# Problems in General Physics → Electrodynamics → Electric Current

 3.147. A long cylinder with uniformly charged surface and cross-sectional radius a = 1.0 cm moves with a constant velocity v = 10 m/s along its axis. An electric field strength at the surface of the cylinder is equal to E = 0.9 kV/cm. Find the resulting convection current, that is, the current caused by mechanical transfer of a charge. Free solution >> 3.148. An air cylindrical capacitor with a dc voltage V = 200 V applied across it is being submerged vertically into a vessel filled with water at a velocity v = 5.0 mm/s. The electrodes of the capacitor are separated by a distance d = 2.0 mm, the mean curvature radius of the electrodes is equal to r = 50 mm. Find the current flowing in this case along lead wires, if d << r. Free solution >> 3.150. Find the resistance of a wire frame shaped as a cube (Fig. 3.35) when measured between points (a) 1-7; (b) 1-2; (c) 1-3. The resistance of each edge of the frame is R. Free solution >> 3.151. At what value of the resistance Rx in the circuit shown in Fig. 3.36 will the total resistance between points A and B be independent of the number of cells? Free solution >> 3.153. There is an infinite wire grid with square cells (Fig. 3.38). The resistance of each wire between neighbouring joint connections is equal to R0. Find the resistance R of the whole grid between points A and B. Instruction. Make use of principles of symmetry and superposition. Free solution >> 3.154. A homogeneous poorly conducting medium of resistivity ρ fills up the space between two thin coaxial ideally conducting cylinders. The radii of the cylinders are equal to a and b, with a < b, the length of each cylinder is l. Neglecting the edge effects, find the resistance of the medium between the cylinders. Free solution >> 3.155. A metal ball of radius a is surrounded by a thin concentric metal shell of radius b. The space between these electrodes is filled up with a poorly conducting homogeneous medium of resistivity ρ. Find the resistance of the interelectrode gap. Analyse the obtained solution at b → ∞. Free solution >> 3.156. The space between two conducting concentric spheres of radii a and b (a < b) is filled up with homogeneous poorly 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 disconnected from an external voltage, decreases η-fold during the time interval Δt. Free solution >> 3.157. Two metal balls of the same radius a are located in a homogeneous poorly conducting medium with resistivity ρ. Find the resistance of the medium between the balls provided that the separation between them is much greater than the radius of the ball. Free solution >> 3.158. A metal ball of radius a is located at a distance l from an infinite ideally conducting plane. The space around the ball is filled with a homogeneous poorly conducting medium with resistivity ρ. In the case of a << l 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. Free solution >> 3.159. Two long parallel wires are located in a poorly conducting medium with resistivity ρ. The distance between the axes of the wires is equal to l, the cross-section radius of each wire equals a. In the case a << l find: (a) the current density at the point equally removed from the axes of the wires by a distance r if the potential difference between the wires is equal to V; (b) the electric resistance of the medium per unit length of the wires. Free solution >> 3.160. The gap between the plates of a parallel-plate capacitor is filled with glass of resistivity ρ = 100 GΩ*m. The capacitance of the capacitor equals C = 4.0 nF. Find the leakage current of the capacitor when a voltage V = 2.0 kV is applied to it. Free solution >> 3.161. Two conductors of arbitrary shape are embedded into an infinite homogeneous poorly conducting medium with resistivity ρ and permittivity ε. Find the value of a product RC for this system, where 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. Free solution >> 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 σ1 = 1.0 pS/m to σ2 = 2.0 pS/m. Each plate has an area S = 230 cm2, and the separation between the plates is d = 2.0 mm. Find the current flowing through the capacitor due to a voltage V = 300 V. Free solution >> 3.164. Demonstrate that the law of refraction of direct current lines at the boundary between two conducting media has the form tan α2/tan α1 = σ2/σ1, where σ1 and σ2 are the conductivities of the media, α2 and α1 are the angles between the current lines and the normal of the boundary surface. Free solution >> 3.165. Two cylindrical conductors with equal cross-sections and different resistivities ρ1 and ρ2 are put end to end. Find the charge at the boundary of the conductors if a current I flows from conductor 1 to conductor 2. Free solution >> 3.167. An inhomogeneous poorly conducting medium fills up the space between plates 1 and 2 of a parallel-plate capacitor. Its permittivity and resistivity vary from values ε1, ρ1 at plate 1 to values ε2, ρ2 at plate 2. A dc voltage is applied to the capacitor through which a steady current I flows from plate 1 to plate 2. Find the total extraneous charge in the given medium. Free solution >> 3.168. The space between the plates of a parallel-plate capacitor is filled up with inhomogeneous poorly conducting 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 η. The gap width equals d. Find the volume density of the charge in the gap if a voltage V is applied to the capacitor. ε is assumed to be 1 everywhere. Free solution >> 3.169. A long round conductor of cross-sectional area S is made of material whose resistivity depends only on a distance r from the axis of the conductor as ρ = α/r2, where α is a constant. Find: (a) the resistance per unit length of such a conductor; (b) the electric field strength in the conductor due to which a current I flows through it. Free solution >> 3.171. A capacitor filled with dielectric of permittivity ε = 2.1 loses half the charge acquired during a time interval τ = 3.0 min. Assuming the charge to leak only through the dielectric filler, calculate its resistivity. Free solution >> 3.172. A circuit consists of a source of a constant emf ξ and a resist ance R and a capacitor with capacitance C connected in series. The internal resistance of the source is negligible. At a moment t = 0 the capacitance of the capacitor is abruptly decreased η-fold. Find the current flowing through the circuit as a function of time t. Free solution >> 3.173. An ammeter and a voltmeter are connected in series to a battery with an emf ξ = 6.0 V. When a certain resistance is connected in parallel with the voltmeter, the readings of the latter decrease η = 2.0 times, whereas the readings of the ammeter increase the same number of times. Find the voltmeter readings after the connection of the resistance. Free solution >> 3.176. N sources of current with different emf's are connected as shown in Fig. 3.40. The emf's of the sources are proportional to their internal resistances, i.e. ξ = αR, where α is an assigned constant. The lead wire resistance is negligible. Find: (a) the current in the circuit; (b) the potential difference between points A and B dividing the circuit in n and N - n links. Free solution >> 3.177. In the circuit shown in Fig. 3.41 the sources have emf's ξ1 = 1.0 V and ξ2 = 2.5 V and the resistances have the values R1 = 10 Ω and R2 = 20 Ω. The internal resistances of the sources are negligible. Find a potential difference φA - φB between the plates A and B of the capacitor C. Free solution >> 3.178. In the circuit shown in Fig. 3.42 the emf of the source is equal to ξ = 5.0 V and the resistances are equal to R1 = 4.0 Ω and R2 = 6.0 Ω. The internal resistance of the source equals R = 0.10 Ω. Find the currents flowing through the resistances R1 and R2. Free solution >> 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 a resistance R. The potentiometer has a length l and a resistance R0, and voltage V0 is applied to its terminals. Find the voltage V fed to the device as a function of distance x. Analyse separately the case R >> R0. Free solution >> 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 ξ1 and ξ2 and internal resistances to R1 and R2. Free solution >> 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 emf's of the sources are equal to ξ1 = 1.5 V and ξ2 = 3.7 V and the resistances are equal to R1 = 10 Ω, R2 = 20 Ω, R = 5.0 Ω. The internal resistances of the sources are negligible. Free solution >> 3.182. In the circuit shown in Fig. 3.45 the sources have emf's ξ1 = 1.5 V, ξ2 = 2.0 V, ξ3 = 2.5 V, and the resistances are equal to R1 = 10 Ω, R2 = 20 Ω, R3 = 30 Ω. The internal resistances of the sources are negligible. Find: (a) the current flowing through the resistance R1; (b) a potential difference φA - φB between the points A and B. Free solution >> 3.183. Find the current flowing through the resistance R in the circuit shown in Fig. 3.46. The internal resistances of the batteries are negligible. Free solution >> 3.184. Find a potential difference φA - φB between the plates of a capacitor C in the circuit shown in Fig. 3.47 if the sources have emf's ξ1 = 4.0 V and ξ2 = 1.0 V and the resistances are equal to R1 = 10 Ω, R2 = 20 Ω, R3 = 30 Ω. The internal resistances of the sources are negligible. Free solution >> 3.185. Find the current flowing through the resistance R1 of the circuit shown in Fig. 3.48 if the resistances are equal to R1 = 10 Ω, R2 = 20 Ω, R3 = 30 Ω, and the potentials of points 1, 2, and 3 are equal to φ1 = 10 V, φ2 = 6 V, and φ3 = 5 V. Free solution >> 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 Δt; (b) decreases down to zero halving its value every Δt seconds? Free solution >> 3.190. A dc source with internal resistance R0 is loaded with three identical resistances R interconnected as shown in Fig. 3.52. At what value of R will the thermal power generated in this circuit be the highest? Free solution >> 3.193. A voltage V is applied to a dc electric motor. The armature winding resistance is equal to R. At what value of current 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? Free solution >> 3.197. In a circuit shown in Fig. 3.54 resistances R1 and R2 are known, as well as emf's ξ1 and ξ2. The internal resistances of the sources are negligible. At what value of the resistance R will the thermal power generated in it be the highest? What is it equal to? Free solution >> 3.199. A capacitor of capacitance C = 5.00 μF is connected to a source of constant emf ξ = 200 V (Fig. 3.55). Then the switch Sw was thrown over from contact 1 to contact 2. Find the amount of heat generated in a resistance R1 = 500 Ω if R2 = 330 Ω. Free solution >> 3.203. The radii of spherical capacitor electrodes are equal to a and b, with a < b. The interelectrode space is filled with homogeneous substance of permittivity ε and resistivity ρ. Initially the capacitor is not charged. At the moment t = 0 the internal electrode gets a charge q0. Find: (a) the time variation of the charge on the internal electrode; (b) the amount of heat generated during the spreading of the charge. Free solution >> 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 connected to a voltage V0 and then at the moment t = 0 was shorted by means of the switch Sw. Find: (a) a current I in the circuit as a function of time t; (b) the amount of generated heat provided a dependence I(t) is known. Free solution >> 3.206. A coil of radius r = 25 cm wound of a thin copper wire of length l = 500 m rotates with an angular velocity ω = 300 rad/s about its axis. The coil is connected to a ballistic galvanometer by means of sliding contacts. The total resistance of the circuit is equal to R = 21 Ω. Find the specific charge of current carriers in copper if a sudden stoppage of the coil makes a charge q = 10 nC flow through the galvanometer. Free solution >> 3.207. Find the total momentum of electrons in a straight wire of length l = 1000 m carrying a current I = 70 A. Free solution >> 3.210. A homogeneous proton beam accelerated by a potential difference V = 600 kV has a round cross-section of radius r = 5.0 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 mA. Free solution >> 3.214. The air between two closely located plates is uniformly ionized by ultraviolet radiation. The air volume between the plates is equal to V = 500 cm3, the observed saturation current is equal to Isat = 0.48 μA. Find: (a) the number of ion pairs produced in a unit volume per unit time; (b) the equilibrium concentration of ion pairs if the recombination coefficient for air ions is equal to r = 1.67*10-6 cm3/s. Free solution >> 3.215. Having been operated long enough, the ionizer producing n'i = 3.5*109 cm-3*s-1 of ion pairs per unit 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*10-6 cm3/s, find how soon after the ionizer's switching off the ion concentration decreases η = 2.0 times. Free solution >>