Capacitance

Two capacitors, each having capacitance 40 μF are connected in series. The space between one of the capacitors is filled with dielectric material of dielectric constant K such that the equivalence capacitance of the system became 24 μF. The value of K will be :

A slab of dielectric constant K has the same cross-sectional area as the plates of a parallel plate capacitor and thickness $\dfrac34d$, where d is the separation of the plates. The capacitance of the capacitor when the slab is inserted between the plates will be :

(Given $C_0$ = capacitance of capacitor with air as medium between plates.)

Two identical thin metal plates has charge $q_1$ and $q_2$ respectively such that $q_1>q_2$. The plates were brought close to each other to form a parallel plate capacitor of capacitance C. The potential difference between them is :

Combination of four parallel plate air capacitors is shown in figure. Separation between the plate P$_1$ & P$_2$ is $\dfrac d3$ and separation between the plates P$_2$ and P$_3$ is d. The equivalent capacitance of the configuration is _______ μF.

(Given $\dfrac{\epsilon_0A}d=4$ μF. Where A = Area of plates)

The magnitude of $q_1$ is ____.

The magnitude of $q_2$ is ____.

A long straight wire carries a current, $I = 2$ ampere. A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances 1 cm and 4 cm from the wire. At time $t = 0$, the rod starts moving on the rails with a speed $v = 3.0$ m/s (see the figure).

A resistor $R = 1.4\ \Omega$ and a capacitor $C_o = 5.0\ \mu$F are connected in series between the rails. At time $t = 0$, $C_o$ is uncharged. Which of the following statement(s) is(are) correct? [$\mu_0 = 4\pi \times 10^{-7}$ SI units. Take $\ln 2 = 0.7$]

Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be :

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800\ \mathrm{kg\,m^{-3}}$ and dielectric constant 21. If the angle between the strings remains the same after the immersion, then

Two capacitors with capacitance values $C_1 = 2000 \pm 10$ pF and $C_2 = 3000 \pm 15$ pF are connected in series. The voltage applied across this combination is $V = 5.00 \pm 0.02$ V. The percentage error in the calculation of the energy stored in this combination of capacitors is ______.

In the circuit shown, initially there is no charge on capacitors and keys $S_1$ and $S_2$ are open. The values of the capacitors are $C_1 = 10\ \mu$F, $C_2 = 30\ \mu$F and $C_3 = C_4 = 80\ \mu$F.

Which of the statement(s) is/are correct?

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 $\epsilon_r$ = ______.

In Process 1, the energy stored in the capacitor $E_C$ and heat dissipated across resistance $E_D$ are related by:

In Process 2, total energy dissipated across the resistance $E_D$ is:

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 :

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

The energy stored in the electric field produced by a metal sphere is 4.5 J. If the sphere contains 4 μC charge, its radius will be : $\left[\text{Take : }\dfrac1{4\pi\epsilon_o}=9\times10^9\text{ N-m}^2/\text{C}^2\right]$

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.

An infinite line charge of uniform electric charge density $\lambda$ lies along the axis of an electrically conducting infinite cylindrical shell of radius $R$. At time $t=0$, the space inside the cylinder is filled with a material of permittivity $\varepsilon$ and electrical conductivity $\sigma$. The electrical conduction in the material follows Ohm's law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density $j(t)$ at any point in the material?