}^{-2} \).]
The difference in the capacitance (in \( \text{F} \)) between the metal plates at \( t = 0 \) and that at \( t = 500\,\text{s} \) is \( (8 - n)\,\varepsilon_0 \), where \( \varepsilon_0 \) is the permittivity of free space. The value of \( n \) is ________.
A tank contains two immiscible liquids of densities \( 6\rho \) and \( 2\rho \). The higher density liquid is filled up to a height \( L/2 \) from the bottom. A thin rod of density \( \rho \) and length \( L \) is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is \( \dfrac{2\pi}{n}\sqrt{\dfrac{L}{g}} \), where \( g \) is the acceleration due to gravity.
The value of \( n \) is: 
