Embibe Experts Solutions for Chapter: Capacitance, Exercise 4: Exercise - 4

Find the potential difference $V_a–V_b$ between the points $a$ and $b$ shown in each part of the figure.

The figure shows two identical parallel plate capacitors connected to a battery with the switch $S$ closed. The switch is now opened and the free space between the plates of the capacitors is filled with a dielectric of dielectric constant (or relative permittivity) $3$. Find the ratio of the total electrostatic energy stored in both capacitors before and after the introduction of the dielectric.

A capacitor is composed of three parallel conducting plates. All three plates are of the same area $A$. The first pair of plates are kept a distance $d_1$ apart, and the space between them is filled with a medium of a dielectric $K_1$. The corresponding data for the second pair are $d_2$ and $K_2$, respectively. The middle plate is connected to the positive terminal of a constant voltage source $V$ and the external plates are connected to the other terminal of $V$.

$\left(\mathrm{a}\right)$ Find the capacitance of the system.

$\left(\mathrm{b}\right)$ What is the surface charge density on the middle plate?

$\left(\mathrm{c}\right)$ Compute the energy density in the medium $K_1$.

Two parallel plate capacitors $A$ and $B$ have the same separation $d=8.85\times {10}^{-4} \mathrm{m}$ between the plates. The plate areas of $A$ and $B$ are $0.04 {\mathrm{m}}^2$ and $0.02 {\mathrm{m}}^2$, respectively. A slab of dielectric constant (relative permittivity $K=9$) has dimensions such that it can exactly fill the space between the plates of the capacitor $B$.

$\left(\mathrm{i}\right)$ The dielectric slab is placed inside $A$ as shown in the figure $\left(\mathrm{a}\right)$. $A$ is charged to a potential difference of $110 \mathrm{V}$. Calculate the capacitance of $A$ and energy stored in it.

$\left(\mathrm{ii}\right)$ The battery is disconnected and then the dielectric also slab is moved from $A$. Find the work done by the external agency in removing the slab from $A$.

$\left(\mathrm{iii}\right)$ The same dielectric slab is now placed inside $B$, filling it completely. The two capacitors $A$ and $B$ are then connected as shown in the figure $\left(\mathrm{c}\right)$. Calculate the energy stored in the system.

Find the equivalent capacitance of the combinations shown in the figure between the indicated points.

A finite ladder circuit is constructed by connecting several sections of $6 \mathrm{\mu F}, 8\mathrm{\mu F}$ capacitor combinations as shown in the figure. Circuit is terminated by a capacitor of capacitance $C$. Find the value of $C$, such that the equivalent capacitance of the ladder between the points $A$ and $B$  becomes independent of the number of sections in between?

The capacitance of a parallel plate capacitor with plate area $A$ and separation is $d$, is $C$. The space between the plates is filled with two wedges of dielectric constant $K_1$ and $K_2$ respectively (figure). Find the capacitance of the resulting capacitor.

Two capacitors of capacitances $2C \& C$ are connected in series with an inductor of inductance L. Initially capacitors have charges such that $V_B-V_A=4V_0\mathit{ }\&\mathit{ }{V}_{\mathit{C}}-V_D=V_0.$ Initial current in the circuit is zero. Find The maximum current that will flow in the circuit and the potential difference across each capacitor at that instant.