— Magnetic field of a point charge $q$ moving with non-relativistic velocity $\mathbf{v}$ :
$$\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{q[\mathbf{v}\mathbf{r}]}{r^3}.$$
— Biot-Savart law:
$$d\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{[\mathbf{j}\mathbf{r}]}{r^3}dV,d\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{I[d\mathbf{l},\mathbf{r}]}{r^3}.$$
— Circulation of a vector $\mathbf{B}$ and Gauss’s theorem for it:
$$\oint \mathbf{B}d\mathbf{r}={\mu }_{0}I,\oint \mathbf{B}d\mathbf{S}=0.$$
— Lorentz force:
— Ampere force:
$$\mathbf{F}=q\mathbf{E}+q[\mathbf{v}\mathbf{B}].$$
$$d\mathbf{F}=[\mathbf{j}\mathbf{B}]dV,d\mathbf{F}=I[d\mathbf{l},\mathbf{B}].$$
— Force and moment of forces acting on a magnetic dipole ${\mathbf{p}}_m=IS\mathbf{n}$ :
$$\mathbf{F}=p_m\frac{\partial \mathbf{B}}{\partial n},\mathbf{N}=\left[{\mathbf{p}}_m\mathbf{B}\right]$$
where $\partial \mathbf{B}/\partial n$ is the derivative of a vector $\mathbf{B}$ with respect to the dipole direction.
— Circulation of magnetization J:
$$\oint \mathbf{J}d\mathbf{r}=I^{\prime },$$
where $I^{\prime }$ is the total molecular current.
— Vector $\mathbf{H}$ and its circulation:
$$\mathbf{H}=\frac{\mathbf{B}}{{\mu }_0}-\mathbf{J},\oint \mathbf{H}d\mathrm{r}=I,$$
where $I$ is the algebraic sum of macroscopic currents.
— Relations at the boundary between two magnetics:
$$B_{1n}=B_{2n},H_{1\tau }=H_{2\tau }\cdot$$
— For the case of magnetics in which $\mathrm{J}=\chi \mathbf{H}$ :
$$\mathbf{B}=\mu {\mu }_0\mathbf{H},\mu =1+\chi .$$
3.219. A current $I=1.00$ A circulates in a round thin-wire loop of radius $R=100\mathrm{mm}$. Find the magnetic induction
(a) at the centre of the loop;
(b) at the point lying on the axis of the loop at a distance $x=$ $=100\mathrm{mm}$ from its centre.
3.220. A current $I$ flows along a thin wire shaped as a regular polygon with $n$ sides which can be inscribed into a circle of radius $R$. Find the magnetic induction at the centre of the polygon. Analyse the obtained expression at $n\to \mathrm{\infty }$.
3.221. Find the magnetic induction at the centre of a rectangular wire frame whose diagonal is equal to $d=16\mathrm{cm}$ and the angle between the diagonals is equal to $\phi ={30}^{\circ }$; the current flowing in the frame equals $I=5.0\mathrm{A}$.
3.222. A current $I=5.0$ A flows along a thin wire shaped as shown in Fig. 3.59. The radius of a curved part of the wire is equal to $R=$ $=120\mathrm{mm}$, the angle $2\phi ={90}^{\circ }$. Find the magnetic induction of the field at the point $O$.
Fig. 3.59.
Fig. 3.60.
3.223. Find the magnetic induction of the field at the point $O$ of a loop with current $I$, whose shape is illustrated
(a) in Fig. $3.60a$, the radii $a$ and $b$, as well as the angle $\phi$ are know;
(b) in Fig. 3.60b, the radius $a$ and the side $b$ are known.
3.224. A current $I$ flows along a lengthy thin-walled tube of radius $R$ with longitudinal slit of width $h$. Find the induction of the magnetic field inside the tube under the condition $h\ll R$.
3.225. A current $I$ flows in a long straight wire with cross-section having the form of a thin half-ring of radius $R$ (Fig. 3.61). Find the induction of the magnetic field at the point $O$.
Fig. 3.61.
Fig. 3.62.
3.226. Find the magnetic induction of the field at the point $O$ if a current-carrying wire has the shape shown in Fig. $3.62a,b,c$. The radius of the curved part of the wire is $R$, the linear parts are assumed to be very long.

3.227. A very long wire carrying a current $I=5.0\mathrm{A}$ is bent at right angles. Find the magnetic induction at a point lying on a perpendicular to the wire, drawn through the polnt of beading, at a distance $t=35\mathrm{cm}$ from it.
3.228. Find the magnetic induction at the point $O$ if the wire carrying a current $I=8.0\mathrm{A}$ has the shape shown in Fig. $3.63\mathrm{a},b,c$.
Fie. 3.63 .

The radius of the curved part of the wire is $R-100\mathrm{mm}$, the linear parts of the wire are very long.
3.229. Find the magnitude and direction of the magnetie induction vector B
(a) of an infinite plane carrying a current of linear density i; the vector $i$ is the same at all points of the plane;
(b) of two parallel infinite planes carrying curreats of linear den: sities I and -i; the vectors i and -i are censtant at all points of the corresponding planes.
3.230. $A$ uniform current of density $/$ Dows inside an infinite plate of thickness $2d$ parallel to its surface. Find the magnetie induction induced by this current as a function of the distance $x$ from the median plane of the plate. The magnetic permeability is assumed to be equal to unity both inside and outside the plate.
3.231. A direct current $I$ flows along a lengthy straight wire. From the point $o$ (Fig. 3.64) the current spreads radially all over an infinite conducting plane perpendiesIar to the wire. Find the magnetic induction

ทie. 3.64 . at all points of space.
3.232. A current $I$ flows along a round loop. Find the integral $\int \mathbf{B}d\mathrm{r}$ along the axis of the loop within the range from $-\mathrm{\infty }$ to $+\mathrm{\infty }$. Explain the result obtained.
3.233. A direct current of density f flows along a round uniform straight wire with cross-section radius $R$. Find the magnetic induction vecter of this current at the point whose position relative to the axis of the wire is defined by a radius vector $\mathbf{r}$. The magnetic permeability is assumed to be equal to unity throughout all the space.
3.234. Inside a long straight uniform wire of round eross-section there is a long round cyliadrical cavity whose axis is parallel to the axis of the wire and displaced from the latter by a distance 1 . A direct current of density $\mathbf{j}$ flows along the wire. Find the magnetic induction inside the cavity. Consider, in particular, the case $\mathbf{I}=0$.
3.235. Find the curreat density as a function of distance $r$ from the axis of a radially symmetrical parallel stream of electrons if the magnetic induction inside the strean varies as $B=b{r}^2$, where $b$ and $a$ are positive constants.
3.236. A single-layer coil (solenoid) has length $t$ and cros-section radius $\mathit{A},\mathrm{A}$ number of turns per unit length is equal to $n$. Find the magnetie induction at the centre of the coil when a current $I$ flows through it.
3.237. A very long straight solenoid has a cross-section radius $\mathit{R}$ and $n$ turn per unit length. A direct current $I$ flows through the solenoid. Suppose that $z$ is the distance from the end of the solenoid, measured along its axis. Find:
(a) the magnetic induction $B$ on the axis as a function of $x$; draw an approximate plot of $B$ vs the ratio $x/R$;
(b) the distance $x_0$ to the point on the axis at which the value of $B$ diflers by $\eta -1⩽$ from that is the middle section of the soleneid.
3.238. A thin conducting strip of width $h=2.0\mathrm{cm}$ is tightly wound in the shape of a very long coil with cross-section radius $A=$ $=2.5\mathrm{cm}$ to make a single-layer straight solenold. A direct curreat $I=5.0\mathrm{A}$ flows through the strip. Find the magnetic induction inside and outside the solenoid as a function of the distance $r$ from its aris.
3.239. $N=2.5\cdot {10}^{\circ }$ wire turns are un iformly wound on a wooden toroidal core of very small cross-section. A current $I$ flows through the wire. Find the ratio $\eta$ of the magnetic induction inside the core to that at the centre of the toroid.
3.240. A direct carrent $I=10\mathrm{A}$ fows in a long straight round conductor. Find the magnetie flux through a half of wire’s crosssection per one metre of its length.
3.251. A very long straight soleneid carries a current $I$. The cross-sectional area of the solenoid is equal to $S$, the number of turns per wait length is equal to $n$. Find the flux of the vector B through the end plane of the solenoid.
3.242. Fig. 3.65 shows a toroidal solenoid whose eross-section is rectangular. Find the magnetie fux through this cross-section if the current through the winding equals $I=1.7\mathrm{A}$, the total Fie. 3.65 . number of turns is $N=1000$, the ratio of the outside diameter to the iaside one is $\eta =1.6$, and the height is equal to $\mathrm{h}-5.0\mathrm{cm}$.
3.243. Find the magnetic moment of a thin round loop with current if the radius of the loop is equal to $R=100\mathrm{mm}$ and the magnetic induction at its centre is equal to $B=6.0\mu \mathrm{T}$.
3.244. Calculate the magnetic moment of a thin wire with a current $I=0.8\mathrm{A}$, wound tightly on half a tore (Fig. 3.66). The diameter of the cross-section of the tore is equal to $d=5.0\mathrm{cm}$, the number of turns is $N=500$.
Fig. 3.66.
Fig. 3.67.
3.245. A thin insulated wire forms a plane spiral of $N=100$ tight turns carrying a current $I=8\mathrm{mA}$. The radii of inside and outside turns (Fig. 3.67) are equal to $a=50\mathrm{mm}$ and $b=100\mathrm{mm}$. Find:
(a) the magnetic induction at the centre of the spiral;
(b) the magnetic moment of the spiral with a given current.
3.246. A non-conducting thin disc of radius $R$ charged uniformly over one side with surface density $\sigma$ rotates about its axis with an angular velocity $\omega$. Find:
(a) the magnetic induction at the centre of the disc;
(b) the magnetic moment of the disc.
3.247. A non-conducting sphere of radius $R=50\mathrm{mm}$ charged uniformly with surface density $\sigma =10.0\mu \mathrm{C}/{\mathrm{m}}^2$ rotates with an angular velocity $\omega =70\mathrm{rad}/\mathrm{s}$ about the axis passing through its centre. Find the magnetic induction at the centre of the sphere.
3.248. A charge $q$ is uniformly distributed over the volume of a uniform ball of mass $m$ and radius $R$ which rotates with an angular velocity $\omega$ about the axis passing through its centre. Find the respective magnetic moment and its ratio to the mechanical moment. 3.249. A long dielectric cylinder of radius $R$ is statically polarized so that at all its points the polarization is equal to $\mathbf{P}=\alpha \mathbf{r}$, where $\alpha$ is a positive constant, and $r$ is the distance from the axis. The cylinder is set into rotation about its axis with an angular velocity $\omega$. Find the magnetic induction $\mathbf{B}$ at the centre of the cylinder.
3.250. Two protons move parallel to each other with an equal velocity $v=300\mathrm{km}/\mathrm{s}$. Find the ratio of forces of magnetic and electrical interaction of the protons.
3.251. Find the magnitude and direction of a force vector acting on a unit length of a thin wire, carrying a current $I=8.0\mathrm{A}$, at a point $O$, if the wire is bent as shown in
(a) Fig. $3.68a$, with curvature radius $R=10\mathrm{cm}$;
(b) Fig. 3.68b, the distance between the long parallel segments of the wire being equal to $l=20\mathrm{cm}$.
3.252. A coil carrying a current $I=10\mathrm{mA}$ is placed in a uniform magnetic field so that its axis coincides with the field direction. The single-layer winding of the coil is made of copper wire with
Fig. 3.68.
Fig. 3.69.
diameter $d=0.10\mathrm{mm}$, radius of turns is equal to $R=30\mathrm{mm}$. At what value of the induction of the external magnetic field can the coil winding be ruptured?
3.253. A copper wire with cross-sectional area $S=2.5{\mathrm{mm}}^2$ bent to make three sides of a square can turn about a horizontal axis $O{O}^{\prime }$ (Fig. 3.69). The wire is located in uniform vertical magnetic field. Find the magnetic induction if on passing a current $I=16\mathrm{A}$ through the wire the latter deflects by an angle $\theta ={20}^{\circ }$.
3.254. A small coil $C$ with $N=200$ turns is mounted on one end of a balance beam and introduced between the poles of an electromagnet as shown in Fig. 3.70. The cross-sectional area of the coil
Fig. 3.70.
is $S=1.0{\mathrm{cm}}^2$, the length of the arm $OA$ of the balance beam is $l=30\mathrm{cm}$. When there is no current in the coil the balance is in equilibrium. On passing a current $I=22\mathrm{mA}$ through the coil the equilibrium is restored by putting the additional counterweight of mass $\mathrm{\Delta }m=60\mathrm{mg}$ on the balance pan. Find the magnetic induction at the spot where the coil is located.
3.255. A square frame carrying $=$ earrent $I=0.90$ A is located in the same plane as a long straight wire carrying a current $I_0=$ $=5.0\mathrm{A}$. The frame side has a length $a=8.0\mathrm{cm}$. The axis of the frame passing through the midpoints of opposite sides is parallel to the wire and is separated from it by the distance which is $\eta =1.5$ times greater than the side of the frame. Find:
(a) Ampere force acting on the frame;
(b) the mechanical work to be performed in order to turn the frame through ${180}^{\circ }$ about its axis, with the currents maintained constant.
3.256. Two long parallel wires of negligible resistance are connected at one end to a resistance $R$ and at the other end to a de voltage source. The distance bet ween the axes of the wires is $\eta =20$ times greater than the cross-sectional radius of each wire. At what value of resistance $\mathit{R}$ does the resultant force of interaction between the wires tum into tero?
3.257. A direct current $I$ flows in a long straight conductor whose cross-section has the form of a thin half-ring of radius $\mathit{R}$. The same current flows in the opposite direction along a thin conductor located on the \»axis\» of the first conduetor (peint $O$ in Fig. 3.61). Find the magnetic interaction force between the given conducters reduced to anit of their length.
3.258. Two long thin parallel conductors of the shape shewn in Fig. 3.71 carry direct currents $I_1$ and $I_4$. The separation between the conductors is $a$, the width of the right-hand conducter is equal to $b$. With both conductors lying in one plane, find the magnetic interaction force between them reduced to a unit of their length.
3.259. A system consists of two parallel planes earrying eurrents produeing a uniform magnetic feld of induction $B$ between the planes. Outside this space there is no magnetic field. Find the magnetic force acting per unit area of each plane.
3.260. A conducting current-carrying plane is placed in an external uniform magnetic field. As a result, the magnetic induction becomes
Fie. 3.72.
equal to $B_1$ on one side of the plase and to $B_0$ on the other. Find the magnetic force acting per unit area of the plane in the cases illustrated in Fig. 3.72. Determine the direction of the current in the plane in each case.
3.251. In an electromagnetic pump designed for transferring molten metals a pipe section with metal is located in a uniform magnetic feld of induction $B$ (Fig. 3.73). A current $I$ is made to flow acress this pipe section in the direction perpendicular both to the vecter $\mathbf{B}$ and to the axis of the pipe. Find the gauge pressure produced by the pump if $B=0.10\mathrm{T},I=100\mathrm{A}$, and $a=2.0\mathrm{cm}$.
3.252. A current $I$ flows in a long thiswalled cylinder of radius $\mathit{A}$. What pressure do the walls of the cylinder experience?
3.263. What pressure does the lateral surface of a long straight seleneid with $n$ turns per unit length experience when a current $I$ flows through it?
3.264. A current $I$ dows in a long single-layer solenoid with cresssectional radius $R$. The number of turns per unit length of the solenoid equals n. Find the limiting current at which the winding may rupture if the tensile strength of the wire is equal to $F_{\text{itw }}$.
3.265. A parallel-plate capacitor with area of each plate equal to $S$ and the separation between them to $d$ is put inte a stream of conducting liquid with resistivity $\rho$. The liquid moves parallel to the plates with a constant velocity $v$. The whole system is located in uniform magnetic field of induction $B$, vector $B$ being parallel to the plates and perpendicular to the stream direction. The capacitor plates are interconnected by means of an external resistance $\mathit{R}$. What amount of power is generated in that resistance? At what value of $R$ is the generated power the highest? What is this highest power equal to?
3.266. A straight round copper conducter of radius $R=5.0\mathrm{mm}$ carries a current $\mathit{I}=50\mathrm{A}$. Find the potential difference between the axis of the cenductor and its surface. The concentration of the conduction electrons in cepper is equal to $n=0.9\cdot {10}^{\mathrm{DH}}{\mathrm{cm}}^{-3}$.
3.267. In Hall effect measurements in a sodium conductor the strength of a transverse field was found to be equal to $E=5.0\mathrm{pV}/\mathrm{em}$ with a current density $f=200\mathrm{A}/{\mathrm{cm}}^2$ and magnetic induction $B=$ $=1.00\mathrm{T}$. Find the concentration of the conduction electrons and its ratio to the total number of atems in the given cenducter.
3.268. Find the mobility of the conduction electrons in a copper conductor if in Hall effect measurements performed in the magnetic field of induction $B=100\mathrm{mT}$ the transverse electric field strength of the given conductor turned out to be $\eta =3.1\cdot {10}^{\circ }$ times less than that of the longitudinal electric field.
3.269. A small curtent-carrying loop is located at a distance $r$ from a long straight condector with current $I$. The magnetic moment
of the loop is equal to P.. Find the magnitude and direction of the force vector applied to the loop if the vector ${\mathrm{P}}_{\mathrm{m}}$
(a) is parallel to the straight conductor;
(b) is oriented along the radius vector $r$;
(c) coincides in direction with the magnetic field produced by the current $I$ at the point where the loop is located.
3.270. A small current-carrying coil having a magnetic moment ${\mathrm{P}}_{\mathrm{m}}$ is located at the axis of a round loop of radius $R$ with current $I$ flowing through it. Find the magnitude of the vector force applied to the coil if its distance from the centre of the loop is equal to $x$ and the vector ${\mathrm{P}}_{\mathrm{m}}$ coincides in direction with the axis of the loop.
3.271. Find the interaction force of twe ceils with magnetic moments $p_1=4.0\mathrm{mA}\cdot {\mathrm{m}}^2$ and $p_1=6.0\mathrm{mA}\cdot {\mathrm{m}}^2$ and collinear axes if the separation between the coils is equal to $l=20\mathrm{cm}$ which exceeds considerably their linear dimensions.
3.272. A permanent magnet has the shape of a sufficiently thin dise magnetized along its axis. The radius of the dise is $R=1.0\mathrm{cm}$. Evaluate the magnitude of a molecular current $I^{\prime }$ flowing along the rim of the dise if the maguetic induction at the peint on the axis of the disc, lying at a distance $x=10\mathrm{cm}$ from its centre, is equal to $B=30\mathrm{HT}$.
3.273. The magnetic induction in vacuum at a plane surface of a uniform isotropic magnetic is equal to B, the vector B forming an angle $a$ with the normal of the surface. The permeability of the magnetic is equal to $\mu$. Find the magnitude of the magnetie induction ‘ $B$ ‘ in the magnetic in the vicinity of its surface.
3.274. The magnetic induction in vacuum at a plane surface of a magnetic is equal to $B$ and the vector $B$ forms an angle $\theta$ with the
Fie. 3.74.
normal $\mathrm{n}$ of the surface (Fig. 3.74). The permeability of the magnetic is equal to $\mu$. Find:
(a) the fux of the vector $\mathbf{H}$ through the spherical surface $S$ of radius $\mathit{A}$, whose centre lies on the surface of the magnetic;
(b) the circulation of the vector $\mathbf{B}$ around the square path $\mathrm{r}$ with side $l$ located as shown in the figure.
3.275. A direct current $I$ flows in a long reund uniform cylindrical wire made of paramagnetie with susceptibility $X$. Find:
(a) the surface molecular current $I_{\text{, }}$ :
(b) the volume molecular current $r_{\ast }$.
How are these currents directed toward each other?
3.276. Half of an infinitely long straight current-carrying solenoid is filled with magnetic substance as shown in Fig. 3.75. Draw the
Fie. 3.75.
appreximate plets of magnetic induction $B$, strength $H$, and magnetization $J$ on the axis as functions of $x$.
3.277. An infinitely long wire with a current $I$ flowing in it is located in the boundary plane between two non-conducting medis with permeabilities ${\mu }_1$ and ${\mu }_r$. Find the modulus of the magnetic induction vector throughout the space as a function of the distance $r$ from the wire. It should be borne in mind that the lines of the vector $\mathbf{B}$ are circles whose centres lie on the axis of the wire.
3.278. A round current-carrying loop lies in the plane boundary between magnetic and vacuum. The permeability of the magnetic is equal to $\mu$. Find the magnetic induction B at an arbitrary point on the axis of the loop if in the absence of the magnetic the magnetic induction at the same point becomes equal to ${\mathbf{B}}_6$. Generalize the ebtained result to all points of the field.
3.279. When a ball made of uniform magnetic is intreduced inte an external uniform magnetic field with induetion ${\mathbf{B}}_3$, it gets uniformIy magnetired. Find the magnetic induction B inside the ball with permeability $\mu$; recall that the magnetic field inside a uniformly mag netired ball is uniform and its strength is equal to ${\mathbf{H}}^{\prime }=-J/3$, where $J$ is the magnetiration.
3.280. $N=300$ turns of thin wire are unifermly wound on a permanent magnet shaped as a cylinder whose length is equal to $I=$ $=15\mathrm{cm}$. When a current $I=3.0\mathrm{A}$ was passed through the wiring the field eutside the magnet disappeared. Find the coercive force ${\mathit{H}}_{\mathrm{e}}$ of the material from which the magnet was manufactured.
3.281. A permanent magnet is shaped as a ring with a narrew gap between the poles. The mean diameter of the ring equals $d=20\mathrm{cm}$. The width of the gap is equal to $b=2.0\mathrm{mm}$ and the magnetie induction in the gap is equal to $B=40\mathrm{mT}$. Assuming that the scattering of the magnetic flux at the gap edges is negligible, find the modulus of the magnetic field strength vector inside the magnet.
3.282. An iron core shaped as a tore with mean radius $R=250\mathrm{mm}$ supports a winding with the total number of turns $N=1000$. The core has a cross-cut of width $b=1.00\mathrm{mm}$. With a current $I=$ $=0.85\mathrm{A}$ nowing through the winding. the magnetic induction in the gap is equal to $B=0.75\mathrm{T}$. Assuming the scattering of the magnetie fux at the gap edges to be negligible, find the permeability of iron under these conditions.
3.283. Fig. 3.76 illustrates a basic magnetization curve of iron (commercial purity grade). Using this plot, draw the permeability
Fig. 3.76.
$\mu$ as a function of the magnetic field strength $H$. At what value of $H$ is the permeability the greatest? What is ${\mu }_{max}$ equal to?
3.284. A thin iron ring with mean diameter $d=50\mathrm{cm}$ supports a winding consisting of $N=800$ turns carrying current $I=3.0\mathrm{A}$. The ring has a cross-cut of width $b=2.0\mathrm{mm}$. Neglecting the scattering of the magnetic flux at the gap edges, and using the plot shown in Fig. 3.76, find the permeability of iron under these conditions.
3.285. A long thin cylindrical rod made of paramagnetic with magnetic susceptibility $\chi$ and having a cross-sectional area $S$ is located along the axis of a current-carrying coil. One end of the rod is located at the coil centre where the magnetic induction is equal to $B$ whereas the other end is located in the region where the magnetic field is practically absent. What is the force that the coil exerts on the rod?
3.286. In the arrangement shown in Fig. 3.77 it is possible to measure (by means of a balance) the force with which a paramagnetic ball of volume $V=41{\mathrm{mm}}^3$ is attrabted to a pole of the electromagFig. 3.77. net $M$. The magnetic induction at the axis of the poleshoe depends on the height $x$ as $B=B_0\exp (-a{x}^2)$, where $B_0=1.50\mathrm{T},a=100{\mathrm{m}}^{-2}$. Find:
(a) at what height $x_m$ the ball experiences the maximum attraction;
146
(b) the magnetic susceptibility of the paramagnetic if the maximum attraction force equals $F_{max}=160\mu \mathrm{N}$.
3.287. A small ball of volume $V$ made of paramagnetic with susceptibility $\chi$ was slowly displaced along the axis of a current-carrying coil from the point where the magnetic induction equals $B$ out to the region where the magnetic field is practically absent. What amount of work was performed during this process?