Capacitors in Series

Question Stem for Question Nos. 15 and 16

A container of height $$2\,\mathrm{m}$$, length $$2\,\mathrm{m}$$ and breadth $$1\,\mathrm{m}$$ 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 $$\sqrt{10}\,\mathrm{cm^2}$$ 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 $$\epsilon_r=15$$ and the right chamber is empty ($$\epsilon_r=1$$). 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 $$\epsilon_r=1$$ 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 $$10\,\mathrm{ms^{-2}}$$.]

The difference in the capacitance (in F) between the metal plates at $$t=0$$ and that at $$t=500\,\mathrm{s}$$ is $$(8-n)\epsilon_0$$, where $$\epsilon_0$$ is the permittivity of free space. The value of $$n$$ is: