Nội dung

705 Solutions to Selected Problems 23. oo=vSyBo 27. (a) I'= ) I,(O 2- (pC1 + ,,= 2s1 I 2) -Mo. 34. (a) 2 H= 2 Moa ---- [COS 2r ir+sin I -d), 35. (a) H.(x) =--(x Dd 36. (a) f, = (( - #i,] 1 I2s f, = 2Do- (b) (b) f, = jioMoDs[Ho+ Mo] 0o)HoDs, Chapter 6 (e) 1. (a) M = Ao[D - JiD-], fr = 3.(d) v(t)=vo 120singlt+cosPt e-'"/2;0 = 2 i(t)= mvowo sin f3t eBobp 2 Bobs p.oNs a 4. (a) M=-" In, M= oN[R-d 27 b 7. (c)f, (b) K(t)=Ko rod dB 9. (a) i =dr, 2 dt 14. (a)v2 vl ] N2 2 2N,' i, X So-da4(dB2 8 ddt1 10. L = oN2 [b 14. (a) -] 3po(IdS)2 32d 32 rd4 8. (a) H.= K(t), 2N, N2 2 16. (a) V,,=JBd. (n+- 2 n-) q(.+n,++p._n) 2 1 oli[D- wo- 2 706 Solutions to Selected Problems olVOI 17. (b),(c) EMF=-= 2,r R InR2 R1 (M -.uo)IVo In R R22 (d) EMF= - R, 27rr wB, 2 2 (b) voc=ýB-(b -a ) 2 18. (a) H=0,B=A oMoii, 20. (b) V >-- 1 Aoo-ND 21. (a) (RT+ Rf) >( G 4Lf (b) Ccrit-= S1 (C w G C > Ccrit(dC), 2; [R,+Rf-Go]2 [LC C < Ccr,i,(ac) 2 2Lf 2Io 22. (b) H,(x,t)= sin- 2 niD , nxrx d e n odd 23. (c) H,(x,t)= Ho- S4Ho . n'Tx •fn d sin - n=l - e-'- n odd (-1)"4Ko 25. (b) Hj(y, t)= -Ko+ ( =0 7r(2n +1) cos (2n + 1)ry I 2 2D -e , i ) y/ 8 e(l+i)ye/8 r [e(l+ (d) / (y)= Ko L[e(+i)D/-8 +e +IS 26. (a) Hz(x)-= -1 R• 27. (a) - 1-e [2eR• i/ -(1 +e R)] xeR";P= e4,(x)=Koe- 1 28. (a) s)(iý-ji.4 2Ko e~-'(.-,[i. iri.] II _I x>s ~ _ _ Solutions to Selected Problems 29. (a) H,= Ho 0 cosh k(x -d/2) ° shkd cosh kd/2 Io Ji[(r/3)(1 -j)] 32. (b) Ho(r)= 2-a J[(a/)(1 2ira J1[(a/8)(1-j)] 33. (a) T=-L1 0cos 2 wot sin 20 34. (c) T = M 0 1o1 2 COs 0, (f) 0(t)= Oo[cost+ a sin 3t] ea " MM=00L 20 t] /.Lox b 35. (a) L(x) =-lIn-, 2r a I (b) f. = 2 Poi2 41r b In a . 29 o) In - - g(b 2 -a 37. h = 47 a 2 ) Chapter 7 4. (b) W=4[PEc+goMHj] jw[oJo sin kd e-'js"'d) 9. (b)(z) nosinJkd -i os kd] d <-d 9.(b)E,(z)= Joi cos hz n COSk< d we [no sin kd -jj cos kd] 10. (b) E=Eoei(ore e/ ) z< ; z<0 = W•e -4 11. (a) tl - t = Y(Z2-Z1), co (b) t -t'2=y(tl-t2), 12. (a) u,= S-vu2 co (c)z2-z' =yL Ux.,= 2,-V -uI-- 1 -vujco 2 15. (b) e(w)= s I+ 2 2 16. (c) k 20 (0.D 0 -- &J J 1c 0w(w F wo)J 20. (a)E=Eo cosW t-cCOS 22. a2-k 2 = -w2, k =1 8 1-X-c 707 708 Solutions to Selected Problems 26. (a) L 1+L 2 =Si sin O6+s, sin ,=hA tan Oe+h2 tan 0, 0 31. 0, e41.7 (n - 33. (a) 2 ;, n 2 '- +a 2] aR (b) R'= -,-)(1 [4n Chapter 8 Vo sin P(z -1) e sin sin Bl - 2. (c) 6i(z) 4. (c) w2 = ,+k 2c2 , "/ (Short circuited end) (d) v(z, t)= 1 5. (b) k 14. (a) V+=-V- - VoZo 2R, 16. (b) tan kl = -XYo 1+4 21. (c) VSWR =5.83 22. (b) VSWR = 2 23. ZL = 170.08 - 133.29j mA nA 24. (a) ll = .137A +-, 12 = .089A +2 2 A mA 11= .279A +-, 12= .411A +2 2 mA ni 25. (a) 1 1=.166A+-, 12 = .411A +2 2 mA nA 11= .077A +-, 12 = .043A +2 2 27. (e) a = 28. (b) 2(w•/a)[b + (a/2)(o. 2 a /lI cg)] wjbko 2oe(bk• +ak ) +k) oa,8kzab(k.2+k ) Vo sin kz cos at sin sin ki Solutions to Selected Problems 29. (a) TE mode: electric field: cos k7x cos ky = const ~' A 5 sin (k;)* const magnetic field: sin sin k,y iT) ( + T)2 31. (b) 32. (a) w • = iT)2 e8L -sEoto Chapter 9 V= Q 1. (c) 4. (a) e- ivl ' 2 sin (adl/2) , a dlAt) *c-"', !(z) =- loa jo sin az ,= 4"reoRS3 o 6. (a) (c) (P)= 2 12rc 12 7rc 7. (a) mid= 27rHOR s 8. (b) sin2cos (t - kr)in ( 9. (a) •. sin oe(r-x jkr 11. 12. (a) S - kr -kr) os = const cos ka 2)] 1 2jKodl7 4E=cose-ik' e sin kL -Lsin 0 cos Fe = • O sin c se [(j C jrkrL cos 2 0 cos cos - 709 _ _ INDEX Addition, vector, 9-10 Admittance, characteristic, 579 A field, 336. See also Vector potential Amber, 50 Ampere, unit, 55 Ampere's circuital law, 334 displacement current correction to, 488 Ampere's experiments, 322 Amperian currents, 348 Analyzer, 518 Angular momentum, 350 Anisotropic media, 516-520 Antennas: long dipole, 687-695 N element array, 685-687 point electric dipole, 667-677 point magnetic dipole, 679-681 two element array, 681-685 Array: broadside, 683 endfire, 685 factor, 683, 685, 687 N element, 685-687 two element, 681-685 Atmosphere, as leaky spherical capacitor, 195-197 Atom, binding energy of, 211-212 Attenuation constant: dielectric waveguide, 646-648 lossy transmission line, 602-606 lossy rectangular waveguide, 644 non-uniform plane waves, 531-532 Autotransformer, 474 Avogadro's number, 136 Axisymmetric solutions to Laplace's equation, 286-288 Backward wave distributed system, 651 Barium titanate, 150 Base units, 55 Batteries due to lightning, 197 Bessel's equation, 280, 482 functions, 281 Betatron, 402-404 oscillations, 404 Bewley, L. V., 433, 475 B field, see Magnetic field Binding energy, of atom, 211-212 of crystal, 205-206 Biot-Savart law, 322-323 Birefringence, 518-520 Bohr atomic model, 111-112 Bohr magneton, 350 Bohr radius, 63 Boltzmann constant, 155 Boltzmann distribution, 156 Boundary conditions: normal component of: current density J, 168-169 displacement field D, 163-164 magnetic field B, 366 polarization P, 165-166 e0 E, 165-166 tangential component of: electric field E, 162-163 magnetic field H, 359-360 magnetization M, 360 Breakdown, electric strength, 93, 223 'electromechanical, 252 Brewster's angle, 540-543 and polarization by reflection, 547 Broadside array, 683 Capacitance: as approximation to short transmission line, 589-592, 601 coaxial cylindrical electrodes, 176-177 concentric spherical electrodes, 176177 energy stored in, 212-213 force on, 219-223 any geometry, 172 isolated sphere, 178, 213 parallel plate electrodes, 173-177 per unit length on transmission line, 570, 572 power flow in, 491-493 reflections from at end of transmission line, 593-594 and resistance, 177 in series or parallel, 242-243 slanted conducting planes, 273 two contacting spheres, 178-181 two wire line, 101-103 Cartesian coordinates, 29-30 Cauchy's equation, 563 Cauchy-Riemann equations, 305 Chalmers, J. A., 293 Characteristic admittance, 579 Characteristic impedance, 579 Charge: by contact, 50 differential elements, 60 distributions, 59-63 711 712 Index and electric field, 56-57 force between two electrons, 56 forces on, 51-52 and Gauss's law, 74-76 polarization, 140-142, 149 Charge relaxation, series lossy capacitor, 184-189 time, 182-184 transient, 182 uniformly charged sphere, 183-184 Child-Langmuir law, 200 Circuit theory as quasi-static approximation, 490 Circular polarization, 515-516 Circulation, 29 differential sized contour, 30 and Stokes' theorem, 35 Coaxial cable, capacitance, 176-177 inductance, 456-458 resistance, 172 Coefficient of coupling, 415 Coercive electric field, 151 Coercive magnetic field, 356-357 Cole-Cole plot, 234 Collision frequency, 154 Commutator, 429 Complex permittivity, 509, 524 Complex Poynting's theorem, 494-496 Complex propagation constant, 530-532 Conductance per unit length, 190 Conduction, 51 drift-diffusion, 156-159 Ohmic, 159-160 superconductors, 160-161 Conductivity, 159-160 of earth's atmosphere, 195 and resistance, 170 Conjugate functions, 305 Conservation of charge, 152-154 boundary condition, 168-169 inconsistency with Ampere's law, 488489 on perfect conductor with time varying surface charge, 537 Conservation of energy, 199 Constitutive laws: linear dielectrics, 143-146 linear magnetic materials, 352, 356 Ohm's law, 159-160 superconductors, 160-161 Convection currents, 182, 194-195 Coordinate systems, 2-7 Cartesian (rectangular), 2-4 circular cylindrical, 4-7 inertial, 417 spherical, 4-7 Coulomb's force law, 54-55 Critical angle, 541-544 Cross (Vector) product, 13-16 and curl operation, 30 Crystal binding energy, 205-206 Curl: Cartesian (rectangular) coordinates, 29-30 circulation, 29-31 curvilinear coordinates, 31 cylindrical coordinates, 31-33 of electric field, 86 of gradient, 38-39 of magnetic field, 333 spherical coordinates, 33-35 and Stokes' theorem, 35-38 Current, 152-154 boundary condition, 168-169 density, 153-154 over earth, 196 between electrodes, 169-170 through lossless capacitor, 178 through series lossy capacitor, 187-189 sheet, as source of non-uniform plane waves, 532-534 as source of uniform plane waves, 500-503 Curvilinear coordinates, general, 46 Cut-off in rectangular waveguides, 638641 Cyclotron, 319-321 frequency, 316 Cylinder: magnetically permeable, 357-359 and method of images, 97-103 permanently polarized, 166-168 surface charged, 80-82 with surface current, 335-336 in uniform electric field, 273-277 perfectly conducting, 278 perfectly insulating, 279 volume charged, 72, 82 with volume current, 336 Cylindrical coordinates, curl, 31 divergence, 24-26 gradient, 17 Debye length, 157-159 Debye unit, 139 Dees, 319 Del operator, 16 and complex propagation vector, 531 and curl, 30 and divergence, 24 ___ Index and gradient, 16 Delta function, 187 Diamagnetism, 349-352 Dichroism, 517 Dielectric, 143 coating, 525-528 constant, 146-147 linear, 146-147 modeled as dilute suspension of conducting spheres, 293 and point charge, 164-165 waveguide, 644-648 Difference equations: capacitance of two contacting spheres, 179-181 distributed circuits, 47-48 self-excited electrostatic induction machines, 227-230 transient transmission line waves, 586587 Differential: charge elements, 60 current elements, 323 cylindrical charge element, 81-82 lengths and del operator, 16-17 line, surface, and volume elements, 4 planar charge element, 68 spherical charge element, 79-80 Diffusion, coefficient, 156 equation, 191 Diode, vacuum tube, 198-201 Dipole electric field: far from permanently polarized cylinder, 168 far from two oppositely charged electrodes, 169, 172 along symmetry axis, 58-59 two dimensional, 231, 274 Dipole moment, electric, 137 magnetic, 345 Directional cosines, 41 Dispersion, complex waves, 531 light, 563 Displacement current, 154, 178 as correction to Ampere's law, 488-489 Displacement field, 143 boundary condition, 163-164 parallel plate capacitor, 175 permanently polarized cylinder, 166168 in series capacitor, 185 Distortionless transmission line, 603 Distributed circuits: backward wave, 650 inductive-capacitive, 47-48 713 resistive-capacitive, 189-194 transmission line model, 575-576 Divergence: Cartesian (rectangular) coordinates, 2324 of curl, 39 curvilinear coordinates, 24 cylindrical coordinates, 24-26 of electric field, 83 of magnetic field, 333 spherical coordinates, 26 theorem, 26-28 and Gauss's law, 82-83 relating curl over volume to surface integral, 44 relating gradient over volume to surface integral, 43 Domains, ferroelectric, 50 ferromagnetic, 356-357 Dominant waveguide mode, 640 Doppler frequency shifts, 507-508 Dot (scalar) product, 11-13 and divergence operation, 24 and gradient operation, 16 Double refraction, 518-520 Double stub matching, 625-629 Drift-diffusion conduction, 156-159 Earth, fair weather electric field, 195 magnetic field, 424-425 Eddy currents, 401 Effective length of radiating electric dipole, 676-677 Einstein's relation, 156 Einstein's theory of relativity, 207 Electrets, 151 force on, 218 measurement of polarization, 239-240 Electric breakdown, 93, 223-224 mechanical, 252 Electric dipole, 136 electric field, 139 moment, 137-140, 231 potential, 136-137 radiating, 667-671 units, 139 Electric field, 56-57 boundary conditions, normal component, 83, 165-166 tangential component, 162-163 of charge distribution, 63-64 of charged particle precipitation onto sphere, 293 of cylinder with, surface charge, 71, 80-82 714 Index volume charge, 72, 82 in conducting box, 269 discontinuity across surface charge, 83 of disk with surface charge, 69-71 due to lossy charged sphere, 183 due to spatially periodic potential sheet, 266 due to superposition of point charges, 57-58 energy density, 208-209 and Faraday's law, 395 of finite length line charge, 89 and gradient of potential, 86 around high voltage insulator bushing, 284 of hoop with line charge, 69 between hyperbolic electrodes, 262 of infinitely long line charge, 64-65 of infinite sheets of surface charge, 6569 line integral, 85-86 local field around electric dipole, 145146 around lossy cylinder, 276 around lossy sphere, 289 numerical method, 298 around permanently polarized cylinder, 166-168 of permanently polarized cylinder, 166168 of point charge above dielectric boundary, 165 of point charge near grounded plane, 107 of point charge near grounded sphere, 106 of radiating electric dipole, 671 in resistive box, 263 in resistor, coaxial cylinder, 172 concentric sphere, 173 parallel plate, 171 of sphere with, surface charge, 76-79 volume charge, 79-80 transformation, 417 between two cones, 286 of two infinitely long opposite polarity line charges, 94 of two point charges, 58-59 of uniformly charged volume, 68-69 Electric field lines: around charged sphere in uniform field, 297 around cylinder in uniform field, 276277 due to spatially periodic potential I sheet, 267 of electric dipole, 139 around high voltage insulator bushing, 284 between hyperbolic electrodes, 262 of radiating electric dipole, 671-673 within rectangular waveguide, 636, 639 around two infinitely long opposite polarity line charges, 95-96 around uncharged sphere in uniform field, 290-291 Electric potential, 86-87 of charge distribution, 87 within closed conducting box, 268, 300 due to spatially periodic potential sheet, 266 and electric field, 86-87 of finite length line charge, 88-89 around high voltage insulator bushing, 282-284 between hyperbolic electrodes, 262 of infinitely long line charge, 94 inside square conducting box, 299-301 of isolated sphere with charge, 109 around lossy cylinder in uniform electric field, 274 around lossy sphere in uniform electric field, 288 within open resistive box, 263 of point charge, 87 of point charge above dielectric boundary, 165 of point charge and grounded plane, 107 of point charge and grounded sphere, 103 of sphere with, surface charge, 90-91 volume charge, 90-91 between two cones, 286 of two infinitely long line charges, 94 between upper atmosphere and earth's surface, 196-197 and zero potential reference, ground, 87 Electric susceptibility, 146 Electromechanical breakdown, 252 Electromotive force (EMF), 395 due to switching, 433 due to time varying number of coil turns, 433-435 in magnetic circuits, 406 Electron, beam injection into dielectrics, 201 charge and mass of, 56 radius of, 207