# Problems in General Physics → Electrodynamics → Constant Magnetic Field. Magnetics

 3.219. A current I = 1.00 A circulates in a round thin-wire loop of radius R = 100 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 mm from its centre. Free solution >> 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 → ∞. Free solution >> 3.221. Find the magnetic induction at the centre of a rectangular wire frame whose diagonal is equal to d = 16 cm and the angle between the diagonals is equal to φ = 30°; the current flowing in the frame equals I = 5.0 A. Free solution >> 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 mm, the angle 2φ = 90°. Find the magnetic induction of the field at the point O. Free solution >> 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 φ are known; (b) in Fig. 3.60b, the radius a and the side b are known. Free solution >> 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 << R. Free solution >> 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. Free solution >> 3.226. Find the magnetic induction of the field at the point O if a current-carrying wire has the shape shown in Fig. 3.62 a, b, c. The radius of the curved part of the wire is R, the linear parts are assumed to be very long. Free solution >> 3.227. A very long wire carrying a current I = 5.0 A is bent at right angles. Find the magnetic induction at a point lying on a perpendicular to the wire, drawn through the point of bending, at a distance l = 35 cm from it. Free solution >> 3.228. Find the magnetic induction at the point O if the wire carrying a current I = 8.0 A has the shape shown in Fig. 3.63 a, b, c. The radius of the curved part of the wire is R = 100 mm, the linear parts of the wire are very long. Free solution >> 3.229. Find the magnitude and direction of the magnetic 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 currents of linear densities i and -i; the vectors i and -i are constant at all points of the corresponding planes. Free solution >> 3.230. A uniform current of density j flows inside an infinite plate of thickness 2d parallel to its surface. Find the magnetic 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. Free solution >> 3.232. A current I flows along a round loop. Find the integral ∫ B dr along the axis of the loop within the range from -∞ to +∞. Explain the result obtained. Free solution >> 3.233. A direct current of density j flows along a round uniform straight wire with cross-section radius R. Find the magnetic induction vector of this current at the point whose position relative to the axis of the wire is defined by a radius vector r. The magnetic permeability is assumed to be equal to unity throughout all the space. Free solution >> 3.234. Inside a long straight uniform wire of round cross-section there is a long round cylindrical cavity whose axis is parallel to the axis of the wire and displaced from the latter by a distance l. A direct current of density j flows along the wire. Find the magnetic induction inside the cavity. Consider, in particular, the case l = 0. Free solution >> 3.235. Find the current density as a function of distance r from the axis of a radially symmetrical parallel stream of electrons if the magnetic induction inside the stream varies as B = brα, where b and α are positive constants. Free solution >> 3.236. A single-layer coil (solenoid) has length l and cross-section radius R. A number of turns per unit length is equal to n. Find the magnetic induction at the centre of the coil when a current I flows through it. Free solution >> 3.237. A very long straight solenoid has a cross-section radius R and n turns per unit length. A direct current I flows through the solenoid. Suppose that x 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 x0 to the point on the axis at which the value of B differs by η = 1% from that in the middle section of the solenoid. Free solution >> 3.240. A direct current I = 10 A flows in a long straight round conductor. Find the magnetic flux through a half of wire's cross-section per one metre of its length. Free solution >> 3.242. Fig. 3.65 shows a toroidal solenoid whose cross-section is rectangular. Find the magnetic flux through this cross-section if the current through the winding equals I = 1.7 A, the total number of turns is N = 1000, the ratio of the outside diameter to the inside one is η = 1.6, and the height is equal to h = 5.0 cm. Free solution >> 3.243. Find the magnetic moment of a thin round loop with current if the radius of the loop is equal to R = 100 mm and the magnetic induction at its centre is equal to B = 6.0 μT. Free solution >> 3.244. Calculate the magnetic moment of a thin wire with a current I = 0.8 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 cm, the number of turns is N = 500. Free solution >> 3.245. A thin insulated wire forms a plane spiral of N = 100 tight turns carrying a current I = 8 mA. The radii of inside and outside turns (Fig. 3.67) are equal to a = 50 mm and b = 100 mm. Find: (a) the magnetic induction at the centre of the spiral; (b) the magnetic moment of the spiral with a given current. Free solution >> 3.246. A non-conducting thin disc of radius R charged uniformly over one side with surface density σ rotates about its axis with an angular velocity ω. Find: (a) the magnetic induction at the centre of the disc; (b) the magnetic moment of the disc. Free solution >> 3.247. A non-conducting sphere of radius R = 50 mm charged uniformly with surface density σ = 10.0 μC/m2 rotates with an angular velocity ω = 70 rad/s about the axis passing through its centre. Find the magnetic induction at the centre of the sphere. Free solution >> 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 ω about the axis passing through its centre. Find the respective magnetic moment and its ratio to the mechanical moment. Free solution >> 3.250. Two protons move parallel to each other with an equal velocity v = 300 km/s. Find the ratio of forces of magnetic and electrical interaction of the protons. Free solution >> 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 A, at a point O, if the wire is bent as shown in (a) Fig. 3.68a, with curvature radius R = 10 cm; (b) Fig. 3.68b, the distance between the long parallel segments of the wire being equal to l = 20 cm. Free solution >> 3.252. A coil carrying a current I = 10 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 diameter d = 0.10 mm, radius of turns is equal to R = 30 mm. At what value of the induction of the external magnetic field can the coil winding be ruptured? Free solution >> 3.253. A copper wire with cross-sectional area S = 2.5 mm2 bent to make three sides of a square can turn about a horizontal axis OO' (Fig. 3.69). The wire is located in uniform vertical magnetic field. Find the magnetic induction if on passing a current I = 16 A through the wire the latter deflects by an angle θ = 20°. Free solution >> 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 is S =1.0 cm2, the length of the arm OA of the balance beam is l = 30 cm. When there is no current in the coil the balance is in equilibrium. On passing a current I = 22 mA through the coil the equilibrium is restored by putting the additional counterweight of mass Δm = 60 mg on the balance pan. Find the magnetic induction at the spot where the coil is located. Free solution >> 3.255. A square frame carrying a current I = 0.90 A is located in the same plane as a long straight wire carrying a current I0 = 5.0 A. The frame side has a length a = 8.0 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 η = 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° about its axis, with the currents maintained constant. Free solution >> 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 dc voltage source. The distance between the axes of the wires is η = 20 times greater than the cross-sectional radius of each wire. At what value of resistance R does the resultant force of interaction between the wires turn into zero? Free solution >> 3.258. Two long thin parallel conductors of the shape shown in Fig. 3.71 carry direct currents I1 and I2. The separation between the conductors is a, the width of the right-hand conductor 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. Free solution >> 3.260. A conducting current-carrying plane is placed in an external uniform magnetic field. As a result, the magnetic induction becomes equal to B1 on one side of the plane and to B2, 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. Free solution >> 3.261. In an electromagnetic pump designed for transferring molten metals a pipe section with metal is located in a uniform magnetic field of induction B (Fig. 3.73). A current I is made to flow across this pipe section in the direction perpendicular both to the vector B and to the axis of the pipe. Find the gauge pressure produced by the pump if B = 0.10 T, I = 100 A, and a = 2.0 cm. Free solution >> 3.263. What pressure does the lateral surface of a long straight solenoid with n turns per unit length experience when a current I flows through it? Free solution >> 3.264. A current I flows in a long single-layer solenoid with cross-sectional 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 Flim. Free solution >> 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 μV/cm with a current density j = 200 A/cm2 and magnetic induction B = 1.00 T. Find the concentration of the conduction electrons and its ratio to the total number of atoms in the given conductor. Free solution >> 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 mT the transverse electric field strength of the given conductor turned out to be η = 3.1*103 times less than that of the longitudinal electric field. Free solution >> 3.269. A small current-carrying loop is located at a distance r from a long straight conductor with current I. The magnetic moment of the loop is equal to pm. Find the magnitude and direction of the force vector applied to the loop if the vector pm (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. Free solution >> 3.270. A small current-carrying coil having a magnetic moment pm 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 pm coincides in direction with the axis of the loop. Free solution >> 3.271. Find the interaction force of two coils with magnetic moments p1m = 4.0 mA*m2 and p2m = 6.0 mA*m2 and collinear axes if the separation between the coils is equal to l = 20 cm which exceeds considerably their linear dimensions. Free solution >> 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 α with the normal of the surface. The permeability of the magnetic is equal to μ. Find the magnitude of the magnetic induction B' in the magnetic in the vicinity of its surface. Free solution >> 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 θ with the normal n of the surface (Fig. 3.74). The permeability of the magnetic is equal to μ. Find: (a) the flux of the vector H through the spherical surface S of radius R, whose centre lies on the surface of the magnetic; (b) the circulation of the vector B around the square path Γ with side l located as shown in the figure. Free solution >> 3.276. Half of an infinitely long straight current-carrying solenoid is filled with magnetic substance as shown in Fig. 3.75. Draw the approximate plots of magnetic induction B, strength H, and magnetization J on the axis as functions of x. Free solution >> 3.277. An infinitely long wire with a current I flowing in it is located in the boundary plane between two non-conducting media with permeabilities μ1 and μ2. 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 B are circles whose centres lie on the axis of the wire. Free solution >> 3.279. When a ball made of uniform magnetic is introduced into an external uniform magnetic field with induction B0, it gets uniformly magnetized. Find the magnetic induction B inside the ball with permeability μ; recall that the magnetic field inside a uniformly magnetized ball is uniform and its strength is equal to H' = -J/3, where J is the magnetization. Free solution >> 3.281. A permanent magnet is shaped as a ring with a narrow gap between the poles. The mean diameter of the ring equals d = 20 cm. The width of the gap is equal to b = 2.0 mm and the magnetic induction in the gap is equal to B = 40 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. Free solution >> 3.282. An iron core shaped as a tore with mean radius R = 250 mm supports a winding with the total number of turns N = 1000. The core has a cross-cut of width b = 1.00 mm. With a current I = 0.85 A flowing through the winding, the magnetic induction in the gap is equal to B = 0.75 T. Assuming the scattering of the magnetic flux at the gap edges to be negligible, find the permeability of iron under these conditions. Free solution >> 3.284. A thin iron ring with mean diameter d = 50 cm supports a winding consisting of N = 800 turns carrying current I = 3.0 A. The ring has a cross-cut of width b = 2.0 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. Free solution >> 3.285. A long thin cylindrical rod made of paramagnetic with magnetic susceptibility χ 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? Free solution >> 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 mm3 is attrabted to a pole of the electromagnet M. The magnetic induction at the axis of the poleshoe depends on the height x as B = B0 exp(-ax2), where B0 = 1.50 T, a = 100 m-2. Find: (a) at what height xm the ball experiences the maximum attraction; (b) the magnetic susceptibility of the paramagnetic if the maximum attraction force equals Fmax = 160 μN. Free solution >>