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3.204. The electrodes of a capacitor of capacitance (^) C = 2.00 tiF carry opposite charges q, = 1.00 mC. Then the electrodes are inter-
connected through a resistance R = 5.0 MQ. Find:
(a) the charge flowing through that resistance during a time inter- val i (^) = 2.00 s; (b) the amount of heat generated in the resistance during the same interval. 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 Vo 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 -r-_F- Sw
C C -r time t; I I (b) the amount of generated heat provided a dependence (^) I (t) is known. 3.206. A coil of radius r = 25 cm wound of a thin Fig. 3.57. copper wire of length 1 = 500 m rotates with an angular velocity co = 300 rad/s about its axis. The coil is connect- ed to a ballistic galvanometer by means of sliding contacts. The
total resistance of the circuit is equal to R = 21 Q. 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 galvano- meter. 3.207. Find the total momentum of electrons in a straight wire
of length 1 = 1000 m carrying a current I = 70 A.
3.208. A copper wire carries a current of density j = 1.0 A/mm2. Assuming that one free electron corresponds to each copper atom, evaluate the distance which will be covered by an electron during its displacement 1 = 10 mm along the wire. 3.209. A straight copper wire of length 1 = 1000 m and cross-
sectional area S = 1.0 mm2carries a current I = 4.5 A. Assuming
that one free electron corresponds to each copper atom, find: (a) the time it takes an electron to displace from one end of the wire to the other; (b) the sum of electric forces acting on all free electrons in the given wire. 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. 3.211. Two large parallel plates are located in vacuum. One of them serves as a cathode, a source of electrons whose initial velocity is negligible. An electron flow directed toward the opposite plate prod- uces a space charge causing the potential in the gap between the plates to vary as cp = ax4/3, where a is a positive constant, and x is the distance from the cathode. Find:
'
(a) the volume density of the space charge as a function
of x;
(b) the current density. 3.212. The air between two parallel plates separated by a distance d = 20 mm is ionized by X-ray radiation. Each plate has an area S = 500 cm2. Find the concentration of positive ions if at a voltage V = 100 V a current^ I =^ 3.0 p,A flows between the plates, which is well below the saturation current. The air ion mobilities are u-ei- = 1.37 cm2/(V•s) and uo = 1.91 cm2/(V•s). 3.213. A gas is ionized in the immediate vicinity of the surface of plane electrode 1 (Fig. 3.58) separated from electrode 2 by a dis- tance 1. An alternating voltage varying with time t (^) as V = Vosin cot is applied to the electrodes. On decreasing the frequency co it was observed that the galvano- meter G indicates a current only at to < co n, where co, is a certain cut-off frequency. Find the mobility of ions reaching electrode 2 under these conditions. 3.214. The air between two closely located (^) V plates is uniformly ionized by ultraviolet radia- tion. The air volume between the plates is equal (^) Fig. 3.58. to V = 500 cm3, the observed saturation current is equal to /sat= 0.48 RA. 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 = 1.67.10-6cm3/s.
3.215. Having been operated long enough, the ionizer producing nt= 3.5.109cm-3•s-1of 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 coeffic- ient r = 1.67-10-6cm3/s, find how soon after the ionizer's switching off the ion concentration decreases ri = 2.0 times. 3.216. A parallel-plate air capacitor whose plates are separated by a distance d = 5.0 mm is first -charged to a potential difference V = 90 V and then disconnected from a de voltage source. Find the time interval during which the voltage across the capacitor de- creases by II = 1.0%, taking into account that the average number of ion pairs formed in air under standard conditions per unit volume per unit time is equal to ni= 5.0 cm-3•s-1and that the given volt- age corresponds to the saturation current. 3.217. The gap between two plane plates of a capacitor equal to d is filled with a gas. One of the plates emits voelectrons per second which, moving in an electric field, ionize gas molecules; this way each electron produces a new electrons (and ions) along a unit length of its path. Find the electronic current at the opposite plate, neglect- ing the ionization of gas molecules by formed ions.
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(b) at the point lying on the axis of the loop at a distance x = = 100 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 oo. 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 q) = 30°; the current flowing in the frame equals I = 5.0 A. 3.222. A current /=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 21:p = 90°. Find the magnetic induction of the field at the point 0.
I
3.223. Find the magnetic induction of the field at the point (^) 0 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 q) are known; (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 mag- netic field inside the tube under the condition (^) h << 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 (^) 0.
(b) (c)
Fig. 3.61. Fig. 3.62.
3.226. Find the magnetic induction of the field at the point 0 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.
(a)
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 per- pendicular to the wire, drawn through the point of bending, at a distance 1 = 35 cm from it. 3.228. Find the magnetic induction at the point 0 if the wire car- rying a current I = 8.0 A has the shape shown in Fig. 3.63 a,^ b, c.
Fig. 3.63.
The radius of the curved part of the wire is R = 100 mm, the linear parts of the wire are very long. 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 den- sities i and —i; the vectors i and —i are constant at all points of the corresponding planes. 3.230. A uniform current of density j flows inside an infinite plate of thickness 2d parallel to its surface. Find the magnetic induc- tion 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 0 (Fig. 3.64) the current spreads radially all (^) 0 over an infinite conducting plane perpendicu- lar to the wire. Find the magnetic induction Fig. 3.64. at all points of space. 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 —00 to +00. Explain the result obtained. 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.
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