A container has a base of $50 \mathrm{~cm} \times 5 \mathrm{~cm}$ and height $50 \mathrm{~cm}$, as shown in the figure. It has two parallel electrically conducting walls each of area $50 \mathrm{~cm} \times 50 \mathrm{~cm}$. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant 3 at a uniform rate of $250 \mathrm{~cm}^3 \mathrm{~s}^{-1}$. What is the value of the capacitance of the container after 10 seconds?

[Given: Permittivity of free space $\epsilon_0=9 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$, the effects of the non-conducting walls on the capacitance are negligible]

One mole of an ideal gas expands adiabatically from an initial state $\left(T_{\mathrm{A}}, V_0\right)$ to final state $\left(T_{\mathrm{f}}, 5 V_0\right)$. Another mole of the same gas expands isothermally from a different initial state $\left(T_{\mathrm{B}}, V_0\right)$ to the same final state $\left(T_{\mathrm{f}}, 5 V_0\right)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $T_{\mathrm{A}} / T_{\mathrm{B}}$ ?

Two satellites $\mathrm{P}$ and $\mathrm{Q}$ are moving in different circular orbits around the Earth (radius $R$ ). The heights of $\mathrm{P}$ and $\mathrm{Q}$ from the Earth surface are $h_{\mathrm{P}}$ and $h_{\mathrm{Q}}$, respectively, where $h_{\mathrm{P}}=R / 3$. The accelerations of $\mathrm{P}$ and $\mathrm{Q}$ due to Earth's gravity are $g_{\mathrm{P}}$ and $g_{\mathrm{Q}}$, respectively. If $g_{\mathrm{P}} / g_{\mathrm{Q}}=36 / 25$, what is the value of $h_{\mathrm{Q}}$ ?

A Hydrogen-like atom has atomic number $Z$. Photons emitted in the electronic transitions from level $n=4$ to level $n=3$ in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is $1.95 \mathrm{eV}$. If the photoelectric threshold wavelength for the target metal is $310 \mathrm{~nm}$, the value of $Z$ is _________.

[Given: $h c=1240 \mathrm{eV}-\mathrm{nm}$ and $R h c=13.6 \mathrm{eV}$, where $R$ is the Rydberg constant, $h$ is the Planck's constant and $c$ is the speed of light in vacuum]