Dielectrics

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

A parallel plate air capacitor has a capacitance $C$. When it is half filled as show in figure with a dielectric constant $K=5$, the percentage increase in the capacitance is ________.

A parallel plate capacitor is having separation between plates 0.885 mm. It has a capacitance of 1 μF when the space between the plates is filled with an insulating material of resistivity $1\times10^{13}$ Ωm and resistance $17.7\times10^{14}$ Ω. Relative permittivity of the insulating material is $\alpha\times10^7$. The value of $\alpha$ is ________.

(Take permittivity of free space $=8.85\times10^{-12}$ F/m)

Two co-axial conducting cylinders of same length $\ell$ with radii $\sqrt2R$ and $2R$ are kept, as shown in Fig. 1. The charge on the inner cylinder is $Q$ and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric constant $\kappa = 5$. Consider an imaginary plane of the same length $\ell$ at a distance $R$ from the common axis of the cylinders. This plane is parallel to the axis of the cylinders. The cross-sectional view of this arrangement is shown in Fig. 2. Ignoring edge effects, the flux of the electric field through the plane is ($\epsilon_0$ is the permittivity of free space):

A container has a base of $50\ \text{cm}\times5\ \text{cm}$ and height 50 cm, as shown in the figure. It has two parallel electrically conducting walls each of area $50\ \text{cm}\times50\ \text{cm}$. 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 $250\ \mathrm{cm^3\,s^{-1}}$. What is the value of the capacitance of the container after 10 seconds?

[Given: Permittivity of free space $\epsilon_0 = 9\times10^{-12}\ \mathrm{C^2N^{-1}m^{-2}}$, the effects of the non-conducting walls on the capacitance are negligible]

A medium having dielectric constant $K > 1$ fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is $d$. The capacitor is connected to a battery of voltage $V$, as shown in Figure (a). Now, both the plates are moved by a distance of $\dfrac d2$ from their original positions, as shown in Figure (b).

In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?

A parallel plate capacitor is formed by two plates each of area $30\pi\ \mathrm{cm^2}$ separated by 1 mm. A material of dielectric strength $3.6\times10^7\ \mathrm{Vm^{-1}}$ is filled between the plates. If the maximum charge that can be stored on the capacitor without causing any dielectric breakdown is $7\times10^{-6}$ C, the value of dielectric constant of the material is :

[Use $\dfrac1{4\pi\varepsilon_0}=9\times10^9\ \mathrm{Nm^2C^{-2}}$]

Two metallic plates form a parallel plate capacitor. The distance between the plates is '$d$'. A metal sheet of thickness $\dfrac d2$ and of area equal to area of each plate is introduced between the plates. What will be the ratio of the new capacitance to the original capacitance of the capacitor?

Given below two statements : One is labelled as Assertion (A) and other is labelled as Reason (R).

Assertion (A) : Non-polar materials do not have any permanent dipole moment.

Reason (R) : When a non-polar material is placed in an electric field, the centre of the positive charge distribution of it's individual atom or molecule coincides with the centre of the negative charge distribution.

In the light of above statements, choose the most appropriate answer from the options given below.

A parallel plate capacitor with plate area A and plate separation d = 2 m has a capacitance of $4\ \mu$F. The new capacitance of the system if half of the space between them is filled with a dielectric material of dielectric constant K = 3 (as shown in figure) will be :

A parallel plate capacitor filled with a medium of dielectric constant 10, is connected across a battery and is charged. The dielectric slab is replaced by another slab of dielectric constant 15. Then the energy of capacitor will :

$C_o$ is the capacitance of a parallel plate capacitor with air as a medium between the plates (as shown in Fig. 1). If half space between the plates is filled with a dielectric of relative permittivity $\varepsilon_r$ (as shown in Fig. 2), the new capacitance of the capacitor will be :

Two parallel plate capacitors of capacity C and 3C are connected in parallel combination and charged to a potential difference 18 V. The battery is then disconnected and the space between the plates of the capacitor of capacity C is completely filled with a material of dielectric constant 9. The final potential difference across the combination of capacitors will be ______ V.

A composite parallel plate capacitor is made up of two different dielectric materials with different thickness ($t_1$ and $t_2$) as shown in figure. The two different dielectric materials are separated by a conducting foil F. The voltage of the conducting foil is ______ V.

A source of potential difference V is connected to the combination of two identical capacitors as shown in the figure. When key 'K' is closed, the total energy stored across the combination is $E_1$. Now key 'K' is opened and dielectric of dielectric constant 5 is introduced between the plates of the capacitors. The total energy stored across the combination is now $E_2$. The ratio $E_1/E_2$ will be :

A parallel plate capacitor with width 4 cm, length 8 cm and separation between the plates of 4 mm is connected to a battery of 20 V. A dielectric slab of dielectric constant 5 having length 1 cm, width 4 cm and thickness 4 mm is inserted between the plates of parallel plate capacitor. The electrostatic energy of this system will be ______ $\varepsilon_0$ J.

(Where $\varepsilon_0$ is the permittivity of free space)

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 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

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]