# Problems in General Physics → Electrodynamics → Electromagnetic Induction. Maxwell's Equations

 3.288. A wire bent as a parabola y = ax2 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 w (Fig. 3.78). Find the emf of electromagnetic induction in the loop thus formed as a function of y. Free solution >> 3.289. A rectangular loop with a sliding connector of length l is located in a uniform magnetic field perpendicular to the loop plane (Fig. 3.79). The magnetic induction is equal to B. The connector has an electric resistance R, the sides AB and CD have resistances R1 and R2 respectively. Neglecting the self-inductance of the loop, find the current flowing in the connector during its motion with a constant velocity v. Free solution >> 3.290. A metal disc of radius a = 25 cm rotates with a constant angular velocity ω = 130 rad/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 mT is directed perpendicular to the disc. Free solution >> 3.293. A long straight wire carrying a current I and a Π-shaped conductor with sliding connector are located in the same plane as shown in Fig. 3.81. The connector of length l and resistance R slides to the right with a constant velocity v. Find the current induced in the loop as a function of separation r between the connector and the straight wire. The resistance of the Π-shaped conductor and the self-inductance of the loop are assumed to be negligible. Free solution >> 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 v. Find the emf induced in the frame as a function of distance x. Free solution >> 3.296. A copper connector of mass m slides down two smooth copper bars, set at an angle α to the horizontal, due to gravity (Fig. 3.84). At the top the bars are interconnected through a resistance R. The separation between the bars is equal to l. The system is located in a uniform magnetic field of induction B, perpendicular to the plane in which the connector slides. The resistances of the bars, the connector 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. Free solution >> 3.297. The system differs from the one examined in the foregoing problem (Fig. 3.84) by a capacitor of capacitance C replacing the resistance R. Find the acceleration of the connector. Free solution >> 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). The inductance and the resistance of the frame are equal to L and R respectively. The frame was turned through 180° about the axis OO' separated from the current-carrying conductor by a distance b. Find the electric charge having flown through the frame. Free solution >> 3.301. A long straight wire carries a current I0. At distances a and b from it there are two other wires, parallel to the former one, which are interconnected by a resistance R (Fig. 3.87). A connector slides without friction along the wires with a constant velocity v. Assuming the resistances of the wires, the connector, the sliding contacts, and the self-inductance of the frame 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. Free solution >> 3.302. A conducting rod AB 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 AB starts moving to the right with an initial velocity v0. Neglecting the resistances of the rails and the rod AB, 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 R during this process. Free solution >> 3.303. A connector AB can slide without friction along a Π-shaped conductor located in a horizontal plane (Fig. 3.89). The connector has a length l, mass m, and resistance R. The whole system is located in a uniform magnetic field of induction B directed vertically. At the moment t = 0 a constant horizontal force F starts acting on 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 Π-shaped conductor are assumed to be negligible. Free solution >> 3.304. Fig. 3.90 illustrates plane figures made of thin conductors which are located in a uniform magnetic field directed away from a reader beyond the plane of the drawing. The magnetic induction starts diminishing. Find how the currents induced in these loops are directed. Free solution >> 3.305. A plane loop shown in Fig. 3.91 is shaped as two squares with sides a = 20 cm and b = 10 cm and is introduced into a uniform magnetic field at right angles to the loop's plane. The magnetic induction varies with time as B = B0 sin ωt, where B0 = 10 mT and ω = 100 s-1 . Find the amplitude of the current induced in the loop if its resistance per unit length is equal to ρ = 50 mΩ/m. The inductance of the loop is to be neglected. Free solution >> 3.307. A Π-shaped conductor is located in a uniform magnetic field perpendicular to the plane of the conductor and varying with time at the rate B' = 0.10 T/s. A conducting connector starts moving with an acceleration w = 10 cm/s2 along the parallel bars of the conductor. The length of the connector is equal to l = 20 cm. Find the emf induced in the loop t = 2.0 s after the beginning of the motion, if at the moment t = 0 the loop area and the magnetic induction are equal to zero. The inductance of the loop is to be neglected. Free solution >> 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 constant velocity I' A/s. Find the magnitude of the eddy current field strength as a function of the distance r from the solenoid axis. Draw the approximate plot of this function. Free solution >> 3.309. A long straight solenoid of cross-sectional diameter d = 5 cm and with n = 20 turns per one cm of its length has a round turn of copper wire of cross-sectional area S = 1.0 mm2 tightly put on its winding. Find the current flowing in the turn if the current in the solenoid winding is increased with a constant velocity I' = 100 A/s. The inductance of the turn is to be neglected. Free solution >> 3.310. A long solenoid of cross-sectional radius a has a thin insulated wire ring tightly put on its winding; one half of the ring has the resistance η times that of the other half. The magnetic induction produced by the solenoid varies with time as B = bt, where b is a constant. Find the magnitude of the electric field strength in the ring. Free solution >> 3.311. A thin non-conducting ring of mass m carrying a charge q can freely rotate about its axis. At the initial moment the ring was at rest and no magnetic field was present. Then a practically uniform magnetic field was switched on, which was perpendicular to the plane of the ring and increased with time according to a certain law B(t). Find the angular velocity ω of the ring as a function of the induction B(t). Free solution >> 3.313. A magnetic flux through a stationary loop with a resistance R varies during the time interval τ as Φ = at(τ - t). Find the amount of heat generated in the loop during that time. The inductance of the loop is to be neglected. Free solution >> 3.317. A coil of inductance L = 300 mH and resistance R = 140 mΩ is connected to a constant voltage source. How soon will the coil current reach η = 50% of the steady-state value? Free solution >> 3.318. Calculate the time constant τ of a straight solenoid of length l = 1.0 m having a single-layer winding of copper wire whose total mass is equal to m = 1.0 kg. The cross-sectional diameter of the solenoid is assumed to be considerably less than its length. Note. The time constant τ is the ratio L/R, where L is inductance and R is active resistance. Free solution >> 3.319. Find the inductance of a unit length of a cable consisting of two thin-walled coaxial metallic cylinders if the radius of the outside cylinder is η = 3.6 times that of the inside one. The permeability of a medium between the cylinders is assumed to be equal to unity. Free solution >> 3.320. Calculate the inductance of a doughnut solenoid whose inside radius is equal to b and cross-section has the form of a square with side a. The solenoid winding consists of N turns. The space inside the solenoid is filled up with uniform paramagnetic having permeability μ. Free solution >> 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. Free solution >> 3.322. Find the inductance of a unit length of a double line if the radius of each wire is η 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 η >> 1. Free solution >> 3.323. A superconducting round ring of radius a and inductance L 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° 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. Free solution >> 3.324. A current I0 = 1.9 A flows in a long closed solenoid. The wire it is wound of is in a superconducting state. Find the current flowing in the solenoid when the length of the solenoid is increased by η = 5%. Free solution >> 3.326. A closed circuit consists of a source of constant emt ξ and a choke coil 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 η times. Find the current in the circuit as a function of time t. Instruction. During a stepwise change of inductance the total magnetic flux (flux linkage) remains constant. Free solution >> 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 Sw is shorted at the moment t = 0. Free solution >> 3.328. In the circuit shown in Fig. 3.94 an emf ξ, a resistance R, and coil inductances L1 and L2 are known. The internal resistance of the source and the coil resistances are negligible. Find the steady-state currents in the coils after the switch Sw was shorted. Free solution >> 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 in the same plane, with the side b being closest to the wire, separated by a distance l from it and oriented parallel to it. Free solution >> 3.330. Determine the mutual inductance of a doughnut coil and an infinite straight wire passing along its axis. The coil has a rectangular cross-section, its inside radius is equal to a and the outside one, to b. The length of the doughnut's cross-sectional side parallel to the wire is equal to h. The coil has N turns. The system is located in a uniform magnetic with permeability μ. Free solution >> 3.331. Two thin concentric wires shaped as circles with radii a and b lie in the same plane. Allowing 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. Free solution >> 3.334. There are two stationary loops with mutual inductance L12. The current in one of the loops starts to be varied as I1 = αt, where α is a constant, t is time. Find the time dependence I2(t) of the current in the other loop whose inductance is L2 and resistance R. Free solution >> 3.335. A coil of inductance L = 2.0 μH and resistance R = 1.0 Ω is connected to a source of constant emf ξ = 3.0 V (Fig. 3.96). A resistance R0 = 2.0 Ω is connected in parallel with the coil. Find the amount of heat generated in the coil after the switch Sw is disconnected. The internal resistance of the source is negligible. Free solution >> 3.341. A thin uniformly charged ring of radius a = 10 cm rotates about its axis with an angular velocity ω = 100 rad/s. Find the ratio of volume energy densities of magnetic and electric fields on the axis of the ring at a point removed from its centre by a distance l = a. Free solution >> 3.347. The space between two concentric metallic spheres is filled up with a uniform poorly conducting medium of resistivity ρ and permittivity ε. 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 located in the medium and enclosing the internal sphere, if at the given moment of time the charge of that sphere is equal to q. Free solution >> 3.352. A point charge q moves with a non-relativistic velocity v = const. Find the displacement current density jd at a point located at a distance r from the charge on a straight line (a) coinciding with the charge path; (b) perpendicular to the path and passing through the charge. Free solution >> 3.356. Demonstrate that the law of electric charge conservation, i.e. ∇⋅j = -∂ρ/∂t, follows from Maxwell's equations. Free solution >> 3.357. Demonstrate that Maxwell's equations ∇xE = -∂B/∂t and ∇⋅B = 0 are compatible, i.e. the first one does not contradict the second one. Free solution >> 3.358. In a certain region of the inertial reference frame there is magnetic field with induction B rotating with angular velocity ω. Find ∇xE in this region as a function of vectors ω and B. Free solution >> 3.361. A long solid aluminum cylinder of radius a = 5.0 cm rotates about its axis in a uniform magnetic field with induction B = 10 mT. The angular velocity of rotation equals ω = 45 rad/s, with ω ↑↑ B. Neglecting the magnetic field of appearing charges, find their space and surface densities. Free solution >> 3.367. In an inertial reference frame K there is only a uniform electric field E = 8 kV/m in strength. Find the modulus and direction (a) of the vector E', (b) of the vector B' in the inertial reference frame K' moving with a constant velocity v relative to the frame K at an angle α = 45° to the vector E. The velocity of the frame K' is equal to a β = 0.60 fraction of the velocity of light. Free solution >> 3.370. In an inertial reference frame K there are two uniform mutually perpendicular fields: an electric field of strength E = 40 kV/m and a magnetic field induction B = 0.20 mT. Find the electric strength E' (or the magnetic induction B') in the reference frame K' where only one field, electric or magnetic, is observed. Instruction. Make use of the field invariants cited in the foregoing problem. Free solution >> 3.371. A point charge q moves uniformly and rectilinearly with a relativistic velocity equal to a β fraction of the velocity of light (β = v/c). Find the electric field strength E produced by the charge at the point whose radius vector relative to the charge is equal to r and forms an angle θ with its velocity vector. Free solution >>