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3.169. (a) 111= 23-ca/S2; (b) E 231a//S2. 3.170. t = -RC In (I - VIVO =^ 0.6 tis. 3.171. p = %leo In 2 = 1.4.1013 5 -2•m. 3.172. I = [(11 - 1) g 1 I1] e-rit/RC.
3.173. V = + 1) =^ 2.0 V.
3.174. (^) (pi (^) - = (el- + R2) - e, =^ -4 V. 3.175. R = R2 - R1, Acp = 0 in the source of current with in- ternal resistance R2. 3.176. (a) I = (^) a; (b) CPA - tP B = 0. 3.177. PA - B = (el - R1/(R1^ + R2) =^ -0.5 V. 3.178. /1 = eR2/(RR1^ RiR 2 R2R)^ = 1.2 A,^ 12 = 11R1/R2= = 0.8 A. 3.179. V = V^ oRx1ER1 R (1 - x) x111;^ for^ R > Ro V V ox11. 3.180. t (^) = (g 1R 2 g2R1)/(Ri + R2), Ri = R1R 2/(R1+ R2). 3.181. I = (R16 2- R261)1(RR1 R1R2+ R2R) =^ 0.02 A, the current is directed from the left to the right (see Fig. 3.44). 3.182. (a) (^) = [R3 (et -+ Rol + gov(R,R, + R2R3 +
- R3R 1 ) = 0.06 A; (b) (pA - cpB = gi 101 = 0.9 V. 3.183. I = [6 (R2+ R3) + 0R 3]1[11 (R2+ R3) + R2R3]. 3.184. TA .- (I) B = (^) [e2R3 (R1 + R2) - g1R1 (R2 +R3)11(14112+ ▪ R 2R3 R3R1)=- -1.0 V. 3.185. /1= [R3 ((Pi - (1)2) -4- R2 (Cl (P3)11(R1R2 R2R ▪ R3R1) = 0.2 A. 3.186. I= Ri+R R2 ( R1 [I ± R2R4 (R1+ R3) /RIR 3 (112 R4)]
1) = 1.0 A.
The current flows from point C to point D. 3.187. R AB = r (r 3R)1(R 3r). 3.188. V = 1/2e (1^ e-2tniC). 3.189. (a) (^) Q = 4/3 q2R1 At;^ (b)^ Q =^^1 /2 In 2.q2R/At. 3.190. R = 3R0.
3.192. Q I (g - v) =^ 0.6^ W, P = -IV =^ -2.0 W.
3.193. I = (^) V12R; Pmax = V2/4R; (^) 11 = 1/2. 3.194. By 2i = 2%. 3.195. T - T o = (1 - e-htic) V2/kR. 3.196. Rx = + R2) = 12 52. 3.197. R = R1R2I(R1 + R2); Qyna. = (e1R2 + F 2111)214R1R2(R1 + R2). 3.198. n 111 n TrIR = 3. 3.199. Q =^1 12C6 21111(R1 + R2) = 60 mJ. 3.200. (a) AW = -1/2CV2i/(1 1) = -0.15 mJ; (b) A = 112CV2111(1- 11) =^ 0.15 mJ.
3.201. AW^ = - 1 / 2 (a - 1)^ CV2 = -0.5 mJ,^ A^ rnech
= 1 /2 (a - 1) Cr^ 0.5 mJ. 3.202. h 1 /2 a0(a - 1) V2/pgd2,^ where p is the density of water. 3.203. (a) q = qoe-1/806P; (b) Q = (1/a - 1/b) Oneo e. 3.204. (a) q = qo (1 - e-t/RC) = 0.18 mC; (b)^ Q = (1 - e -Inic)q/2C = 82 mJ.
3.205. (a) I = (V 0IR) e-2tIRC- '
(b) (^) Q = 1/ 3.206. elm = lcor/qR = 1.8.1011C/kg. 3.207. p = lImle = 0.40 RN • s. 3.208. s = enl (v)11, 107m, where n is the concentration of free electrons, (v) is the mean velocity of thermal motion of an electron.
3.209. (a) t = en1SII = 3 Ms; (b) F = enlpl = 1.0 MN,
where p is the resistivity of copper.
3.210. E (I12:-ce0r) ym/2eV=32 V/m, 6,9=(//4neo) V m/2eV =
= 0.80 V.
3.211. (a) p (x)= — 419E0ex-213; (b) j =^ 4/9E,a3/ 2 V 2e/m. 3.212. n = Idle (u: ii-(;) VS =^ 2.3.108 cm-3. 3.213. uo = (1)0/2/2Vo.
3.214. (a) ni = (^) I sat IeV = 6.109 cm--3 .s -1; (b) n = V nilr 6.107 cm-3.
3.215. t= 1)/ V rni = 13 ms.
3.216. t = EAU/enid2= 4.6 days.
3.217. I = ev aead. 3.218. j = (ecui — 1) enila. 3.219. (a) B = μ,I/2R = 6.3 p.T; (b) B = 1A0 R2I12(112^ x2)312= = 2.3 RT. 3.220. B = ntio/ tan (n/n)/2nR, for n^ oo^ B =^ I.10//2R 3.221. B = 414l/ltd sin 9 = 0.10 mT. 3.222. B = (n — 9 tan 9) Ro//2nR = 28ta.
3.223. (a) B = P 4 ' 91. ( 2n:(1) + b) ) ; (b) B = 43.t1 (4: + .17b 2. 3.224. B Roh//1-t2Rr, where^ r^ is the distance from the cut. 3.225. B = RoIln2R. 3.226. (a) B = (p,0/4n) (nER); (b) B =^ (p,0/41t) (1 + 3n/2)^ PR; (c) B = (p,0/411) (2 n)^ IIR. 3.227. B= (110 /4n)1y 2/i = 2.01LT.
3.228. (a) B = (110/4n) V4 - I- n2 I1R = 0.30 RT; (b) B = (p.0/4n) X
x 1/2 +2n + n2 PR= 0.34p,T; (c) B =^ (p.0/4n) V 2^ 1/R=^ P.11 ELT.
3.229. (a) B = Ni/2; (b)^ B = Rot^ between the planes and B = 0 outside the planes.
3.230. B=^
ttoix inside the plate, Roid outside the plate. 3.231. In the half-space with the straight wire, B =^ p,o//2nr, r is the distance from the wire. In the other half-space B =^ 0. 3.232. The given integral is equal to Rol.
3.233. B
1 /2u0 [Jr]for r-<„,11,
1 /2p,o[jr] R2/rz for 3.234. B = 1 /2p.0UM i.e. field inside the cavity is uniform. 3.235. j (r) = (b I p, 0) (1 a)
3.264. (^) him = V 2F11in/RonR. 3.265. P = v2B2d2RI(R pd/S)2; when^ R = pd/S,^ the power is (^) P = P max = 114v2B2dS1p. 3.266. (^) U = 1/4110/2/n2R2ne = 2pV. 3.267. n = jBleE = 2.5.1028 m--1; almost 1 : 1. 3.268. u0= 1/1B = 3.2.10-3 m2/(V•s). 3.269. (a) F = 0; (b) F = (110/4n) 2/pm/r2, F1-1. B; (c)^ F = (p,0/4n) 2Ip,,Ir2, F r. 3.270. F = (R0l4n) 6nR2Ipmxl(R2+ x2)512. 3.271. (^) F = 312R0PimP2,1n14 = 9 nN. 3.272. 2Bx3/R9R2= 0.5 kA. 3.273. B' =B R2 sine cos2 a. 3.274. (a) %, H dS = nR2B cos (^9) 1)/N-10;
(b) ic;IB dr = (1— [) B1 sin O. 3.275. (a) Isu,. = (^) (b) 1;ot = xI; in opposite directions. 3.276. See Fig. 24. 3.277. (^) B — R0111R2^ I 111+14 nr • 3.278. B = 2B0[11( 3.279. B = 3B9[t/( 3.280. H, = NIll = 6 kA/m. 3.281. H^ bBliuond = =0.10 kA/m. 3.282. When b << R,^ the per- meability is 12 2aRBI(p,0NI bB) = 3.7-103. 3.283. H = 0.06 kA/m, Rmaxz 1.0.104. 3.284. From the theorem on circulation of the vector H we obtain B tio N I b 11°3-cd^ H^ = 1.51 — 0.987H (kA/m). Besides, B and H are interrelated as shown in Fig. 3.76. The requir- ed values of H and B must simultaneously satisfy both relations. Solving this system of equations by means of plotting, we obtain H z 0.26 kA/m, B z 1.25 T, and p, = B/R0H z 4.103. 3.285. F z 112xSB21110•
3.286. (a) xn, = 1/ V 47i; (b) X = RoFmax V e/a/VB 3.6.10-4.
3.287. A 1 /2xVB2/R0•
3.288. e 1 = By178wIa.
3.289. I = Byll(R 111,,), where Rp, = M2/(RI. + (^) B2). 3.290. (a) Acp = (^) 1 12(o2a2mle = 3.0 nV; (b) 1 /2o1Ba2 = = 20 mV. c 3.291..c E dr = —^1 /2c0Bd2 = —10 mV.
Fig. 24.
A
Fig. 25.
3.292. gi= 1/2(-1)- Bafit, where n = 1, 2,... is the num- ber of the half-revolution that the loop performs at the given mo-
ment t. The plot gi (t) is shown in Fig. 25 where t„, = V2nn/13.
3.293. /ind = a/r, where a =
3.294. gi 4nμu_^. 2 ( s±Ia2va). 3.295. ei 1 /2(wa3/33 + 2mg sin tot)laB.
3.296. v
mgR sin a B2/2 • 3.297. w g^ sin a. 1+12B2cim • 3.298. (P) =^^1 /2(no.ia2B)2/R. 3.299. B = '12qRINS = 0.5 T. 3.300. q=--
1.;orta in bb+aa i.e. is indepen-
dent of L.
3.301. (a) / = 1j2° 511 :;,i'v In 1 2 ;; (b) F=i-(1^1 :^ ln -bw ).
3.302. (a) s = vom11112B2, (b) Q = 1.12mv:.
3.303. v= am— (1— e-at), where (^) ar =B2/2/mR. 3.304. (a) In the round conductor the current flows clockwise, there is no current in the connector; (b) in the outside conductor, clockwise; (c) in both round conductors, clockwise; no current in the connector, (d) in the left-hand side of the figure eight, clockwise. 3.305. I = ,o)B f, (a —^ b)/p = 0.5 A. 3.306. cm. = 1/3na2/1Ta)Bo. 3.307. gi = 3 /2W/ Bt2 =12 mV.
3.308. E=
{ 1/2p,onir for r <a,
1 /2I-Lonia2/r for^ r> a. 3.309. I = 1 /41.t onSdi/p = 2 mA, where p is the resistivity of copper. 3.310. E = 1 /tab (ri — + 1).
3.311. co= — -26+-n B (t).
3.312. Ft max^ fZaRTb:.
3.313. Q = i/3a2t3/R. 3.314. I =114(b2 a2)nh/p. 3.315. /=1/-4n/oL/R0= 0.10 km.
3.316. L = 2 4n-2- Ippo where p and poare the resistivity and the density of copper. 3.317. t= --R in (1 = 1.5 s.
