Problems in General Physics → Electrodynamics → Electric Current 
3.147. A long cylinder with uniformly charged surface and crosssectional 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.
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.
3.150. Find the resistance of a wire frame shaped as a cube (Fig. 3.35) when measured between points
3.151. At what value of the resistance R_{x} in the circuit shown in Fig. 3.36 will the total resistance between points A and B be independent of the number of cells?
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 R_{0}. Find the resistance R of the whole grid between points A and B.
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.
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 → ∞.
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 ηfold during the time interval Δt.
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.
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:
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 crosssection radius of each wire equals a. In the case a << l find:
3.160. The gap between the plates of a parallelplate 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.
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.
3.163. The gap between the plates of a parallelplate capacitor is filled up with an inhomogeneous poorly conducting medium whose conductivity varies linearly in the direction perpendicular to the plates from σ_{1} = 1.0 pS/m to σ_{2} = 2.0 pS/m. Each plate has an area S = 230 cm^{2}, 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.
3.164. Demonstrate that the law of refraction of direct current lines at the boundary between two conducting media has the form tan α_{2}/tan α_{1} = σ_{2}/σ_{1}, where σ_{1} and σ_{2} are the conductivities of the media, α_{2} and α_{1} are the angles between the current lines and the normal of the boundary surface.
3.165. Two cylindrical conductors with equal crosssections and different resistivities ρ_{1} and ρ_{2} 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.
3.167. An inhomogeneous poorly conducting medium fills up the space between plates 1 and 2 of a parallelplate capacitor. Its permittivity and resistivity vary from values ε_{1}, ρ_{1} at plate 1 to values ε_{2}, ρ_{2} 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.
3.168. The space between the plates of a parallelplate 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.
3.169. A long round conductor of crosssectional area S is made of material whose resistivity depends only on a distance r from the axis of the conductor as ρ = α/r^{2}, where α is a constant. Find:
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.
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 ηfold. Find the current flowing through the circuit as a function of time t.
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.
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:
3.177. In the circuit shown in Fig. 3.41 the sources have emf's ξ_{1} = 1.0 V and ξ_{2} = 2.5 V and the resistances have the values R_{1} = 10 Ω and R_{2} = 20 Ω. The internal resistances of the sources are negligible. Find a potential difference φ_{A}  φ_{B} between the plates A and B of the capacitor C.
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 R_{1} = 4.0 Ω and R_{2} = 6.0 Ω. The internal resistance of the source equals R = 0.10 Ω. Find the currents flowing through the resistances R_{1} and R_{2}.
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 R_{0}, and voltage V_{0} is applied to its terminals. Find the voltage V fed to the device as a function of distance x. Analyse separately the case R >> R_{0}.
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 ξ_{1} and ξ_{2} and internal resistances to R_{1} and R_{2}.
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 ξ_{1} = 1.5 V and ξ_{2} = 3.7 V and the resistances are equal to R_{1} = 10 Ω, R_{2} = 20 Ω, R = 5.0 Ω. The internal resistances of the sources are negligible.
3.182. In the circuit shown in Fig. 3.45 the sources have emf's ξ_{1} = 1.5 V, ξ_{2} = 2.0 V, ξ_{3} = 2.5 V, and the resistances are equal to R_{1} = 10 Ω, R_{2} = 20 Ω, R_{3} = 30 Ω. The internal resistances of the sources are negligible. Find:
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.
3.184. Find a potential difference φ_{A}  φ_{B} between the plates of a capacitor C in the circuit shown in Fig. 3.47 if the sources have emf's ξ_{1} = 4.0 V and ξ_{2} = 1.0 V and the resistances are equal to R_{1} = 10 Ω, R_{2} = 20 Ω, R_{3} = 30 Ω. The internal resistances of the sources are negligible.
3.185. Find the current flowing through the resistance R_{1} of the circuit shown in Fig. 3.48 if the resistances are equal to R_{1} = 10 Ω, R_{2} = 20 Ω, R_{3} = 30 Ω, and the potentials of points 1, 2, and 3 are equal to φ_{1} = 10 V, φ_{2} = 6 V, and φ_{3} = 5 V.
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
3.190. A dc source with internal resistance R_{0} 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?
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?
3.197. In a circuit shown in Fig. 3.54 resistances R_{1} and R_{2} are known, as well as emf's ξ_{1} and ξ_{2}. 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?
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 R_{1} = 500 Ω if R_{2} = 330 Ω.
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 q_{0}. Find:
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 V_{0} and then at the moment t = 0 was shorted by means of the switch Sw. Find:
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.
3.207. Find the total momentum of electrons in a straight wire of length l = 1000 m carrying a current I = 70 A.
3.210. A homogeneous proton beam accelerated by a potential difference V = 600 kV has a round crosssection 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.
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 cm^{3}, the observed saturation current is equal to I_{sat} = 0.48 μA. Find:
3.215. Having been operated long enough, the ionizer producing n'_{i} = 3.5*10^{9} 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} cm^{3}/s, find how soon after the ionizer's switching off the ion concentration decreases η = 2.0 times.
