Electromagnetic Field Theory: A Problem Solving Approach Part 73 docx

Problems 695 which is plotted versus kL in Fig. 9-14. This result can be checked in the limit as L becomes very small (kL << 1) since the radiation resistance should approach that of a point dipole given in Section 9-2-5. In this short dipole limit the bracketed terms in (14) are sin kL (kL) 2 l kL 6 lim (kL) 2 (15) )tL• i coS kL 1 2 kLSi(kL) - (kL) " so that (14) reduces to lim R (kL) 2 23L- 2 = 8 0 L(2 L 2 (16) AL'< 2i 3 3\A A Er which agrees with the results in Section 9-2-5. Note that for large dipoles (kL >> 1), the sine integral term dominates with Si(kL) approaching a constant value of 7r/2 so that lim R -7kL= 60 •- r 2 (17) kL>1 4 Er A PROBLEMS Section 9-1 1. We wish to find the properties of waves propagating within a linear dielectric medium that also has an Ohmic conductivity or. (a) What are Maxwell's equations in this medium? (b) Defining vector and scalar potentials, what gauge condition decouples these potentials? (c) A point charge at r = 0 varies sinusoidally with time as Q(t) = Re (( e'"). What is the scalar potential? (d) Repeat (a)-(c) for waves in a plasma medium with constitutive law = w eE at 2. An infinite current sheet at z = 0 varies as Re [K 0 e ( ' - k"-)ix]. (a) Find the vector and scalar potentials. (b) What are the electric and magnetic fields? 696 Radiation (c) Repeat (a) and (b) if the current is uniformly dis- tributed over a planar slab of thickness 2a: jo eij(9-kXi, , -a<z<a J 0, Izj >a 3. A sphere of radius R has a uniform surface charge dis- tribution oy= Re (&o e"•' ) where the time varying surface charge is due to a purely radial conduction current. (a) Find the scalar and vector potentials, inside and outside the sphere. (Hint: rep=r 2 +R 2 -2rR cos 0; rQp drQ= rR sin 0 dO.) (b) What are the electric and magnetic fields everywhere? Section 9.2 4. Find the effective lengths, radiation resistances and line charge distributions for each of the following current dis- tributions valid for I zI <dl/2 on a point electric dipole with short length dl: (a) I(z) = Io cos az (b) f(z) = Io e - *1 1 (c) I(z)= Io cosh az 5. What is the time-average power density, total time-average power, and radiation resistance of a point magnetic dipole? 6. A plane wave electric field Re (Eo e i ' ) is incident upon a perfectly conducting spherical particle of radius R that is much smaller than the wavelength. (a) What is the induced dipole moment? (Hint: See Section 4-4-3.) (b) If the small particle is, instead, a pure lossless dielectric with permittivity e, what is the induced dipole moment? (c) For both of these cases, what is the time-average scat- tered power? 7. A plane wave magnetic field Re (Ho e••) is incident upon a perfectly conducting particle that is much smaller than the wavelength. (a) What is the induced magnetic dipole moment? (Hint: See Section 5-7-2ii and 5-5-1.) (b) What. are the re-radiated electric and magnetic fields? (c) What is the time-average scattered power? How does it vary with frequency? 8. (a) For the magnetic dipole, how are the magnetic field lines related to the vector potential A? (b) What is the equation of these field lines? Section 9.3 9. Two aligned dipoles if dl and i2 dl are placed along the z axis a distance 2a apart. The dipoles have the same length i · I Problems 697 y while the currents have equal magnitudes but phase difference X. (a) What are the far electric and magnetic fields? (b) What is the time-average power density? (c) At what angles is the power density zero or maximum? (d) For 2a = A/2, what values of X give a broadside or end-fire array? (e) Repeat (a)-(c) for 2N+ 1 equally spaced aligned dipoles along the z axis with incremental phase difference Xo. 10. Three dipoles of equal length dl are placed along the z axis. (a) Find the far electric and magnetic fields. (b) What is the time average power density? (c) For each of the following cases find the angles where the power density is zero or maximum. (i) = Io, 12= 21o (ii) 1 = I , Il2= -21o (iii) Is = -Is = Io, 12 = 2jIo 2ar 'I I 1 A di I dl' li di ýp Y 698 y (a) Find the far fields from this current sheet. (b) At what angles is the power density minimum or maximum? Section 9.4 12. Find the far fields and time-average power density for each of the following current distributions on a long dipole: (a) i(z) Io ( 1 - 2z / L), O<z<L/2 SIo(1+2z/L), -L/2<z<0 Hint: C e az Z eaz dz = -(az - 1) f a (b) I(z)= Iocos 1z/L, -L/2<z <L/2 Hint: zi az (a cos pz + p sin pz) e cos pz dz = e (a2+ p2) (c) For these cases find the radiation resistance when kL << 1. Radiation 11. Many closely spaced point dipoles of length dl placed along the x axis driven in phase approximate a z-directed current sheet Re (Ko e'"'i) of length L. _C_ _ __ __ Solutions to Selected Problems 699 SOLUTIONS TO SELECTED PROBLEMS Chapter 1 1. Area = •a 2 3. (a) A + B = 6ix - 2i, -6i, (b) A.B=6 (c) AxB=-14i + 12i,- 18i, 5. (b) Bi1= 2(-i. x+2i,-iý), B, = 5i + i, - 3i 7. (a) A - B = -75 (b) AxB=-100i, (c) 0 = 126.870 12. (a) Vf = (az + 3bx 2 y)i + bx 3 i, + axi 14. (a) V A=3 17. (b) ' = 2abc 18. (a) VxA=(x-y 2 )ix-yi,-xi. 1 af. + af 1I af. h. au h, av h, a-w (c) dV = h/h,h, du dv dw 1 8 a (d) VA-A= - (h,h.Au)+ (huhA)+ (huhAw) huhh, au av aw 1 8a(hA4) 8(hA,) (V x A), = [ a-hAw) a(A hh, av aw 25. (a) rQp = i, (b) iQ= rQ i - 5i, + 2i rQP -30 5i. + iy (c)n -+ Chapter 2 4 wR'pg 3. Eo = 4 3 q 4. Q2= 2reod Mg 4. Q•,= 5. (a) o QIQ2 '1/2 L47eR E m 700 Solutions to Selected Problems mlm 2 7. (a) m - (b) -qq 2 1 1 (b) v = + 2 2-eom r ro) r /2/2 (d) t = ro L J qEoL 2 8. h = 2 mv 6V3 10. (b) q =- Q 12. (a) q = 2Aoa, (b) q = irpoa , (c) q = 2ooabr 15. 0 = tan - 1 2 EoMg] AL 16. (a) E,= 2reor 27reo L 2+X2) 2 18. (a) E,= -Xoa 2 7r Eo[Z2 + a2] 3 / 2 a ++Z2 (b) E - + In Aoa 2 20. (a) E, - 7ro(a22 23/ 21. E = o(a2 23/2 27reo(a +z ) 22. (a) QT= 4rEOAR 4 23. (c) Po (-d 2 ) 2 <d E,,= 2Eod 0 Jxl>d 25. (c) Por 2 r<a 3eoa E, = 2 poa r>a 3eor>a pod. 26. E = 1 2EO 2 27. W=- A 4Eo Solutions to Selected Problems 28. (a) vo- ,• Q (b) r = 4R 2rEsoRm' 29. (a) E=-2Axi., pf= -2Aeo 31. (a) Av =oa 5o Q 32. (a) dq= dz' R 33. (c) V oa cos0, (d) r =rosin 0 4reor qV, 34. (d) q - V, 36. (a) E,- 2 InrE - 21qeoq 38. (a) xo= q6oEo' (b) vo> Itýq o/4 (c) W= 161reod 43. (e) A= , a = R 2 2 44. (g) qT= - 4 7 r oR - Chapter 3 2. (a) p = AoL 2 , (e) p. =QR 3Q 4. (a) po = R- 41rvoR Eo 7. (a) d= Q Q 8. (b) -=2 rEoEo L R S 10. (a) Pind=PD Vo sinhx/ld 12. (a) V(x) = 2 sinhl/ld 2 sinhl/1d mmRAo" 15. (b) Q = q 701 702 Solutions to Selected Problems A 17. (a) Dr - 27rr A(e 2 -E1) 2E 2 A 19. (a) A'=-A " = ( e - E l) , A'= 12E el+e2 e1+82 23. (a) Por r<R EoR E,= 0 r>R 02 s In- 26. (a) R = ID(o 2 - l) 31. C= 2rl(e 2 a -elb) e£a (b -a) In 82 elb 33. oa(r=al)=-a( 1 -e ); T= Ie/ 3a, 35. pf=po e-ar/(3sA) Vo sinh V2RG(z - 1) 38. (a) v(z)= sinh NRl 41. (b) 2e[E() - E 2 (O)] + ed = J(t)l 2 dt V o /l 212 (c) E(1)) T 21(1-e - 2 ) It Vo' Vo 1 212 Vo \P I 42. (c) E?= Vo = I Ro-R) 2reel 43. (a) W=- p2- E 2P 44. W = 12ER 47. (a) W= 8weoR 48. (a) Wi.i, = 2C V o, (b) W 4na= 0CVo 49. (b) W=-pE(cos0-1) 50. h = 2(e - Eo) 2 P.Ks _I Solutions to Selected Problems 703 1 roA +Pod 2 52. (b) f =- (s + d V • o • 2 (s+d)Lso TV 0 54. (b) f = ( - eo) In - a 1 Eod v 55. V 2 s 1 , 2 dC -NVTR'eo 56. (c) T = -v -d 2 dO s o'Uwt 57. (a) v(t)= - 41reoR 58. (a) p,= Po e-o'lU 1 2 nCi 1 3 nCi 59. (a) nCi> + (c) > -, wo - R RL 2 R 2 C Chapter 4 2. (a) (oo _ > cos aye x> 0 2Ea V= 2 cos aye x <O 2ea s nry nr(I -x) sin sinh 4. (a) V = 1 d "r n=1 .HTI n odd n sinh d Po 7. (a) V,= 2 sin ax e 0 a 12. P 2 -P Ercos 0-5 2eo V(r', )= [_-Eor_+ cos4 r>a 2eor 13. (a) E= Eo 1 + L) cos + t) -Eo -) sin i S( ar 21er r r A (t) (b) cos 4 < taE (c) Amax=47reaEo 41TeaEo' V In r I In a 15. (a) V(r, z) a Voz 17. (b) Eo - 8p R 27E 22. V(2, 2) = V(3, 2) = V(2, 3) = V(3, 3) = -4. 23. (a) V(2, 2)= -1.0000, V(3, 2)= 5000, V(2, 3) = 5000, V(3, 3) = .0000 (b) V(2, 2) = 1.2500, V(3, 2) = 2500, V(2, 3) =.2500, V(3, 3)= -1.2500 Chapter 5 2. (b) B >2mV es (e) -mg 3. Bo= -mg qvo eE_ 4. (d) e= E, 2 m RBo 8. (c) J = o(E+vx B) 2lAol(a 2 + b)1/2 10. (a) B. = b Mrab B2> 8b2mVo e(b - a 2 ) 2 (c) B. = tan - 2va n PoKolr 12. (a) B. - 4 clolL 13. (a) B# I /olod 15. (b) B,= jod 2 ,-o-a • a) lyl <a 17. (b) B=. 2a 0 0 lyl> Yo 18. (d) y = at x = -o 21. (a) m, = qwaa 704 Solutions to Selected Problems I _ . au h, av h, a- w (c) dV = h/h,h, du dv dw 1 8 a (d) VA -A= - (h,h.Au)+ (huhA)+ (huhAw) huhh, au av aw 1 8a( hA4) 8(hA,) (V x A) , = [ a- hAw) a( A hh, av aw 25. (a) rQp. placed along the z axis a distance 2a apart. The dipoles have the same length i · I Problems 697 y while the currents have equal magnitudes but phase difference X. (a) What. waves propagating within a linear dielectric medium that also has an Ohmic conductivity or. (a) What are Maxwell's equations in this medium? (b) Defining vector and