Series and parallel combinations

A parallel plate capacitor of capacitance $C$ has spacing $d$ between two plates having area $A$. The region between the plates is filled with $N$ dielectric layers, parallel to its plates, each with thickness $\delta = \dfrac dN$. The dielectric constant of the $m^{th}$ layer is $K_m = K\left(1 + \dfrac mN\right)$. For a very large $N\ (> 10^3)$, the capacitance $C$ is $\alpha\left(\dfrac{K\epsilon_0A}{d\ln2}\right)$. The value of $\alpha$ will be ______.

[$\epsilon_0$ is the permittivity of free space]

Three identical capacitors $C_1$, $C_2$ and $C_3$ have a capacitance of $1.0\ \mu$F each and they are uncharged initially. They are connected in a circuit as shown in the figure and $C_1$ is then filled completely with a dielectric material of relative permittivity $\epsilon_r$. The cell electromotive force (emf) $V_0 = 8$ V. First the switch $S_1$ is closed while the switch $S_2$ is kept open. When the capacitor $C_3$ is fully charged, $S_1$ is opened and $S_2$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on $C_3$ is found to be $5\ \mu$C. The value of $\epsilon_r$ = ______.

A capacitance of 2 μF is required in an electrical circuit across a potential difference of 1.0 kV. A large number of 1 μF capacitors are available which can withstand a potential difference of not more than 300 V.

The minimum number of capacitors required to achieve this is :

A combination of parallel plate capacitors is maintained at a certain potential difference.

When a 3 mm thick slab is introduced between all the plates, in order to maintain the same potential difference, the distance between the plates is increased by 2.4 mm. Find the dielectric constant of the slab.

A combination of capacitors is set up as shown in the figure. The magnitude of the electric field, due to a point charge Q (having a charge equal to the sum of the charges on the 4 μF and 9 μF capacitors), at a point distant 30 m from it, would equal :

Three capacitors each of 4 μF are to be connected in such a way that the effective capacitance is 6 μF. This can be done by connecting them :

Figure shows a network of capacitors where the numbers indicates capacitances in micro Farad. The value of capacitance C if the equivalent capacitance between point A and B is to be 1 μF is :

A parallel plate capacitor having plates of area $S$ and plate separation $d$, has capacitance $C_1$ in air. When two dielectrics of different relative permittivities ($\varepsilon_1=2$ and $\varepsilon_2=4$) are introduced between the two plates as shown in the figure, the capacitance becomes $C_2$. The ratio $\dfrac{C_2}{C_1}$ is

In the given circuit, charge $Q_2$ on the 2μF capacitor changes as C is varied from 1μF to 3μF. $Q_2$ as a function of 'C' is given properly by : (figures are drawn schematically and are not to scale)

In figure is shown a system of four capacitors connected across a 10 V battery. Charge that will flow from switch S when it is closed is :

For the LCR circuit, shown here, the current is observed to lead the applied voltage. An additional capacitor C', when joined with the capacitor C present in the circuit, makes the power factor of the circuit unity. The capacitor C', must have been connected in :

A parallel plate capacitor has a dielectric slab of dielectric constant $K$ between its plates that covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor is $C$ while that of the portion with dielectric in between is $C_1$. When the capacitor is charged, the plate area covered by the dielectric gets charge $Q_1$ and the rest of the area gets charge $Q_2$. The electric field in the dielectric is $E_1$ and that in the other portion is $E_2$. Choose the correct option/options, ignoring edge effects.

Three capacitances, each of 3 μF, are provided. These cannot be combined to provide the resultant capacitance of :

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance $C$. The switch $S_1$ is pressed first to fully charge the capacitor $C_1$ and then released. The switch $S_2$ is then pressed to charge the capacitor $C_2$. After some time, $S_2$ is released and then $S_3$ is pressed. After some time,

In the given circuit, a charge of +80 μC is given to the upper plate of the 4 μF capacitor. Then in the steady state, the charge on the upper plate of the 3 μF capacitor is

A 2 μF capacitor is charged as shown in figure. The percentage of its stored energy dissipated after the switch S is turned to position 2 is

A circuit is connected as shown in the figure with the switch S open. When the switch is closed, the total amount of charge that flows from Y to X is