In the given circuit diagram when the current reaches a steady state in the circuit, the charge on the capacitor of capacitance $C$ will be

In an $LCR$ circuit as shown below both switches are open initially. Now the switch $S_1$ is closed, $S_{2 }$kept open. ($q$ is the charge on the capacitor and $\tau =RC$ is capacitive time constant). Which of the following statement is correct?

A capacitor having a capacitance of $100 \mathrm{\mu F}$ is charged to a potential difference of $24 \mathrm{V}$. The charging battery is disconnected and the capacitor is connected to another battery of emf $12 \mathrm{V}$ with the positive plate of the capacitor joined with the positive terminal of the battery

$\left(\mathrm{a}\right)$ Find the charges on the capacitor before and after the reconnection in steady-state.

$\left(\mathrm{b}\right)$ Find the net charge flown through the $12 \mathrm{V}$ battery

$\left(\mathrm{c}\right)$ Is work done by the battery or is it done on the battery? Find its magnitude.

$\left(\mathrm{d}\right)$ Find the decrease in electrostatic field energy.

$\left(\mathrm{e}\right)$ Find the heat developed during the flow of charge after reconnection.

The circular plates $A$ and $B$ of a parallel plate air capacitor have a diameter of $0.1 \mathrm{m}$ and are $2\times {10}^{-3} \mathrm{m}$ apart. The plates $C$ and $D$ of a similar capacitor have a diameter of $0.12 \mathrm{m}$ and are $3\times {10}^{-3} \mathrm{m}$ apart. The plate $A$ is earthed. Plates $B$ and $D$ are connected together. Plate $C$ is connected to the positive pole of a $120 \mathrm{volt}$ battery whose negative is earthed. Calculate

$\left(\mathrm{i}\right)$ The combined capacitance of the arrangement and

$\left(\mathrm{ii}\right)$ The energy stored in it.

A capacitor is made of a flat plate of the area $A$ and a second plate having a stair-like structure as shown in the figure. The width of each stair is $a$ and the height is $b$. Find the capacitance of the assembly.

Calculate the capacitance of a parallel plate condenser, with plate area $A$ and distance between plates $d$, when filled with a dielectric whose dielectric constant varies as;

$\in \left(x\right)={\in }_0+k\left(x\right)$ $\left(0<x<\frac{d}{2}\right)$; $\in \left(x\right)={\in }_0+k\left(d-x\right)$ $\left(\frac{d}{2}<x<d\right)$ 

$\left(\mathrm{a}\right)$ Find the current in the $20 \mathrm{\Omega }$ resistor shown in the figure.

$\left(\mathrm{b}\right)$ If a capacitor of capacitance $4 \mathrm{\mu F}$ is joined between the points $A$ and $B$, what would be the electrostatic energy stored in it.