– Capacitance of a parallel-plate eapaciter:
\[
c=\epsilon_{0} s / 4
\]
– Interaction energy of a systen of point chargs:
\[
w=\frac{1}{2} \sum \text { ง* }
\]
– Total electric energy of a system with continuess charge distributions
\[
W=\frac{1}{2} \int w d V \text {. }
\]
– Total eletric energy of two charged beties 1 and $z$
\[
w=w_{1}+w_{1}+w_{1}
\]
where $W_{4}$ and $W_{4}$ are the melfenergies of the bedies, and $W_{a}$ is the intersetien energy.
– Esergy of a charged capacitor:
\[
\mathrm{w}=\frac{\mathrm{V}}{2}=\frac{\mathrm{c}^{\mathrm{C}}}{2 \mathrm{C}}-\frac{\mathrm{CV}}{2}
\]
– Volume density of electric feld energy:
\[
=-\frac{\mathrm{ED}}{2}-\frac{u_{0} \mathrm{~F}}{2}
\]
3.101. Find the capacitance of an isolated ball-shaped conductor of radius $R_{1}$ surrounded by an adjacent concentrie layer of dielectrie with permittivity $\varepsilon$ and outside radies $R_{8}$.
3.102. Two parallel-plate air capacitors, each of capacitance $C$, were connected in series to a battery with eaf 8 . Then one of the capacitor was filled up with uniform dielectric with permittivity $\varepsilon$. How many times did the electric field strength in that capacitor decrease? What amount of charge nows through the battery?
3.103. The space between the plates of a paraliel-plate capacitor is filled consecutively with two dielectric layers 1 and 2 having the thicknesses $d_{1}$ and $d_{2}$ and the permittivities $\varepsilon_{1}$ and $\varepsilon_{2}$ respectively. The area of each plate is equal to $S$. Find:
(a) the capacitance of the capaciter;
(b) the density $\sigma^{\prime}$ of the bound charges on the boundary plane if the voltage across the capacitor equals $\boldsymbol{V}$ and the electric field is directed from layer $\boldsymbol{I}$ to layer 2.
3.104. The gap between the plates of a parallel-plate capacitor is filled with isotropie dielectrie whose permittivity \& varies linearly from $\varepsilon_{1}$ to $\varepsilon_{2}\left(\varepsilon_{2}>\varepsilon_{1}\right)$ in the direction perpendicular to the plates. The area of each plate equals $S$, the separation between the plates is equal to d. Find:
(a) the capacitance of the capacitor;
(b) the space density of the bound charges as a function of $\varepsilon$ if the charge of the capaciter is $q$ and the field $\mathbf{E}$ in it is directed toward the growing $\varepsilon$ values.
3. 105 . Find the capacitance of a spherical capacitor whose electrodes have radii $R_{1}$ and $H_{2}>R_{1}$ and which is filled with isotropic dielectric whose permittivity varies as $\varepsilon=a / r$, where $a$ is a constant, and $r$ is the distance from the centre of the capaciter.
3.106. A cylindrical capacitor is filled with two cylindrical layen of dielectric with permittivities $\varepsilon_{1}$ and $\varepsilon_{2}$. The inside radii of the layers are equal to $R_{1}$ and $R_{1}>R_{1}$. The maximum permissible values of electric field strength are equal to $E_{1}=$ and $E_{2}$. for these dielectrics. At what relationship between $\varepsilon, h$, and $E_{0}$ will the voltage increase result in the field strength reaching the breakdown value for both dielectries simultaneously?
3.107. There is a double-layer cylindrical capacitor whose parameters are shown in Fig. 3.16. The breakdown field strength values for these dielectrics are equal to $E_{1}$ and $E_{2} \mathrm{r}$ spectively. What is the breakdown voltage of this capacitor if $\varepsilon_{1} R_{1} E_{1}<\varepsilon_{2} R_{2} E_{2}$ ?
3.108. Two long straight wires with equal cross-sectional radil a are located parallel to each other in air. The distance between their axes equals b. Find the mutual capacitance of the wires per unit length under the condition $b>a$.
3.109. A long straight wire is located parallel to an infinite conducting plate. The wire cross-sec-
Fis. 3.18 . tional radius is equal to $\mathrm{a}$, the distance between the axis of the wire and the plane equals b. Find the mutual capacitance of this system per unit length of the wire under the condition $a<b$.
3.110. Find the capacitance of a system of two identical metal balls of radius $a$ if the distance between their centres is equal to $b$, with $b>a$. The system is located in a uniform dielectric with permittivity $\varepsilon$.
3.111. Determine the capacitance of a system consisting of a metal ball of radius $a$ and an infinite conducting plase separated from the centre of the ball by the distance $l$ if $t>a$.
3.t12. Find the capacitance of a system of identical capacitors between points $A$ and $B$ shown in
(a) Fig. 3.17a; (b) Fig. 3.17b.
118
Fig. 3.17.
3.113. Four identical metal plates are located in air at equal distances $d$ from one another. The area of each plate is equal to $S$. Find the capacitance of the system between points $A$ and $B$ if the plates are interconnected as shown
(a) in Fig. 3.18a; (b) in Fig. 3.18b.
Fig. 3.18.
3.114. A capacitor of capacitance $C_{1}=1.0 \mu \mathrm{F}$ withstands the maximum voltage $V_{1}=6.0 \mathrm{kV}$ while a capacitor of capacitance $C_{2}=2.0 \mu \mathrm{F}$, the maximum voltage $V_{2}=4.0 \mathrm{kV}$. What voltage will the system of these two capacitors withstand if they are connected in series?
3.115. Find the potential difference between points $A$ and $B$ of the system shown in Fig. 3.19 if the emf is equal to $\mathscr{E}=110 \mathrm{~V}$ and the capacitance ratio $C_{2} / C_{1}=\eta=2.0$.
Fig. 3.19.
3.116. Find the capacitance of an infinite circuit formed by the repetition of the same link consisting of two identical capacitors, each with capacitance $C$ (Fig. 3.20).
Fig. 3.20.
Fig. 3.21.
3.117. A circuit has a section $A B$ shown in Fig. 3.21. The emf of the source equals $\mathscr{E}=10 \mathrm{~V}$, the capacitor capacitances are equal to $C_{1}=1.0 \mu \mathrm{F}$ and $C_{2}=2.0 \mu \mathrm{F}$, and the potential difference $\varphi_{A}-$ – $\varphi_{B}=5.0 \mathrm{~V}$. Find the voltage across each capacitor.
3.118. In a circuit shown in Fig. 3.22 find the potential difference between the left and right plates of each capacitor.
120
3.119. Find the charge of each capacitor in the circuit shown in Fig. 3.22.
Fig. 3.22.
Fig. 3.23.
3.120. Determine the potential difference $\varphi_{A}-\varphi_{B}$ between points $A$ and $B$ of the circuit shown in Fig. 3.23. Under what condition is it equal to zero?
3.121. A capacitor of capacitance $C_{1}=1.0 \mu \mathrm{F}$ charged up to a voltage $V=110 \mathrm{~V}$ is connected in parallel to the terminals of a circuit consisting of two uncharged capacitors connected in series and possessing the capacitances $C_{2}=2.0 \mu \mathrm{F}$ and $C_{3}=3.0 \mu \mathrm{F}$. What charge will flow through the connecting wires?
3.122. What charges will flow after the shorting of the switch $S w$ in the circuit illustrated in Fig. 3.24 through sections 1 and 2 in the directions indicated by the arrows?
Fig. 3.24.
Fig. 3.25.
3.123. In the circuit shown in Fig. 3.25 the emf of each battery is equal to $\mathscr{E}=60 \mathrm{~V}$, and the capacitor capacitances are equal to $C_{1}=2.0 \mu \mathrm{F}$ and $C_{2}=3.0 \mu \mathrm{F}$. Find the charges which will flow after the shorting of the switch $S w$ through sections 1,2 and 3 in the directions indicated by the arrows.
3.124. Find the potential difference $\varphi_{A}-\varphi_{B}$ between points $A$ and $B$ of the circuit shown in Fig. 3.26.
Fig. 3.26.
Fig. 3.27.
3.125. Determine the potential at point 1 of the circuit shown in Fig. 3.27, assuming the potential at the point $O$ to be equal to zero.
Using the symmetry of the formula obtained, write the expressions for the potentials. at points 2 and 3.
3.126. Find the capacitance of the circuit shown in Fig. 3.28 between points $A$ and $B$.
Fig. 3.28.
3.127. Determine the interaction energy of the point charges located at the corners of a square with the side $a$ in the circuits shown in Fig. 3.29.
Fig. 3.29.
3.128. There is an infinite straight chain of alternating charges $q$ and $-q$. The distance between the neighbouring charges is equal to $a$. Find the interaction energy of each charge with all the others.

Instruction. Make use of the expansion of $\ln (1+\alpha)$ in a power series in $\alpha$.
3.129. A point charge $q$ is Iocated at a distance $l$ from an infinite tonducting plane. Find the interaction energy of that charge with chose induced on the plane.
3.130. Calculate the interaction energy of two balls whose charges $q_{1}$ and $q_{2}$ are spherically symmetrical. The distance between the centres of the balls is equal to $l$.

Instruction. Start with finding the interaction energy of a ball and a thin spherical layer.
3.131. A capacitor of capacitance $C_{1}=1.0 \mu \mathrm{F}$ carrying initially a voltage $V=300 \mathrm{~V}$ is connected in parallel with an uncharged capacitor of capacitance $C_{2}=2.0 \mu \mathrm{F}$. Find the increment of the electric energy of this system by the moment equilibrium is reached. Explain the result obtained.
122
3.132. What amount of heat will be generated in the circuit shown in Fig. 3.30 after the switch $S w$ is shifted from position 1 to position 2 ?
Fig. 3.30.
Fig. 3.31.
3.133. What amount of heat will be generated in the circuit shown in Fig. 3.31 after the switch $S w$ is shifted from position 1 to position 2 ?
3.134. A system consists of two thin concentric metal shells of radii $R_{1}$ and $R_{2}$ with corresponding charges $q_{1}$ and $q_{2}$. Find the selfenergy values $W_{1}$ and $W_{2}$ of each shell, the interaction energy of the shells $W_{12}$, and the total electric energy of the system.
3.135. A charge $q$ is distributed uniformly over the volume of a ball of radius $R$. Assuming the permittivity to be equal to unity, find:
(a) the electrostatic self-energy of the ball;
(b) the ratio of the energy $W_{1}$ stored in the ball to the energy $W_{2}$ pervading the surrounding space.
3.136. A point charge $q=3.0 \mu \mathrm{C}$ is located at the centre of a spherical layer of uniform isotropic dielectric with permittivity $\varepsilon=3.0$. The inside radius of the layer is equal to $a=250 \mathrm{~mm}$, the outside radius is $b=500 \mathrm{~mm}$. Find the electrostatic energy inside the dielectric layer.
3.137. A spherical shell of radius $R_{1}$ with uniform charge $q$ is expanded to a radius $R_{2}$. Find the work performed by the electric forces in this process.
3.138. A spherical shell of radius $R_{1}$ with a uniform charge $q$ has a point charge $q_{0}$ at its centre. Find the work performed by the electric forces during the shell expansion from radius $R_{1}$ to radius $R_{2}$.
3.139. A spherical shell is uniformly charged with the surface density $\sigma$. Using the energy conservation law, find the magnitude of the electric force acting on a unit area of the shell.
3.140. A point charge $q$ is located at the centre $O$ of a spherical uncharged conducting

Fig. 3.32. layer provided with a small orifice (Fig. 3.32). The inside and outside radii of the layer are equal to $a$ and $b$ respectively. What amount of work has to be performed to slowly transfer the charge $q$ from the point $O$ through the orifice and into infinity?
3.141. Each plate of a parallel-plate air capacitor has an area $S$. What amount of work has to be performed to slowly increase the distance between the plates from $x_{1}$ to $x_{1}$ if
(a) the capacitance of the capacitor, which is equal to q, or (b) the voltage across the capacitor, which is equal to $\boldsymbol{V}$, is kept constant in the proces?
3.142. Inside a parallel-plate capacitor there is a plate parallel to the outer plates, whose thickness is equal to $\eta=0.60$ of the gap width. When the plate is absent the capacitor capacitance equals $c=20 \mathrm{nF}$. First, the capacitor was connected in parallel to a constant voltage source producing $V=200 \mathrm{~V}$, then it was disconected from it, after which the plate was slowly removed from the gap. Find the work performed during the removal, if the plate is
(a) made of metal; (b) made of glass.
3.143. A parallel-plate capacitor was lowered inte water in a horizontal position, with water filling up the gap between the plates $d=1.0 \mathrm{~mm}$ wide. Then a constant voltage $\boldsymbol{V}=500 \mathrm{~V}$ was applied to the capacitor. Find the water pressure increment in the gap.
3.144. A parallel-plate capacitor is located horizontally so that ene of its plates is submerged inte liquid while the other is over its surface (Fig. 3.33). The permittivity of the liquid is equal to $\varepsilon$, its density is equal to $\rho$. To what height will the level of the liquid in the capacitor rise after its plates get a charge of surface dessity o?
3.145. A cylindrical layer of dielectric with permittivity $\varepsilon$ is inserted into a cylindrical capacitor to fill up all the space between the electrodes. The mean radius of the electredes equals $R$, the gap between them is equal to $d$, with $d \ll R$. The constant voltage $V$ is applied across the electrodes of the capacitor. Find the magnitude of the electric force pulling the dielectrie inte the capacitor.
3.146. A capacitor consists of two stationary plates shaped as a semi-cirele of radius $R$ and a movable plate made of dielectrie with permittivity $\varepsilon$ and capable of rotating about an axis $O$ between the stationary plates (Fig. 3.34). The thickness of the movable plate is equal to $d$ which is practically the separation between the stationary plates. A potential diflerence $V$ is applied to the capacitor. Find the magnitude of the moment of forces relative to the axis $O$ acting on the movable plate in the position shown in the figure.