– Faraday’s law of electromagnetic induction:
\[
\mathscr{E}_{t}=-\frac{d \Phi}{d t}
\]
– In the case of a solenoid and doughnut coil:
\[
\Phi=N \Phi_{1},
\]
where $N$ is the number of turns, $\Phi_{1}$ is the magnetic flux through each turn.
– Inductance of a solenoid:
\[
L=\mu \mu_{0} n^{2} V .
\]
– Intrinsic energy of a current and interaction energy of two currents:
\[
W=\frac{L I^{2}}{2}, \quad W_{12}=L_{12} I_{1} I_{2} .
\]
– Volume density of magnetic field energy:
\[
w=\frac{B^{2}}{2 \mu \mu_{0}}=\frac{\mathbf{B H}}{2} .
\]
– Displacement current density:
\[
\mathbf{j}_{d t s}=\frac{\partial \mathrm{B}}{\partial t} .
\]
– Maxwell’s equations in differential form:
\[
\begin{array}{ll}
\boldsymbol{
abla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, & \boldsymbol{
abla} \cdot \mathbf{B}=0, \\
\boldsymbol{
abla} \times \mathbf{H}=\mathbf{j}+\frac{\partial \mathrm{D}}{\partial t}, & \boldsymbol{
abla} \cdot \mathbf{D}=\rho,
\end{array}
\]
where $
abla \times \equiv$ rot (the rotor) and $
abla \cdot \equiv \operatorname{div}$ (the divergence).
– Field transformation formulas for transition from a reference frame $K$ to a reference frame $K^{\prime}$ moving with the velocity $v_{0}$ relative to it.
In the case $v_{0} \ll c$
In the general case
\[
\begin{array}{ll}
\mathbf{E}_{\|}^{\prime}=\mathbf{E}_{\|}, & \mathbf{B}_{\|}^{\prime}=\mathbf{B}_{\|}, \\
\mathbf{E}_{\perp}^{\prime}=\frac{\mathbf{E}_{\perp}+\left[\mathbf{v}_{0} \mathbf{B}\right]}{\sqrt{1-\left(v_{0} / c\right)^{2}}}, & B_{\perp}^{\prime}=\frac{\mathbf{B}_{\perp}-\left[\mathbf{v}_{0} \mathbf{E}\right] / c^{2}}{\sqrt{1-\left(v_{0} / c\right)^{2}}},
\end{array}
\]
where the symbols $\|$ and $\perp$ denote the field components, respectively parallel and perpendicular to the vector $\mathrm{v}_{0}$.
3.258. A wire bent as a parabola $y=a x^{2}$ is located in a uniform magnetic field of induction $B$, the vector $B$ being perpendicular to the plane $x, y$. At the moment $t=0$ a connector starts sliding translationwise from the parabola apex with a constant acceleration a (Fig. 3.78). Find the emf of electromagnetic induction in the loop thus formed as a function of $y$.
ค. 3.78 .
Fie. 3.79.
3.289. A rectangular loop with a sliding consecter of length $l$ is located in a uniform magnetic field perpendicular to the loop plane (Fig. 3.79). The magnetie induction is equal to B. The connector has an electric resistance $R$, the sides $A B$ and $C D$ have resistances $R_{1}$ and $\boldsymbol{R}_{3}$ respectively. Neglecting the self-inductance of the loop. find the current flowing in the connector during its motion with a constant velocity $v$.
3.290. A metal dise of radius $a=25 \mathrm{~cm}$ rotates with a constant angular velocity o- $-130 \mathrm{rad} / \mathrm{s}$ about its axis. Find the potential difference between the centre and the rim of the disc if
(a) the external magnetic field is absent;
(b) the external uniform magnetic field of induction $B=5.0 \mathrm{mT}$ is directed perpendicular to the disc.
3.291. A thin wire $A C$ shaped as a semi-eircle of diameter $d=$ $-20 \mathrm{~cm}$ rotates with a constant angular velocity $0-100 \mathrm{rad} / \mathrm{s}$ in a uniferm magnetic field of induction $B=5.0 \mathrm{mT}$, with a If B. The rotation axis passes through the end $\boldsymbol{A}$ of the wire and is perpendicular to the diameter $\boldsymbol{A} C$. Find the value of a line integral $\int \mathrm{E} d r$ along the wire from point $A$ to point $C$. Generalize the obtained result.
3.292. A wire loop enclosing a semi-circle of radius a is located on the boundary of a uniform magnetic field of induction $B$ (Fig. 3.80). At the moment $t=0$ the loop is set inte rotation with – constant angular acceleration $\beta$ about as axis $O$ coinciding with a line of vecter $\mathbf{B}$ on the boundary. Find the omf induced in the loop as a function of time $t$. Draw the appreximate plet of this function. The arrew in the figure shows the emf direction taken to be positive.
3.293. A long straight wire carrying a curreat I and a n-shaped conductor with sliding connector are located in the same plane as
shown in Fig. 3.81. The connecter of length $l$ and resistance $R$ slides to the right with a constant velocity $\mathrm{v}$. Find the current induced in
Fie. 3.80
Fie. 3.81.
the loop as a function of separation $r$ between the connector and the straight wire. The resistance of the II-shaped conducter and the selfinductance of the loop are assumed to be negligible.
3.294. A square frame with side $a$ and a long straight wire carrying a current $I$ are located in the same plane as shown in Fig. 3.82. The frame translates to the right with a constant velocity 8 . Find the emf induced in the frame as a function of distance $x$.
Fie. 3.82 .
Fie. 3.83 .
3.295. A metal rod of mass $m$ can rotate about a horizental axis $O$, sliding aleng a circular conductor of radius a (Fig. 3.83). The arrangement is located in a uniform magnetie field of induction $B$ directed perpendicular to the ring plane. The axis and the ring are connected to an emf source to form a circuit of resistance $\boldsymbol{R}$. Neglecting the friction, circuit inductance, and ring resistance, find the law according to which the source emf must vary to make the rod rotate with a constant angular velocity $\oplus$.
3.296. A copper connector of mass $m$ slides down $t w e$ smeeth cepper bars, set at an angle $\alpha$ to the horizontal, due to gravity (Fig. 3.84). At the top the bars are intereonnected through a resistance $R$. The separation between the bars is equal to $L$. The system is lecated in a uniform magnetic feld of induction $B$, perpendicular to the plane in which the connector slides. The resistances of the bars, the censector and the sliding contacts, as well as the self-inductance of the loop. are assumed to be negligible. Find the steady-state velocity of the connector.
3.297. The system differs from the one exanined in the foregoing problem (Fig. 3.84) by a capacitor of capacitance $C$ replacing the resistance $R$. Find the acceleration of the connector.
3.298. A wire shaped as a semi-cirele of radius a rotates about an axis $O O^{\prime}$ with an angular velocity o in a uniform magnetic field of induction $B$ (Fig. 3.85). The rotation axis is perpendicular to the field direction. The total resistance of the cireuit is equal to $R$. Neglecting the magnetie field of the induced current, find the mean amount of thermal power being generated in the loop during . rotation period.
3.299. A small coil is intreduced between the poles of an electremagnet so that its axis coincides with the magnetic field direction. The cross-sectional area of the coil is equal to $S=3.0 \mathrm{~mm}^{2}$, the number of tarns is $N=60$. When the coil turns through $180^{\circ}$ about its diameter, a ballistie galvanometer connected to the coil indicates a charge $q=4.5 \mu \mathrm{C}$ flowing through it. Find the magnetic induetion magnitude between the poles provided the total resistance of the electrie circuit equals $R^{\prime}=40 \Omega$.
3.300. A square wire frame with side a and a straight conductor carrying a constant current $I$ are located in the same plane (Fig. 3.86).
ne. 3.s6.
ne. 3.87.

The inductance and the resistance of the frame are equal to $L$ and $R$ respectively. The frame was tarned threugh $180^{\circ}$ about the axis $O O^{\prime}$ separated froa the current-carrying conducter by a distance 6 . Find the electric charge having llown through the frame.
3.301. A long straight wire carries a current $I_{\text {f }}$. At distances a and $b$ from it there are two other wires, parallel to the former one, which are interconsected by a resistance $R$ (Fig. 3.87). A connector
150
slides without friction aleng the wires with a constant velocity $v$. Assuming the resistances of the wires, the connector, the sliding contacts, and the self-inductance of the freme to be negligible, find:
(a) the magnitude and the direction of the current induced in the connector;
(b) the force required to maintain the connector’s velocity constant.
3.302. A conducting red $A B$ of mass $m$ slides without friction over two long conducting rails separated by a distance $l$ (Fig. 3.88). At the left end the rails are interconnected by a resistance $R$. The system is located in a uniform magnetic field perpendicular to the plane of the loop. At the moment $t=0$ the rod $A B$ starts moving to the right with an initial velocity $v_{0}$. Neglecting the resistances of the rails and the $\operatorname{rod} A B$, as well as the self-inductance, find:
(a) the distance covered by the rod until it comes to a standstill;
(b) the amount of heat generated in the resistance $\boldsymbol{R}$ during this process.
Vig. 3.ss,
ne. 3.89 .
3.303. A connector $A B$ can slide without friction along a IIshaped conductor located in a horizental plane (Fig. 3.89). The connector has a length $I$, mase $m$, and resistance $R$. The whele system is located in a uniform magnetic field of induction $B$ directed vertically. At the moment $t=0$ a constant herizontal force $F$ starts acting en the connector shifting it translationwise to the right. Find how the velocity of the connector varies with time $t$. The inductance of the loop and the resistance of the II-shaped conductor are assumed to be negligible.
3.304. Fig. 3.90 illustrates plane figures made of thin conductors which are located in a uniform magnetic field directed away from a
Fis. 2.90 .
reader beyend the plane of the drawing. The magnetic induction starts diminishing. Find how the currents induced in these loops are directed.
3.305. A plane loop shown in Fig. 3.91 is shaped as twe squares with sides $a=20 \mathrm{~cm}$ and $b=10 \mathrm{~cm}$ and is introduced inte aniform magnetic field at right angles to the loop’s plane. The magnetic induction varies with time as $B=B_{0} \sin$ ot, where $B_{0}=10 \mathrm{mT}$ and $\varphi=100 \mathrm{~s}^{-1}$. Find the amplitude of the current induced in the loop if its resistance per unit length is equal to $\rho=$ $50 \mathrm{~m} \Omega \mathrm{m}$. The inductance of the loop is to be neglected.
3.306. A plane spiral with a great number $\boldsymbol{N}$ of turns weund tightly to one another is located in a uniform magnetic field perpendicular to the spiral’s plane. The outside

Fie. 3.91. radius of the spiral’s turns is equal to a. The magnetic induction varies with time as $B-B_{0}$ sin $\omega t$, where $B_{0}$ and $\oplus$ are constants. Find the amplitude of emf induced in the spiral.
3.307. A II-shaped conductor is located in a uniform magnetic field perpendicular to the plane of the conductor and varying with time at the rate $\dot{B}=0.10 \mathrm{~T} / \mathrm{s}$. A conducting connecter starts moving with an acceleration $u=10 \mathrm{~cm} / \mathrm{s}^{2}$ along the parallel bars of the conductor. The length of the connector is equal to $l=20 \mathrm{~cm}$. Find the emf induced in the loop $t=2.0 \mathrm{~s}$ after the beginning of the motion, if at the moment $t=0$ the loep area and the magnetic induetion are equal to rero. The inductance of the loop is to be neglected.
3.308. In a long straight solenoid with cross-sectional radius a and number of turns per unit length $n$ a current varies with a sonstant velocity $i \mathrm{~A} / \mathrm{s}$. Find the magnitude of the eddy current field strength as a function of the distance $r$ from the soleneid axis. Draw the approximate plet of this function.
3.309. A long straight soleneid of eress-sectional diameter $d=$ $=5 \mathrm{~cm}$ and with $n=20$ turns per ene $\mathrm{cm}$ of its length has a round turn of copper wire of cross-sectional area $S=1.0 \mathrm{~mm}$ ‘ tightly put on its winding. Find the current flowing in the turn if the current in the soleneid winding is increased with a constant velocity $i=$ $-100 \mathrm{~A} / \mathrm{s}$. The inductance of the turn is to be neglected.
3.310. A long soleneid of cross-sectional radius $a$ has a thin insolated wire ring tightly put on its winding; one half of the ring has the resistance $n$ times that of the other half. The magnetic induction produced by the soleneid varies with time as $B=b t$, where $b$ is a constant. Find the magnitude of the electric field strength in the ring.
3.311. A thin non-conducting ring of mass $m$ carrying a charge $q$ ean freely rotate about its axis. At the initial moment the ring was at rest and ne magnetic field was present. Then a practically uniform magnetie field was switched on, which was perpendicular to the plane
152
of the ring and increased with time accerding to a certain law B (t). Find the angular velocity of the ring as a function of the induction B $(t)$.
3.312. A thin wire ring of radius $a$ and resistance $r$ is located inside a long solenoid so that their axes coincide. The length of the selenoid is equal to $i$, its cross-sectional radius, to b. At a certain moment the solenoid was connected to a source of a constant voltage $\boldsymbol{V}$. The total resistance of the circuit is equal to $\boldsymbol{R}$. Assuming the inductance of the ring to be negligible, find the maximum value of the radial force acting per unit length of the ring.
3.313. A magnetic flux through a stationary loop with a resistance $R$ varies during the time interval $\tau$ as $\Phi=a t(x-8)$. Find the ameunt of heat generated in the loep during that time. The inductance of the loop is to be neglected.
3.314. In the middle of a long soleneid there is a coaxial ring of square cross-section, made of conducting material with resistivity $\rho$. The thickness of the ring is equal to $h$, its inside and outside radii are equal to $a$ and $b$ respectively. Find the current induced in the ring if the magnetic induction produced by the solenoid varies with time as $B=B t$, where $B$ is a constant. The inductance of the ring is to be neglected.
3.315. How many metres of a thin wire are required to manufacture a solenoid of length $l_{0}=100 \mathrm{~cm}$ and inductance $L=1.0 \mathrm{mH}$ if the solenoid’s cross-sectienal diameter is considerably less than its length?
3.316. Find the inductance of a solenoid of length $t$ whose winding is made of copper wire of mass $m$. The winding resistance is equal to $\boldsymbol{R}$. The solenoid diameter is considerably less than its length.
3.317. A coil of inductance $L=300 \mathrm{mH}$ and resistance $R=$ $=140 \mathrm{~m} \Omega$ is connected to a constant veltage source. How seon will the coil current reach $\eta=50 \%$ of the steady-state value?
3.318. Calculate the time constant $\mathrm{r}$ of a straight solenoid of length $l=1.0 \mathrm{~m}$ having a single-layer winding of copper wire whose total mass is equal to $m=1.0 \mathrm{~kg}$. The cress-sectional diameter of the solenoid is assumed to be considerably less than its length.

Note. The time constant $\mathrm{t}$ is the ratio $L / R$, where $L$ is inductance and $\boldsymbol{A}$ is active resistance.
3.319. Find the inductance of a unit length of a cable cousisting of twe thin-walled coaxial metallic cylinders if the radius of the outside cylinder is $\eta=3.6$ times that of the inside one. The permeability of a medium between the cylinders is assumed to be equal to unity.
3.320. Calculate the inductance of a doughnut soleneid whese inside radius is equal to $b$ and cross-section has the form of a square with side 6 . The solenoid winding consists of $N$ turns. The space inside the solenoid is filled up with uniform paramagnetic having permeability $\mu$.
3.321. Calculate the inductance of a unit length of a double tape line (Fig. 3.92) if the tapes are separated by a distance $h$ which is considerably less than their width b, namely, $b / h=50$.
3.322. Find the inductance of a unit length of a double line if the radius of each wire is $n$ times less than the distance between the axes of the wires. The field inside the wires is to be neglected, the permeability is assumed to be equal to unity throughout, and $\eta \gg 1$.
3.323. A superconducting round
Fie. 2.92 ring of radies a and inductence $L$

Fie. 2.92 was located in a uniform magnetic field of induction $B$. The ring plane was parallel to the vector B, and the current in the ring was equal to zero. Then the ring was turned through $90^{\circ}$ so that its plane became perpendicular to the field. Find:
(a) the current induced in the ring after the turn:
(b) the work performed during the turn.
3.324. A current $I_{0}=1.9 \mathrm{~A}$ flows in a long closed solenoid. The wire it is wound of is in a superconducting state. Find the current flowing in the soleaoid when the length of the solenoid is increased by $\eta=5 \%$.
3.325. A ring of radius $a=50 \mathrm{~mm}$ made of thin wire of radius $b=1.0 \mathrm{~mm}$ was located in a uniform magnetic field with induction $B=0.50 \mathrm{mT}$ so that the ring plase was perpendicular to the vecter $B$. Then the ring was cooled down to a superconducting state, and the magnetic field was switched ofl. Find the ring current after that. Note that the inductance of a thin ring along which the surface current Dows is equal to $L=\mu_{0} a\left(\ln \frac{80}{b}-2\right)$.
3.326. A elosed circuit consists of source of censtant emt $\boldsymbol{\delta}$ and a choke coll of inductance $L$ connected in series. The active resistance of the whole circuit is equal to $R$. At the moment $t=0$ the choke coil inductance was decreased abruptly $\eta$ times. Find the curreat in the circuit as a function of time $t$.

Instruction. During a stepwise change of induatance the total magnetie flux (fux linkage) remains constant.
3.327. Find the time dependence of the current flowing through the inductance $L$ of the circuit shown in Fig. 3.93 after the switch $S w$ is shorted at the moment $t=0$.
3.328. In the circuit shown in Fig. 3.94 an ent 8 , a resistance $R$, and ceil inductances $L_{1}$ and $L_{2}$ are known. The internal resistance of the source and the coil resistances are negligible. Find the steadystate currents in the coils after the switch $S v$ was shorted.
3.329. Calculate the mutual inductance of a long straight wire and a rectangular frame with sides $a$ and $b$. The frame and the wire lie IS6
in the same plane, with the side b being closest to the wire, reparated by a distance $l$ from it and orieated parallel to it.
Fig. 3.93.
Fig. 3.94 .
3.330. Determine the zutual isductance of a deughnut coil and an infinite straight wire passing along its axis. The coil has a rectangular crose-section, its inside radius is equal to e and the outside ene, to b. The length of the doughnut’s cross-sectional side parallel to the wire is equal to $h$. The coil has $\boldsymbol{N}$ turns. The system is located in a uniform magnetie with peraeability $\mu$.
3.331. Two thin concentric wires shaped as circles with radii a and $b$ lie in the same plane. Allewing for $a<b$, find:
(a) their mutual inductance;
(b) the magnetic flux through the surface enclosed by the outside wire, when the inside wire carries a current $I$.
3.332. A small eylindrical magnet $M$ (Fig. 3.95) is placed in the centre of a thin coil of radius a consisting of $N$ turns. The ceil is connected to a ballistic galvanometer. The active resistance of the whole circuit is equal to $\hat{A}$. Find the magnetic moment of the magnet if its removal from the coil results in a charge $q$ flowing through the galvanometer.
3.333. Find the approximate formula expressing the mutual inductance of two thin coaxial loops of the same radius a if their centres are separated by a distance $i$, with $l>a$.
Fie. 3.95.
Fie. 3.98
3.334. There are two stationary loops with mutual inductance $L_{\mathrm{p}}$. The current in one of the loeps starts to be varied as $I_{1}=a t$, where $a$ is a constant, $t$ is time. Find the time dependence $I_{2}(t)$ of the current in the ether loop whose inductance is $L_{2}$ and resistance $\boldsymbol{R}$.
3.335. A coil of inductance $L=2.0 \mu \mathrm{H}$ and resistance $R=1.0 \mathrm{\Omega}$ is connected to a source of constant emf $8=3.0 \mathrm{~V}$ (Fig. 3.96). A resistance $\boldsymbol{R}_{0}=2.0 \Omega$ is connected in parallel with the ceil. Find the amount of heat generated in the coil after the switch $S u$ is disconnected. The internal resistance of the source is segligible.
3.336. An iren tere supports $N=500$ turns. Find the magnetie field energy if a current $I=2.0 \mathrm{~A}$ produces a magnetic flux across the tore’s cross-section equal to $\Phi=1.0 \mathrm{mWb}$.
3.337. An iron core shaped as a doughnut with round cross-section of radius $a=3.0 \mathrm{~cm}$ carries a winding of $N=1000$ turns through which a current $I=1.0 \mathrm{~A}$ flows. The mean radius of the doughnut is $b=32 \mathrm{~cm}$. Using the plot in Fig. 3.76, find the magnetie energy stored up in the core. A field strength $\boldsymbol{H}$ is supposed to be the same throughout the cross-section and equal to its magnitude in the centre of the cross-section.
3.338. A thin ring made of a magnetic has a mean diameter $d=30 \mathrm{~cm}$ and supports a winding of $N=800$ tums. The cross. sectional area of the ring is equal to $S=5.0 \mathrm{~cm}^{2}$. The ring has a cross-cut of width $b=2.0 \mathrm{~mm}$. When the winding carries a certain current, the permeability of the magnetic equals $\mu=1400$. Neglecting the dissipation of magnetic fux at the gap edges, find:
(a) the ratio of magnetic energies in the gap and in the magnetic;
(b) the inductance of the system; do it in two ways: using the flux and using the energy of the field.
3.339. A long cylinder of radius a carrying a uniform surface charge retates about its axis with an angular velecity a. Find the mag. netic field energy per unit length of the cylinder if the linear charge density equals $\lambda$ and $\mu=1$.
3.340. At what magnitude of the electric field strength in vacuum the velume energy density of this field is the same as that of the magnetie field with induction $B=1.0 \mathrm{~T}$ (alse in vacuum).
3.341. A thin uniformly charged ring of radius $a=10 \mathrm{~cm}$ rotates about its axis with an angular velocity $\omega=100 \mathrm{rad} / \mathrm{s}$. Find the $\mathrm{ra}$ tio of volume energy densities of magnetic and electric fields on the axis of the ring at a point remeved from its centre by a distance $l=c$.
3.342. Using the expression for volume density of magnetic energy, demonstrate that the amount of work contributed to magnetization of a unit volume of para- or diamagnetic, is equal to $A$ $=-\mathrm{JB} / 2$.
3.343. Two identical coils, each of inductance $L$, are interconnected (a) in series, (b) in parallel. Assuming the mutual inductance of the coils to be negligible, find the inductance of the system in both cases.
3.344. Two solenoids of equal length and almost equal cross sectional area are fully inserted inte ene anether. Find their mutual inductance if their inductances are equal to $L_{1}$ and $L_{2}$.
3.345. Demonstrate that the magnetic energy of interaction of two current-carrying loops located in vacuum can be represented as $W_{i_{0}}=\left(1 / \mu_{4}\right) \int \mathbf{B}_{1} B_{1} d V$, where $B_{1}$ and $\mathbf{B}_{2}$ are the magnetie inductions 156
within a velume element $d V$, produced individually by the currents of the first and the second loop respectively.
3.346. Find the interaction energy of twe loops carrying currents $I_{1}$ and $I_{3}$ if both loops are shaped as circles of radii $a$ and $b$, with $a \& b$. The loops’ centres are located at the same point and their planes form an angle $\theta$ between them.
3.367. The space between two concentric metallic spheres is filled up with a uniform poorly condueting medium of resistivity $p$ and permittivity \&. At the moment $t=0$ the inside sphere obtains a certain charge. Find:
(a) the relation between the vectors of displacement current density and conduction current density at an arbitrary point of the medium at the same moment of time;
(b) the displacement current across an arbitrary closed surface wholly lecated in the medium and enclosing the internal sphere, if at the given moment of time the charge of that sphere is equal to $\mathrm{q}$.
3.348. A parallel-plate capseiter is formed by twe dises with uniform poorly conducting medium between them. The capacitor was initially charged and then disconnected from a voltage source. Neglecting the edge effects, show that there is no magnetic field between capacitor plates.
3.369. A parallel-plate air condenser whose each plate has an ares $S=100 \mathrm{~cm}^{2}$ is connected in series to an ac circuit. Find the electric field strength amplitude in the capacitor if the sinusoidal current amplitude in lead wires is equal to $I_{\mathrm{m}}=1.0 \mathrm{~mA}$ and the current frequency equals $0=1.6 \cdot 10^{7} \mathrm{~s}^{-1}$,
3.350. The space between the electrodes of a parallel-plate capacitor is filled with a uniform poorly conducting medium of conductivity $a$ and peraittivity . The capacitor plates shaped as round discs are separated by a distance $d$. Neglecting the edge effects, find the magnetic field strength between the plates at a distance $r$ from their axis if an ac voltage $V=V_{n}$ cos of is applied to the capacitor.
3.351. A leng straight solenoid has $n$ turns per unit length. An alternating current $I=I_{2}$ sin of flows through it. Find the displacement current density as a function of the distance $r$ from the solenoid axis. The cross-sectional radius of the solenoid equals $R$.
3.352. A point charge q moves with a non-relativistic veloeity v – const. Find the displacement current density $f_{4}$ at a point located at a distance $r$ from the charge on a straight line
(a) coinciding with the charge path;
(b) perpeadicular to the path and passing through the charge.
3.353. A thin wire ring of radies a carrying a charge $q$ approaches the observation point $P$ so that its centre moves rectilinearly with a constant velocity $\mathrm{v}$. The plane of the ring remains perpendicular to the motion direction. At what distance $x_{n}$ from the point $P$ will the ring be located at the moment when the displacement current density at the point $P$ becomes maximum? What is the magnitude of this maximum density?
3.354. A point charge $q$ moves with a non-relativistic velocity $\mathbf{v}=$ const. Applying the theorem for the circulation of the vector $\mathbf{H}$ around the dotted circle shown in Fig. 3.97, find $H$ at the point $A$ as a function of a radius vector $r$ and velocity $v$ of the charge.
3.355. Using Maxwell’s equations, show that
(a) a time-dependent magnetie field cansot exist without an electrie field;
(b) a uniform electric field cannet exist in the presence of a timedependent magnetic field:
(c) iaside an empty cavity a uniform electric (or magnetic) field can be time-dependent.
3.356. Demonstrate that the law of electric charge censervation. i.e. $\boldsymbol{
abla} \cdot \mathbf{j}=-\partial \mathrm{p} / \partial t$, follows from Maxwell’s equations.
3.357. Demonstrate that Maxwell’s equations $\mathbf{
abla} \times \mathbf{E}=-\partial \mathbf{B} / \partial t$ and $\boldsymbol{
abla} \cdot \mathbf{B}=0$ are compatible, i.e. the first one does not contradict the second one.
3.358. In a certain region of the inertial reference frame there is magnetic field with induction $B$ rotating with angular velocity $\oplus$. Find $\mathbf{v} \times \mathbf{E}$ in this region as a function of vectors $\boldsymbol{\omega}$ and $\mathbf{B}$.
3.359. In the inertial reference frame $\boldsymbol{K}$ there is a uniform magnetic field with induction B. Find the electric field strength in the frame $\boldsymbol{K}^{\prime}$ which moves relative to the frame $\boldsymbol{K}$ with a nen-relativistic velocity $\mathbf{v}$, with $\mathbf{v} \perp \mathbf{B}$. To solve this problem, censider the forees acting on an imaginary charge in beth reference frames at the moment when the velocity of the charge in the frame $K^{\prime}$ is equal to zero.
3.360. A large plate of non-ferromagnetic material moves with a constant velocity $v=90 \mathrm{~cm} / \mathrm{s}$ in a uniform mangnetie field with indection $B=50 \mathrm{mT}$ as shewn in Fig. 3.98. Find the surface density of electric charges appearing on the plate as a result of its motion.
3.361. A long solid aluminum cylinder of radius $e=5.0 \mathrm{~cm}$ rotates about its axis in a uniform magnetic fleld with induction $B=10 \mathrm{mT}$. The angular velocity of rotation equals $\omega=45 \mathrm{rad} / \mathrm{s}$, with o tf B. Neglecting the magnetic feld of appearing charges, find their space and surface densities.
3.362. A non-relativistic point charge $q$ moves with a constant velocity $\mathbf{v}$. Using the field transformation formulas, find the magnetic induction B produced by this charge at the point whose position relative to the charge is determined by the radius vecter $\mathrm{f}$.
3.363. Using Eqs. (3.6h), demenstrate that if in the inertial reforence frame $K$ there is only electric or only magnetic feld, in any other inertial frame $\boldsymbol{K}^{\prime}$ both electric and magnetic felds will coexist simultaneously, with $\mathbf{E}^{\prime} \perp \mathbf{B}^{\prime}$.
3.364. In an inertial reference frame $K$ there is only magnetic feld with induction $B=b(y i-x j) /\left(x^{2}+y^{n}\right)$, where $b$ is a constant, $i$ and $j$ are the unit vectors of the $x$ and $y$ axes. Find the electric field strength $\mathrm{E}^{\prime}$ in the frame $K^{\prime}$ moving relative to the frame $K$ with a constant nen-relativistic velocity $\mathbf{v}=u \mathbf{k} ; \mathbf{k}$ is the unit vector of the $z$ axis. The $z^{\prime}$ axis is assumed to coincide with the $z$ axis. What is the shape of the field $\mathbf{E}^{\prime}$ ?
3.365. In an inertial reference frame $K$ there is only electric field of strength $\mathbf{E}=a(x i+y j) /\left(x^{2}+y^{2}\right)$, where $a$ is a constant, $\mathbf{i}$ and fare the unit vectors of the $x$ and $y$ axes. Find the magnetic induction $\mathbf{B}^{\prime}$ in the frame $\boldsymbol{K}^{\prime}$ moving relative to the frame $\boldsymbol{K}$ with a constant non-relativistie velocity $\mathbf{v}=\mathbf{k} ; \mathbf{k}$ is the unit vector of the $z$ axis. The $z^{\prime}$ axis is assumed to coincide with the $z$ axis. What is the shape of the magnetic induction $\mathbf{B}^{\prime}$ ? .
3.366. Demonstrate that the transformation formulas (3.6h) follow from the formulas $(3.6 i)$ at $v_{0} \ll c$.
3.367. In an inertial reference frame $K$ there is only a uniform electric field $E=8 \mathrm{kV} / \mathrm{m}$ in strength. Find the modulus and direction
(a) of the vector $\mathbf{E}^{\prime}$, (b) of the vector $\mathbf{B}^{\prime}$ in the inertial reference frame $K^{\prime}$ moving with a constant velocity v relative to the frame $\boldsymbol{K}$ at an angle $a=45^{\circ}$ to the vector $\mathbf{E}$. The velocity of the frame $\boldsymbol{K}^{\prime}$ is equal to a $\beta=0.60$ fraction of the velocity of light.
3.368. Solve a problem differing from the foregoing one by a magnetic field with induction $B=0.8 \mathrm{~T}$ replacing the electric field.
3.369. Electromagnetie field has twe invariant quantities. Using the transformation formulas (3.6i), demonstrate that these quantities are
(a) EB; (b) $E^{4}-c^{2} B^{2}$
3.370 . In an inertial reference frame $K$ there are $t w o$ uniform $\mathrm{mu-}$ twally perpendicular fields: an electric field of strength $E=40 \mathrm{kV} / \mathrm{m}$ and agnetic field isduction $B=0.20 \mathrm{mT}$. Find the electric strength $E^{\prime}$ (or the magnetic induction $\boldsymbol{B}^{\prime}$ ) in the reference frame $K^{\prime}$ where enly ene field, electric er magnetic, is ebserved.
Instruction. Make use of the field invariants eited in the foregoing problem.
3.371. A point charge $q$ meves uniformly and rectilinearly with a relativistic velocity equal to a $\beta$ fraction of the velocity of light $(\beta-$ vle). Find the electric field strength $\mathbf{E}$ produced by the charge at the point whose radius vector relative to the charge is equal to $r$ and forms an angle $\theta$ with its velocity vecter.