– Ohm’s law for an inhomogeneos negment of a circuit:
where $V_{11}$ is the voltage drop acrose the megant.
– Dillerential form of obs’s law:
\[
\mathrm{J}=\epsilon^{(\mathbf{E}+\mathbf{E} *)}
\]
where $\mathrm{E}^{*}$ is the strength of a field produced by oxtraneous ferces.
– Kirchbolt’s laws (for an electrie eircuit):
\[
\Sigma t_{k}=0, \quad \Sigma I_{k} n_{4}=\Sigma t_{k} .
\]
– Poser $P$ of eurrust and thermal power $Q$ :
\[
P=V T=\left(\varphi_{1}-\varphi_{1}+\gamma_{1}\right) T, \quad Q=R T \text {, }
\]
– Speciste power $P_{s p}$ of eurrent asd specise thermal pewer Qup:
\[
P_{o p}=J(\mathbf{E}+\mathbf{E}), \quad Q_{n p}=p \mathbf{j}^{r}
\]
– Carrent denaity in a metal:
1- nns.
where o is the avarag velecity of carriers.
– Number of ions recombising per unit volune of gas per unit times
\[
i,=r n^{2} \text {. }
\]
where $r$ is the reombination coelteient.
3.147. A long cylinder with uniformly charged surface and crosssectional radius $a=1.0 \mathrm{~cm}$ moves with a constant velocity $v=$ $=10 \mathrm{~m} / \mathrm{s}$ along its axis. An electric field strength at the surface of the cylinder is equal to $\boldsymbol{B}=0.9 \mathrm{kV} / \mathrm{em}$. Find the resulting convection current, that is, the current caused by mechanical transfer of a charge.
3.148. An air cylindrical capacitor with a de voltage $V=200 \mathrm{~V}$ applied across it is being submerged vertically into a vessel filled with water at a velocity $v=5.0 \mathrm{~mm} / \mathrm{s}$. The electrodes of the capacitor are separated by a distance $d=2.0 \mathrm{~mm}$, the mean curvature radius of the electrodes is equal to $r=50 \mathrm{~mm}$. Find the current flowing in this case along lead wires, if $d \ll r$.
3.149. At the temperature $0^{\circ} \mathrm{C}$ the electric resistance of conductor 2 is $\pi$ times that of cosductor 1 . Their temperature coeflicients of resistance are equal to $a_{3}$ and $a_{1}$ respectively. Find the temperature coefficient of resistance of a circuit segment consisting of these two conductors when they are connected
(a) is series; (b) in parallel.
3.150. Find the resistance of a wire frame
R.3. 3.35. shaped as a cube (Fig. 3.35) when measured between points (a) $1-7 ;$ (b) $1-2 ;$ (c) 13 .
The resistance of each edge of the frame is $A$
3.151. At what value of the resistance $R_{\text {, }}$ in the eirevit show in Fig. 3.36 will the total resistance between points $A$ and $B$ be independent of the number of cells?
Fie. 3.3 .
3.152. Fig. 3.37 shows an infinite circuit formed by the repetition of the same link, consisting of resistance $R_{1}=4.0 \Omega$ and $R_{3}=3.0 \Omega$. Find the resfstance of this efreult between points $A$ and $B$.
Fig. 3.37,
3.153. There is an infinite wire grid with square cells (Fig. 3.38). The resistance of each wire between aeighbouring joint connections is equal to $\boldsymbol{R}_{6}$. Find the resistance $\boldsymbol{R}$ of the whole grid between points $A$ and $B$.

Instruetion. Make use of priseiples of symmetry and superposition.
3.154. A homogeneous poorly conducting medium of resistivity $p$ fills op the space between two this coaxial ideally conducting eylinders. The radil of the eylinders are equal to $a$ and $b$, with $a<b$, the length of each eylinder is $t$. Neglecting the edge eflects, find the resistance of the medium between the cylisders.
3.155. A metal ball of radius $a$ is surrounded by a thin concentric metal shell of radies b. The space between these electrodes is filled up with a poorly conducting homogeneous medium of resistivity $P$; Find the resistance of the interelectrode gap. Analyse the obtained solution at $b \rightarrow \infty$.
3.156. The space between two conducting concentrie spheres of radif $a$ and $b(a<b)$ is filled up with homogeneous peorly conducting medium. The capacitance of such a system equals $C$. Find the resistivity of the medium if the potential difference between the spheres, when they are disconected from an external voltage, decreases $\eta$-fold during the time isterval $\Delta t$.
3.157. Two metal balls of the same radius $a$ are located in a homogeneoss poorly conducting medium with resistivity p. Find the resistance of the medium between the balls provided that the separation between them is much greater than the radius of the ball.
3.158. A metal ball of radius $a$ is located at a distance $t$ from an infinite ideally conducting plane. The space around the ball is filled with a homogeneous poorly conducting medium with resistivity $p$. In the case of $a<t$ find:
(a) the current density at the conducting plane as a function of distance $r$ from the ball if the potential difference between the ball and the plane is equal to $V$ :
(b) the electric resistance of the medium between the ball and the plane.
3.159. Two long parallel wires are located in a poorly conducting medium with resistivity $\rho$. The distance between the axes of the wires is equal to $b$, the cross section radius of each wire equals $a$. In the case a $i$ find:
(a) the current density at the point equally removed from the axes of the wires by a distance $r$ if the poteatial difference between the wires is equal to $\boldsymbol{V}_{\text {: }}$
(b) the electrie resistance of the medium per unit length of the wires.
3.160. The gap between the plates of a parallel-plate capacitor is filled with glass of resistivity $\rho=100 \mathrm{G} \Omega \cdot \mathrm{m}$. The capacitance of the capacitor equals $C=4.0 \mathrm{aF}$. Find the leakage curreat of the capacitor when a voltage $\boldsymbol{V}-2.0 \mathrm{kV}$ is applied to it.
3.161. Two conductors of arbitrary shape are embedded into an infinite homogeneous poorly conducting medium with resistivity $\rho$ and permittivity $\varepsilon$. Find the value of a product $\boldsymbol{R} G$ for this system, where $\boldsymbol{R}$ is the resistance of the medium between the conductors, and $C$ is the mutual capacitance of the wires in the presence of the medium.
3.162. A conductor with resistivity $p$ bousds on a dielectrie with permittivity $\varepsilon$. At a certain point $\boldsymbol{A}$ at the conductor’s surface the electrie displacement equals $D$, the vector D being directed away from the conductor and forming an angle a with the normal of the surlace. Find the surlace density of charges on the conductor at the poist $A$ and the current density in the conducter in the viciaity of the same point.
3.163. The gap between the plates of a parallel-plate capacitor is filled up with an inhomogeneous poorly conducting medium whose conductivity varies linearly in the direction perpendicular to the plates from $\sigma_{1}=1.0 \mathrm{pS} / \mathrm{m}$ to $\sigma_{2}=2.0 \mathrm{pS} / \mathrm{m}$. Each plate has an area $S=230^{\circ} \mathrm{cm}^{2}$, and the separation between the plates is $d=$ $=2.0 \mathrm{~mm}$. Find the current flowing through the capacitor due to a voltage $\boldsymbol{y}=300 \mathrm{~V}$.
3.164. Demonstrate that the law of refraction of direct curreat lines at the boundary between two conducting media has the form $\tan \alpha_{2} / \tan \alpha_{1}-\sigma_{v} / \sigma_{1}$, where $\sigma_{1}$ and $\sigma_{3}$ are the conductivities of the media, $\alpha_{1}$ and $a_{1}$ are the angles bet ween the current lines and the normal of the boundary surface.
3.165. Two eylindrical conductors with equal eross-sections and different resistivities $\rho_{1}$ and $\rho_{1}$ are put end to end. Find the charge at the boundary of the conductors if a current $I$ flows from conductor $t$ to conductor 2 .
3.166. The gap between the plates of a parallel-plate capacitor is fllled up with two dielectric layers $I$ and 2 with thicknesses $d_{1}$ and $d_{p}$ permittivities $\varepsilon_{1}$ and $\varepsilon_{p}$ and resistivities $\rho_{l}$ and $\rho_{2} . A$ de voltage $V$ is applied to the capacitor, with electric field directed from layer $\boldsymbol{I}$ to layer 2. Find $\sigma$, the surface density of extraneous charges at the boundary between the dielectrie layers, and the condition ander which $\sigma=0$.
3.167. An inhomogeneous poorly conducting medium fills up the space between plates $I$ and 2 of a parallel-plate capacitor. Its permittivity and resistivity vary from values $\varepsilon_{2}, p_{1}$ at plate $I$ to values $\varepsilon_{2}, \rho_{2}$ at plate 2 . A de voltage is applied to the capacitor through which a steady current $I$ flows from plate $I$ to plate 2 . Find the total extraneous charge in the given medium.
3.168. The space between the plates of a parallel-plate capacitor is filled up with inhomogeneous poorly cosducting medium whose resistivity varies linearly in the direction perpendicular to the plates. The ratio of the maximum value of resistivity to the minimum one is equal to $\eta$. The gap width equals $d$. Find the volume density of the charge in the gap if a voltage $\boldsymbol{V}$ is applied to the capacitor. $c$ is assumed to be Ieverywhere.
3.169. A long round conductor ot cross-sectional area $S$ is made of material whose resistivity depends ouly on a distance $r$ from the axis of the conductor as $\rho=a / r^{2}$, where $\alpha$ is a constant. Find:
(a) the resistance per unit length of such a cenductor;
(b) the electric field strength in the conductor due to which a current $\boldsymbol{I}$ flows through it.
3.170. A capacitor with capacitance $C=400 \mathrm{pF}$ is consected vin a resistance $\boldsymbol{R}=650 \Omega$ to source of constant voltage $\boldsymbol{V}_{\mathbf{b}}$. How soon will the voltage developed across the capacitor reach a value $\boldsymbol{V}=0.90 \quad \boldsymbol{V}_{\mathrm{s}}$ ?
3.171. A capacitor filled with dielectrie of permittivity $\varepsilon=2.1$ loses half the eharge acquired during a time interval $\mathrm{\tau}=3.0 \mathrm{~min}$. Assuming the charge to leak only through the dielectric filler, calculate its resistivity.
3.172. A eireuit consists of a souree of a constant $\mathrm{em}$ I $\mathcal{E}$ and a resist ance $A$ and a capacitor with capacitance $C$ consected in series. The internal resistance of the source is aegligible. At a moment $t=0$ the capacitance of the capacitor is abruptly decreased $\eta$-fold. Find the current flowing through the cireuit as a function of time $t$.
3.173. An ammeter and a veltmeter are connected in series to a battery with an emt $\delta=6.0 \mathrm{~V}$. When a certain resistance is connected
128
in parallel with the veltmeter, the readings of the latter decrease $\eta=2.0$ times, whereas the readings of the ammeter increase the same number of times. Find the voltaneter readings after the connection of the resistance.
3.174. Find a potential diflerence $\varphi_{1}-\varphi_{1}$ between points $I$ and 2 of the circuit show in Fig. 3.39 if $R_{2}=10 \Omega, R_{2}=20 \Omega, \varepsilon_{1}=$ $=5.0 \mathrm{~V}$, and $\boldsymbol{\delta}_{2}=2.0 \mathrm{~V}$. The internal resistances of the current sources are negligible.
3.175. Two sources of current of equal emf are consected in series and have diflerent internal resistances $R_{1}$ and $\boldsymbol{R}_{2}\left(R_{1}>R_{1}\right)$. Find the external resistance $R$ at which the potential diflerence across the terminals of one of the sources (which one in particular?) be-

Fe. 3.39 . comes equal to zero.
3.176. $N$ sources of current with diflerent emf’s are consected as shown in Fig. 3.49. The emf’s of the sources are proportional to
Fie. 3.40 .
Fie. 3.4 .
their internal resistances, i.e. $\boldsymbol{\delta}-\alpha \boldsymbol{R}$, where $\alpha$ is an assigned constant. The lead wire resistance is negligible. Find:
(a) the current in the circuit;
(b) the potential difleresce between points $A$ and $B$ dividing the eircuit in $n$ and $N-n$ links.
3.177. In the circuit shown in Fig. 3.41 the sources have emf’s $\boldsymbol{8}_{1}=1.0 \mathrm{~V}$ and $\boldsymbol{\delta}_{1}=2.5 \mathrm{~V}$ and the resistances have the values $R_{1}=10 \Omega$ and $R_{2}=20 \Omega$. The internal resistances of the sources are negligible. Find a potential difference $\varsigma_{A}-
abla_{0}$, betwees the plates $A$ and $B$ of the capacitor $C$.
3.178. In the circuit shown in Fig. 3.42 the emf of the source is equal to $\varepsilon=5.0 \mathrm{~V}$ and the resistances are equal to $R_{1}=4.0 \mathrm{\Omega}$ and $R_{1}=6.0 \Omega$. The internal resistance of the source equals $R=$ $=0.10 \Omega$. Find the currents flowing through the resistances $\bar{R}_{1}$ and $\boldsymbol{H}_{\boldsymbol{r}}$.
3.179. Fig. 3.43 illustrates a potentiometric circuit by means of which we can vary a voltage $V$ applied to a certain device possessing – resistance $\boldsymbol{R}$. The potentiometer has a length $l$ and a resistance $R_{0}$, and voltage $V_{0}$ is applied to its terminals. Find the voltage $V$ fed to the device as a function of distance $x$. Analyse separately the case $R \gg R_{0}$.
Fig. 3.42.
Fig. 3.43.
3.180. Find the emf and the internal resistance of a source which is equivalent to two batteries connected in parallel whose emf’s are equal to $\mathscr{E}_{1}$ and $\mathscr{E}_{2}$ and internal resistances to $R_{1}$ and $R_{2}$.
3.181. Find the magnitude and direction of the current flowing through the resistance $R$ in the circuit shown in Fig. 3.44 if the
Fig. 3.44.
Fig.3.45.
emf’s of the sources are equal to $\mathscr{E}_{1}=1.5 \mathrm{~V}$ and $\mathscr{E}_{2}=3.7 \mathrm{~V}$ and the resistances are equal to $R_{1}=10 \Omega, R_{2}=20 \Omega, R=5.0 \Omega$. The internal resistances of the sources are negligible.
3.182. In the circuit shown in Fig. 3.45 the sources have emf’s $\mathscr{E}_{1}=1.5 \mathrm{~V}, \quad \mathscr{E}_{2}=2.0 \mathrm{~V}, \mathscr{E}_{3}=2.5 \mathrm{~V}$, and the resistances are equal to $R_{1}=10 \Omega, R_{2}=20 \Omega, R_{3}=30 \Omega$. The internal resistances of the sources are negligible. Find:
(a) the current flowing through the resistance $R_{1}$;
(b) a potential difference $\varphi_{A}-\varphi_{B}$ between the points $A$ and $B$.
3.183. Find the current flowing through the resistance $R$ in the circuit shown in Fig. 3.46. The internal resistances of the Fig. 3.46. batteries are negligible.
3.184. Find a potential difference $\varphi_{A}-\varphi_{B}$ between the plates of a capacitor $C$ in the circuit shown in Fig. 3.47 if the sources have emf’s $\mathscr{E}_{1}=4.0 \mathrm{~V}$ and $\mathscr{E}_{2}=1.0 \mathrm{~V}$ and the resistances are equal to $R_{1}=10 \Omega, R_{2}=20 \Omega$, and $R_{3}=30 \Omega$. The internal resistances of the sources are negligible.
130
3.185. Find the current flowing through the resistance $R_{1}$ of the circuit shown in Fig. 3.48 if the resistances are equal to $R_{1}=10 \Omega$, $R_{2}=20 \Omega$, and $R_{3}=30 \Omega$, and the potentials of points 1,2 , and 3 are equal to $\varphi_{1}=10 \mathrm{~V}, \varphi_{2}=6 \mathrm{~V}$, and $\varphi_{3}=5 \mathrm{~V}$
Fig. 3.47.
Fig. 3.48.
3.186. A constant voltage $V=25 \mathrm{~V}$ is maintained between points $A$ and $B$ of the circuit (Fig. 3.49). Find the magnitude and
Fig. 3.49.
Fig. 3.50.
direction of the current flowing through the segment $C D$ if the resistances are equal to $R_{1}=1.0 \Omega, R_{2}=2.0 \Omega, R_{3}=3.0 \Omega$, and $R_{4}=$ $=4.0 \Omega$.
3.187. Find the resistance between points $A$ and $B$ of the circuit shown in Fig. 3.50.
3.188. Find how the voltage across the capacitor $C$ varies with time $t$ (Fig. 3.51) after the shorting of the switch $S w$ at the moment $t=0$.
Fig. 3.51.
Fig. 3.52.
3.189. What amount of heat will be generated in a coil of resistance $R$ due to a charge $q$ passing through it if the current in the coil
(a) decreases down to zero uniformly during a time interval $\Delta t$;
(b) decreases down to zero halving its value every $\Delta t$ seconds?
3.190. A de source with internal resistance $R_{0}$ is loaded with three identical resistances $R$ interconnected as shown in Fig. 3.52.
At what value of $\boldsymbol{R}$ will the thermal power generated in this eireuit be the highest?
3.194. Make sure that the eurrent distribution over $t$ mo resistances $\boldsymbol{A}_{1}$ and $\boldsymbol{R}_{2}$ connected in parallel corresponds to the minimum thermal power generated is this circuit.
3.192. A storage battery with emf $\varepsilon=2.6 \mathrm{~V}$ loaded with an external resistance produces a current $I=1.0 \mathrm{~A}$. In this case the potential difference between the terminals of the stomge battery equals $V=2.0 \mathrm{~V}$. Find the thermal power geaerated in the battery and the power developed in it by electric lorces.
3.193. A voltage $V$ is applied to a de electrie motor. The armature winding resistance is equal to $\boldsymbol{R}$. At what value of curreat flowing through the winding will the useful power of the motor be the highest? What is it equal to? What is the motor efficiency in this case?
3.194. Ilow much (in per cent) has a filameat diameter decreased due to evaporation if the maintenance of the previous temperature required an increase of the voltage by $\eta=1.0 \%$ ? The amount of heat transferred from the filament into surrounding space is assumed to be proportional to the filament surlace area.
3.195. A conductor has a temperature-independent resistance $\boldsymbol{A}$ and a total heat capacity $C$. At the moment $i=0$ it is consected to a de voltage $\boldsymbol{V}$. Find the time dependence of a conductor’s temperature $r$ asruming the thermal power diseipated inte surrounding space to vary as $q-k\left(T-T_{0}\right)$, where $k$ is a constant, $T$, is the environmental temperature (equal to the conductor’s temperature at the initial moment).
3.196. A cireuit shows in Fig. 3.53 has resistances $R_{1}=20 \Omega$ and $R_{2}=30 \Omega$. At what value of the resistance $R_{x}$ will the thermal
Fig. $\mathbf{3 . 5 3}$
Fie. 2.54.
power generated in it be practically independent of small variations of that resistance? The voltage between the points $\boldsymbol{A}$ and $B$ is supposed to be constant in this case.
3.197. In a circuit shown in Fig. 3.54 resistances $R_{1}$ and $R_{2}$ are known, as well as emf’s $\boldsymbol{\varepsilon}_{1}$ and $\boldsymbol{\varepsilon}_{1}$. The internal resistances of the sources are negligible. At what value of the resistance $\boldsymbol{R}$ will the thermal power generated in it be the highest? What is it equal to?
3.198. A series-parallel combination battery consisting of a large number $N=300$ of identical cells, each with as internal resistance
132
$r=0.3 \Omega$, is loaded with an external resistance $R=10 \Omega$. Find the number $n$ of parallel groups consisting of an equal number of cells consected in series, at which the external resistance generates the highest thermal power.
3.199. A capacitor of capacitance $C=5.00 \mu \mathrm{F}$ is consected to a source of constant emf $8=200 \mathrm{~V}$ (Fig. 3.55). Then the switch $S_{w}$ was thrown over from contact $I$ to contact 2. Find the amount of heat generated in a resistance $R_{1}=500 \Omega$ if $R_{2}=330 \Omega$.
3.200. Between the plates of a parallel-plate capacitor there is a metallie plate whose thickness takes up $\eta=0.60$ of the capacitor
Fie. 3.s5.
Fie. 3.56.
gap. When that plate is absent the capacitor has a capacity $C=$ $=20 \mathrm{nF}$. The capacitor is connected to a de voltage source $V=$ $=100 \mathrm{~V}$. The metallie plate is slowly extracted from the gap. Find:
(a) the eaergy iscrement of the capacitor;
(b) the mechanical work performed in the process of plate extraction.
3.201. A glass plate totally fills up the gap between the electrodes of a parallel-plate capacitor whose capacitance in the absence of that glass plate is equal to $C=20 \mathrm{nF}$. The capacitor is connected to a de voltage source $V=100 \mathrm{~V}$. The plate is slowly, and without friction, extracted from the gap. Find the capacitor energy increment and the mechanical work performed in the process of plate extraetion.
3.202. A cylindrical capacitor consected to a de voltagesource $\boldsymbol{V}$ touches the surface of water with its end (Fig. 3.56). The separation $d$ between the capacitor electrodes is substantially less than their mean radius. Find a height $h$ to which the water level in the gap will rise. The capillary eflects are to be neglected.
3.203. The radif of spherical capacitor electrodes are equal to $a$ and $b$, with $a<b$. The interelectrode space is filled with homogeneous substance of permittivity $\varepsilon$ and resistivity $p$. Initially the capacitor is not charged. At the moment $t=0$ the internal electrode gets a charge qp. Find:
(a) the time variation of the charge on the internal electrode:
(b) the amount of heat geserated during the spreading of the charge.
3.204. The electrodes of a capacitor of capacitance $C=2.00 \mu \mathrm{F}$ carry opposite charges $q_{0}=1.00 \mathrm{mC}$. Then the electrodes are interconnected through a resistance $R=5.0$ M . Find:
(a) the charge flowing through that resistance during a time interval $\mathrm{x}=2.00 \mathrm{~s}$;
(b) the amount of heat generated in the resistance during the same interval.
3.205. In a circuit shown in Fig. 3.57 the capacitance of each capacitor is equal to $C$ and the resistance, to $R$. One of the capacitors was consected to a voltage $V$, and then at the moment $t=0$ was shorted by means of the switeh Sus. Find:
(a) a current $I$ in the eircuit as a function of time $\mathrm{f}$ :
(b) the amount of generated heat provided a dependence $I(f)$ is known.
3.206. A coil of radius $r=25 \mathrm{~cm}$ wound of thin copper wire of length $t=500 \mathrm{~m}$ rotates with an angular velocity $\omega=300 \mathrm{rad} / \mathrm{s}$ about its axis. The coil is conected to a ballistic galvanometer by means of sliding contacts. The total resistance of the circuit is equal to $R=21 \Omega$. Find the specific charge of current carriers in copper if a suddes stoppage of the coil makes a charge $q=10$ nC flow through the galvanometer.
3.207. Find the total momentum of electrons in a straight wire of length $l=1000 \mathrm{~m}$ carrying a current $I=70 \mathrm{~A}$.
3.206. A copper wire carries a current of density $j=1.0 \mathrm{~A} / \mathrm{mm}^{2}$, Assuming that one free electron corresponds to each copper atom, evaluate the distance which will be covered by an electron during its displacement $t=10 \mathrm{~mm}$ along the wire.
3.209. A straight copper wire of length $l=1000 \mathrm{~m}$ and crosssectional area $S=1.0 \mathrm{~mm}^{2}$ carries a current $I=4.5 \mathrm{~A}$. Asuming that one free electron corresponds to each copper atom, find:
(a) the time it takes an electron to displace from one end of the wire to the ether:
(b) the sum of electric forces acting on all free electrons in the given wire.
3.210. A homogeneous proton beam accelerated by a poteatial difference $\boldsymbol{y}=600 \mathrm{kV}$ has a rouad cros-section of radius $r=$ $=5.0 \mathrm{~mm}$. Find the electric field strength on the surface of the beam and the potential difference between the surface and the axis of the beam if the beam current is equal to $I=50 \mathrm{~mA}$.
3.21t. Two large parallel plates are located in vacuum. One of them serves as a cathede, a source of electrons whose initial velocity. is negligible. An electron flow directed toward the opposite plate produces a space charge causing the potential in the gap between the plates to vary as $\psi=a x^{3,3}$, where $a$ is a positive constant, and $x$ is the distance from the cathode. Find:
(a) the volume density of the space charge as a function of $x$;
(b) the current deasity.
3.212. The air between two parallel plates separated by a distance $d=20 \mathrm{~mm}$ is ionized by $\boldsymbol{X}$-ray radiation. Each plate has an area $S=500 \mathrm{~cm}^{2}$. Find the concentration of positive ions if at a voltage $V=100 \mathrm{~V}$ : current $I=3.0 \mu \mathrm{A}$ flows between the plates, which is well below the saturation current. The air ios mobilities are ut = $=1.37 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $\mathrm{w}_{5}=1.91 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$.
3.213. A gas is ionized in the immediate vicinity of the surface of plane electrode $I$ (Fig. 3.58) separated from electrode 2 by a distance $L$. An alternating voltage varying with time $t$ as $V=V$, sin et is applied to the electrodes. On decreasing the frequency o it was observed that the galvanometer $G$ indicates a current only at $e<\omega_{0}$. where $\omega_{0}$ is a certain cut-off frequency. Find the mobility of ions reaching electrode 2 under these conditions.
3.214. The air betwees two closely located plates is uniformly lonited by ultraviolet radia: tios. The air volume betwees the plates is equal to $\boldsymbol{V}=500 \mathrm{~cm}^{2}$, the observed saturation current is equal to $I_{z=t}=0.48 \mu \mathrm{A}$. Find:
(a) the number of ion pairs produced in a unit volume per unit time:
(b) the equilibrium concentration of ios pairs if the recombination coefficient for air ions is equal to $r=1.67 \cdot 10^{-4} \mathrm{~cm}^{3} / \mathrm{s}$.
3.215. Having been operated long enough, the ionizer producing $\dot{n}_{1}=3.5 \cdot 10^{\circ} \mathrm{cm}^{-3} \cdot \mathrm{s}^{-1}$ of ion pairs per uait volume of air per unit time was switched off. Assuming that the only process tending to reduce the number of ions in air is their recombination with coefficient $r=1.67 \cdot 10^{-4} \mathrm{~cm}^{3} / \mathrm{s}$, find how soon after the ionizer’s switebing of the ion concentration decreases $\eta=2.0$ times.
3.216. A parallel-plate air capacitor whose plates are separated by a distance $d=5.0 \mathrm{~mm}$ is first charged to a potential difference $V=90 \mathrm{~V}$ and then disconsected from a de voltage source. Find the time interval during which the voltage across the capacitor decreases by $\eta=1.0 \%$, taking into account that the average number of ios pairs formed is air under standard conditions per unit volume per unit time is equal to $\dot{n}_{4}=5.0 \mathrm{~cm}^{-3}, \mathrm{~s}^{-1}$ and that the given voltage corresponds to the saturation current.
3.217. The gap between two plane plates of a capacitor equal to $d$ is filled with a gas. One of the plates emits $v_{0}$ electrons per second which, moving in an electric field, ionize gas molecules; this way each electron produces $a$ new electrons (and ions) along a unit length of its path. Find the electronic curreat at the opposite plate, aeglect. ing the ionization of gas molecules by formed ions.
3.218. The gas between the capacitor plates separated by a distance $d$ is uniformly ionized by ultraviolet radiation so that $n_{i}$ electrons per unit volume per second are formed. These electrons moving in the electric field of the capacitor ionize gas molecules, each electron producing $\alpha$ new electrons (and ions) per unit length of its path. Neglecting the ionization by ions, find the electronic current density at the plate possessing a higher potential.