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Answers

CHAPTER 1

1. (a) Ax = 5,Ay = -3,Az =-1. (b) Ax = 1, Ay = 0, Az = 1. (c) v1s. 2. (a) v = £9.821 + N9.821. (b) 84 hrs. 51 mins. (c) 4242.64 km. 3. (a) 82°.49' from north, 7°.11' from west. (b) 220.48 mls. (c) 22,678 s. 4. (a) 5.916. (b) Xl + ys - B. (c) is - yz + z. (d) -is + yz - Z. (e) (-is +18 - 7:8)/M. 5. (b) x +YZ + 7:8, -x+y6+z4,x5 -Z2,X7 -f4+Z2, -x+y6+z4, -x + y6 + z. 6. (a) +16,000kmlh (before), +15, OOOkmlh (after). (b) k = 0.9375. 7. (a) v = (X150+YZ5-z50)/160.08. (b) 160.08. 8. <X9+13-7:3)/llr. 9. (-X3-Y4-z1)/~. 10.(a) (-x-y-Z)/.J3. (b) (i8-Y6+Z2)/v104. (c) (in+fb-i)/Ja2+b2+1. 13.(a) 1800 ± 57.688°. (b) x - 13. 14. (a) 9Iv17N. 15. (a) 27.04. 17. (a) 3.13. (b) 0.911. (c) 0.738. 18. (b) (1,2) and (0,2) 21. (a) P1(.J2,45°, 1), P2(.J2,45°,0), P3(1,90°,I). (b) y = 1. (c) rsin¢ = 1. (d) Rsin(9)sin(¢) = 1. 22. (a) x2+i+z2= a2. (b) r2 + z2 = a2. (c) R = a.

23. (a) P (a sin 9' cos ¢', sin 9' sin¢', a cos 9')

(tl) P (,9',¢'). 24. A = R6.742 - ih.482 -+1.1094. 25. (a) A = x(3 eos2 ¢ + 2r1l2 sin¢) + y(3 cos ¢ sin ¢ - 2rl12 eos¢) + zr¢. (b) AR = 3 cos ¢ sin 9 + r¢ cos 9, A8 = 3eos¢eos9-r¢sin9,A¢ = _2r1l2. 26. (a) rl = Xa+Yb+Zi:. (b) r2 = Xa'+Yb'+"Zi:'.

(c) R = x(a' - a) + y(b' - b) + z(c' - c). 27. (a) rl = 7:3, r2 = xU + Y1.5 + 7:3.J2I2. (b) xU + Y1.5 + z(3.J2 - 6)/2.

CHAPTER 2

1. (a) tan-I (b/a). (b) ran-l (b/a). 3. (a) 12. (b) 12. 4. zero. 5. 71'2/12 -71'(.J3 - 2). 6.3. 7. 371'a2. 8. (a) 771'a4d/6+7I'd3a2/12. (b) 4a571'/5 + a571'2/4. 9. (1007l'ho/9)(600a5 + 600ba4 + a3 h~ + 600a3 b2 + 3a2 bh~ + 600a2 b3 + 6ah~b2+600ab4+10h~b3+600b5) 10. (a) 7:813. (b) x16+18+7:32/3. 11. x50/3+Y7512m1s. 12. xl0/3+Yl0. 13. zero. 14. (a) IVPI =

2J(x - 2)2 + (y - 2)2 + (z + 1)2, VP = (X(x-2)+Y(y-2)+z(z+I)/IVPI. (b) 2(z+l) (c) .J2(x + y - 4). 15. (a) reos2 ¢ + +( -2r cos ¢ sin ¢ + z cos ¢)/r + z sin ¢. (b) x(x3 + 2xy2 - xyz)/(x2 + i)312 + y( _x2y _ 2x;y + x2Z)/(x2 + i)312 + zy/(x2 + i)312. (c) R(eos2¢sin9+ cos 9 sin ¢)+ 9(eos2 ¢eos9 - sin9sin¢) - +(2 eos¢sin ¢ -cos e cos ¢/ sine) 16. (a) x5/.J35 + 13/v1s + zllv1s. (b) x4/~ - 13/~ + zl/~. (c) (in + yb - zl)/Ja2 + b2 + 1.

X3y +y(3x +z) + z(y + 1) 17. (a) .

J9y2 + (3x + Z)2 + (y + 1)2 xl - yzy - Z2z X2x + yzy + Z2z

(b) . (c) . Jl + 4y2 + 4Z2 J4x2 + 4y2 + 4z2

18. (a) 2x. (b) 2z2/r. (c) X/Jx2 +z2 +

1203

1204 ANSWERS

y/Jx2 +y2. 19.4. 20.4. 21. 3.6

24. (a) 3. (b) 3. 25. (a) -i2y. (b) +(4z +

6r) + ilO. (c) yzlJx2 +Z2 + ix/Jx2 +y2. 26. (a) zero. 27. (a) X150xsin(wt + 50z) + Z3 cos(wt + 50z). (b) No. 28. (a) -i2(x + l)y + yz2 + z(2yz - 2y). (b) -z6 sin cf>. 30. (a) 2z. (b) -z5. (c) -20n. 31. (b2c3 + a2b3+a3c2)/6Jb2c2 +a2b2 +a2c2. 33.(a) 2(y-2)2(Z + 1)2 + 2(x - 2)2(Z + Ii + 2(x - 2)2(y - 2)2. (b) 12z. 34. (a) 'il2 A = Z2. (b) 'il2 A = r(6/r)coscf>. 36. (a) X4x+Y1. (b) 10x2 +4y. (c) 4. (d) 'il2R = O. (e) x.z-Y4xz+z(4xy-x). 37. (a) 3. (b) 1 and 5.

CHAPTER 3

1. (a) 10.548 x 1010 C. 2. 2.3 X 1012 N. 3. 15.4 mm 4. (a) 1.168 x 10-11 C. S. (a) ax = -X"4qQxhrmeo(4x2 + d2)3/2. 3m""

occurs at x = ±dl2.fi. 6. (a) to x = 0.414d from QI. 7. (b) 0.1 m to the right of negative

charge. 9. (a) E = _q_ ( 2R _ R - dl2 _ 4neo IRP IR - dl213

R + dl2 ) 3pd (~ 2 IR + dl2P . (b) E = 8neoR4 Rcos (}-

8 sin (} cos (}). 13. 3.82.10-13 C/hr/m2.

14. (a) 0.0605L from bottom of triangle and at infinity. 15. (a) 6.125 x 1010 N/C. (b) 3.0625 x 1010 N/C. 16. (a) 438.15 mls 17. (a) 9.81 x 10-7 C. (b) 9.81 mls. (c) 1.53 X 107 s. 18. EI = y.fiP/2neoL, E2 = XPI(..J5 - 1 )12neoL..J5 + y p/neoL..J5, E3 = Xp/3neoL. 19. (a) E = zPlahl2eoW + a2)3/2. 20. 1.797 x 10-5 N. 21. 1.78 x 10-4 N

(attraction). 22. E = zPsh [! _ 1 ] 2eo h Jh2 +a2 .

23. E = zPsh [In a +..fii2+1i2 _ a ] 2eo h Jh2 +a2 .

24. E = zp/28o. 25. El = 0, E2 = zPv L( 5 - .JI7)/8eOl... E3 = zPv L( 4+..J5 - .J37)/8eo. 26. (a) E(R) = -R1.44 x 1O-9/R2 Vim. (b) E(R) = -ih.44 x 1O-9/R2 Vim. 28. E = -Rpoa4120eoR2. 29. (a) XS.988 x 10-7 N. (b) x1.09~ X 10-8 N. 30. (a) D = Rq/4nR2. (b) E~= Rq/4neoR2 for R < RI and R > R2, E = Rq/4neR2 for Rl < R < R2. (c) 4> = q. (d) 4> = q. (e) 4> = q.

CHAPTER 4

1. (a) p(x) = 3kex2. 2. (a) -8.854 x 10-26

Clm3. 4. (a) E = ZPo128 for z > 0, E = -ipol2e for z < O. (b) E = zPol2eo for z > 0, E = -ZPo1280 for z < O. S. E = -ypva/eO,y < -b. E = y(Ps - pva)/eo, -b < y < -a. E = -YPvlxl/eo, -a < y < O. E = YPvX/80, 0 < Y < a. E = y(Ps + pva)/eo, a < y < b. E = ypva/eo, b < y. 6. (a) E t = xp/2et + yp/2et, E2 = -xp/2eo + f"p/280, E3 = -XP/2e2 -YP/2e2, E4 = Xp/2e3 +yp/283' (b) F2 = -xqp/2eo + yqp/2eo (in 2nd quadrant). 7. (a) E = RRl6eo. (b) E = R4/3eoR2. (c) E = -R(60+R3)/3eoR2. (d) E = -RI48/3R2. 8. E = xPox212eo for 0< x < dl2, E = -xPox21280 for -dl2 < x < 0, E = xPod2/8eO for x ::: dl2, E = -xPod2/8eO for x < dl2. 9. (a) E = Y PlY +

- 2neo(y2 + Z2) ~[PIZ Ps ] r Z 2 (y2 2) + -2 lor z > ZO, neO +Z 80 E = Y PlY + zf PIZ _ .!!!...]

2neo(y2 + z2) '"'l2neo(y2 + Z2) 2eo for z > zoo (b) D(O, 0, 1) = z(p/2n ± p/2) (the sign depends on the values of zo, PI>

and Ps). 10. (a) E = r~(a - b), reo

(c) PI = -21rps(a - b) = 2nps(b - a) C/m. 11. (a) zero. (b) -y14, 029.6 Vim.

~ Q 12. (a) E = R -4 R2 for R < a,

neo ~(a2ps Q)

E = R eoR2 + 4neoR2 for R > a.

(b) Ps = -4Q 2 C/m2• 13. (a) Pv = ey. na

(b) E = ed2/8eo. 14. (a) Pv = 4eoerb. ~ 2 4

(b) E = Rb /4. (c) 4> = eoerb n. 15. 2.65 x 10-5 Clm2• 16. -8 V. 17. (a) E = -xp/eo between plates, E = 0 outside. (b) V = Psdleo between plates. V = 0 to the left of the left plate, V = Psdleo to the right of the right plate. 18. (a) V(R) = _R2 Pv/6eo + Q/4neoR +a2 Pv12eo, R ~ a. 19. (a) Psb = Vllbeo/b In(b/a). (b) PSII = Vllbeo/a In(a/b). 20. V = Po( J a2 + d2 - d)12eo. 21. (a) E =

~ ~ PI (J2 -1) yp/4'\1 5neo. (b) V = -4 In --;;:;- . n80 '\17 - 3

22. zero. 23. (a) E = xpox/e (the x = 0 plane is midway between the

plates). (b) E = xpoalEo for x > a, E = -xpoalso for x < -a. 24. E = x50x. 25. (a) Pv = -6s. (b) E = -R2R inside the sphere, E = -R2sb3/soR2 outside. 26. (a) E = u - Y2 + z4. (b) zero. (c) -u + Y2 - z4. (d) -20 x 10-6 J. 28. (a) Vyl2b (y is the distance from lower plate). (b) VyI(2b - 2c), 0 < Y ::::: d and Vd/(2b - 2c) + V(y - (d + 2c»/(2b - 2c), d + 2c < Y ::::: 2b. 29. Q/47(soR.

V", 30. PSC = )' c2(lIasl - lIbsl + lIbs2 - lIcs2

Vca PSII = a2(lIasl - lIbsl + lIbs2 - lIcs2)' 31. (a) E = 0, V = -kQlb, (a < R ::::: b), E = -RkQIR2, V = -kQIR (R > b).

~ 2 32. E = -Rq/47(sor4, V = -q/47(sOr4' 33. P = RQ(S2 - SO)I27(S2r. 34. (a) 120 Vim. (b) P = 3.187 X 10-9 C/m2•

( ) D ~[PIX Ps ] ~ PlY 35. a = x 27(x2 + y2) +"2 +y 27r(x2 + yl)'

1 D ~[PIX Ps] + ~ PlY x >, = X 27(x2 + y2) - "2 Y 27(x2 + y2) ,

X < 1. (b) D = x6.367 X 10-7 Clm. (c) P = ~[PIX(Sr - 1) Pisr - 1)] ~ PIY(Sr - 1) x 27r(x2 + y2) + 2 + y 27(x2 + y2)'

x> 1, P = x[PIX(Sr -1) _ Ps(sr -1)] + 27r(x2 + y2) 2

~ PIY(Sr - 1) () 198 76 Y 2 2' X < 1. 36. a . 27(x + y )

Vim in air. (b) 198.76 Vim in air. 37. (a) 26.55 x 10-6 Clm2• (b) 3 x 106a V. 38. 1.367 x 1010 C. 39. (a) 4.577 x lOs V. (b) 4.899 X lOs V. 40. D2 = x5so + y6so. 41. (a) E2 = -xsoElsl> E3 = 0, ~ = -xsoElsl> Es = -XE. (b) V = 2soEdlsl. 42. (a) E2 = -rEI sinO - X(solsl)EI cosO, E3 = -yEI sinO - X(SOIS2)EI cosO, E4 = EI = -yEI sinO - XEI cosO.

(b) V = ElasocosO(S2 +sl)/sIS2).

43. (a) PI/(Sl - so) = P2/(S2 - so), SIPln/(SI - so) = S2P2n/(S2 - so) for Ps = 0, PI/(Sl - so) = P2/(S2 - so), SIPlnl(SI - so) - S2P2n/(S2 - so) = Ps for Ps #- O. (b) PlnlPlt = (P2nIP2t)(S2/sl). 44. Ctotal =

27(so 60so/(20d2 + 3dl ). 45. C = In(alb) F/m.

46. 2.78x 10-12 F. 47. (a) 0.02 m. (b) 0.0385 m. 48. C = 47(sl s2bca(sla(c - b) + s2c(b - a».

49. W = ~[~ __ 1 __ ~]. 50. (a) Win = 47(so a a + b b

ANSWERS 1205

27(p;a4/45so. (b) Wout = 47(p;as /18so. (c) Wtotal = 47(p;as 115so. 51. W = 27rsoaV2. 52. (a) VW = sV2al2d(d - a). 53. LlW =

Q2(sr - 1) [! _ !]. 54. (a) w = l 1327(2 soR\ 87(sosr a b

o < R < a, w = (q + 47(a2 PO)2 1327(2 soR\ a < R < b, w = (q + 47(a2 Po - 47(b2 PO)2 1327(2 SOR4,

R > b. (b) W = (q + 47(a2 PO)2 [! _ !]. 87(so a b

55. W = soV2b2(1 - sr)l2d. 56. W = pi In(bla)/47(s. 57. p = soV2ld2. 58. V = dJ2PIslb. 59. F = -sosrV2bl2d in the direction opposite the displacement.

CHAPTER 5

2. k = -O.6yz. V(x) = _pox212so + (Void + Podl2so)x. 5. V(x) = -9411.94x4 + 37.64777x3 + (50000 - 7.52955 x lO-s)x. 6. (a) V(r) = 23.083 In r + 191.45 V. (b) E = -r23.083/r. 7. V(x,y) = -(2Vo/7() tan-I (ylx) + Vo.

~ 2 8.(a) V(R) = 2.75IR-500v. (b) E = R2.751R . 9. (a) P = -1.99 X 10-10 Clm2•

(b) P = -1.424 X 10-10 Clm2•

11. (a) Ps = -Plhl7(x2 + h2). 12. E = -yp/so between sheet and conductor. E = 0 elsewhere. 13. (a) V(2d,y) =

~ In y'4d2 + (y + d)2. (b) E(x,y) = 27(so y'4d2 + (y - d)2

~ 4dpi (1 1) x 27(so 4d2 + (y _ d)2 - 4d2 + (y + d)2 .

q.fi[ b-a 14. (b) P = ± 47( «x _ a)2 + (x _ b)2)3/2 +

b+a b-a «x + a)2 + (x - b)2)3/2 - «x + a)2 + (x - b)2)3/2

b-a ] «x +a)2 + (x + b)2)3/2 ' where b = 0.3827d,

Il = O.9239d. 15. (a) E = -x~ X

LI2 (-li+1 (x-a cos(ISO+(i-I)300» _

i=1 J (x-a cos(IS°+(i-1)300»2+(y-a sin(lso+(i-I)300»2 ~ -.!L ,,12 (-li+1(y-asin(ISo+(i-I)300»

y 411"80 ~i=1 J(x-acos(ISo+(i-I)300»)Z+(y-asin(ISo+(i-I)300»2

16. (b) V(d/2,0) = 3.284q/47(sod. 17. (a) V(x,y) =

q N 1

47(so i=l.fu3 .... y'(x + (i - l)dl2 + 3d/4)2 + y2 -

1206 ANSWERS

N 1

;=2,&'14,00. J(X + idl2 - dl4)2 + y2 -N 1

;=3,&'15,00. J(X + (i - 3)dl2 + 3d14)2 + yZ + N 1

. L /(x + (i - 2)d,/2)2 +y2 ,=4,S,12,16,00. v 22. E(O,O) = X17.284 Vim. 23. E = ~ PI ( (x - 2r) (x - rl2) x- - +

21!'so «x - 2r)2 + yZ) «x - rl2)2 + y2) (x + rl2) (x + 2r) )

--'-----'--~ - + «x + rl2)2 + y2) «x + 2r)2 + y2) ~ PlY (1 1 Y 21I'so «x - 2r)2 + y2) - «x - rl2)2 + y2) +

1 1) «x + rl2)2 + yZ) «x + 2r)2 + yZ) 24. (a) E(o, 0, 0) = -Xl q/81!'s002 . (b) zero. 25. (b) ER(r,9) =

q _02 - 20r 41!'sor (r2 + (0 + r)2 - 2r(0 + r) cos 9)3/2 . (c) pk,9) =

q 02 +20r - 41!'r (r2 + (0 + r)2 - 2r(0 + r) cos 9)3/2 •

~ 2 26. (b) E = RaVo/R , R ~ o. 27. (b) E(r,o) = i(Ps -

So q (02 +20r) )

41!'sor (r2 + (0 + r)2 - 2r(0 + r) cos 9)3/2 •

4Vo ~ 1 . m1!'y ""'" 29. V(x,y) = - L., - sm -b-e-.-.

1!' m=I,3,5,00. m 30. (a) V(x,y) =

4Vo ~ ~ sin m1!'X sinh(7). 1!' L., m 0 sinh _b

m=1,3,S,... II

31. V(x,y) = . Vo b sinh(1!'Y) sin 1!'x. sinh 1L 0 0

II

32. V(x,y) = 4Vo ~ ~ sin ~ (inh(m1!'(o - X»)

L., inh tmrll s b + 1!' m=I,3,5,00. m s T

sinh(m;x)). 33. V(x,y,z) = 16Vo 00 00 sin ( ~) 7 L L . / 2

m=I,3,5 n=I,3,5 mn sinh ( 1!'0 V ( ~) + n2) x

[sin m_1t_x sinh ( 1tZJ m2 + n2) +

M7 Onh( .. 8'+ {)]

2Vo 00 1 (r)k . 34. (a) V(r,¢) = ~ L k; smk¢,

*=1,3,5'00'

2Vo 00 1 (0)* . r < 0, V(r,¢) = ~ L k; smk¢,

*=1,3,5'00'

2Vo 00 1 (r)*-I[ r>o. (b) E= 01!' L k; rsink¢-

*=1,3,5'00'

+cosk¢], r > o. 35. (a) V(r,¢) =

4(VI + Vi) ~ 1 (r)*[. k L., -- sm¢+ 1!' k 0 *=1,3,5, ...

sink(¢+ i)], r < 0, V(r,¢) =

4(VI: V2) t ~ (;)* {sink¢+ sink (¢+ i)}, *=1,3,5'00'

r > o. (b) E =

4(VI + Vi) ~ 1 (r)*-I[~. k L., - - rsm ¢ + 01!' k 0

*=1,3,5'00'

+cosk(¢ + 1!'12)], r < 0,

E 4(VI + V2) ~ 1 (O)k-I [~. k = - L., - - rsm ¢-01!' k r

*=1,3,5,00'

+ cos k(¢ + 1!'12)], r > 0

CHAPTER 6

1. (a) VI = 0, V2 = 3, V3 = 6, V4 = 9, V5 = 12 V. (b) VI = 0, V2 = 1.5, V3 = 3, V4 = 4.5, V5 = 6, V6 = 7.5, V7 = 9, Vs = 10.5, V9 = 12 V. (c) V(x) = 12x. 2. (a) VI = 0, Vi = 3.09375, V3 = 6.125, V4 = 9.09375 V. 3. (b) V(x) = 1.4118 X 104(x - x2), E = -1.4118 x 104(1 - 2x). 7. 1.342 to 5.588 pF. 9. (a) 2.5116 pF. (b) 1.5547 pF. 10. (a) ~ 3 pF. 11. (a) 4.97 x 10-16 F. (b) 4.355 X 10-16 F. 12. (a) NI = (X2 - X)/(X2 - XI), N2 = (x - xI)/(X2 - XI). (b) ¢(x) = (X2 - x)¢I/(X2 - XI) + (x -xI)</JzI(X2 - XI)' 13. (c) V(x) = -Pox4/12s + PoX3d16s + 10xld - PoXd3/12s. 14. (c) E = -x(-poXls + 10Id + Podl2s). 16. (a) 1.42 V. (b) 11.l/Lm. 17. (a) 3240.0V. (b) 1480 V.

CHAPTER 7

1. (a) 360,000 C. (b) 48,700 km2•

2. (a) 1667 S/m. (b) 3.125 x 1021 carriers. 3. (a) 10.294 x 106 A. (b) 205.88 X 106 A. 4. 0.553 X 10-3 mls. S. (a) 5.584 x 10-2 Vim. (b) 2.487 x 10-4 mls 6. Inside beam: E = -r7.19 x 107r Vim, outside beam: E = -rJ.9.97Ir Vim. 7. (a) E = -rVlr In(bla). (b) J = -rO'Vlr In(bla). (c) I = 21rLO'VI In(bla). (d) R = In(bla)I27rLO'. 8. 7.368 x 105 S/m. 9. (a) 5.1473 x 10-5 Q.

(b) -0.198% 10. (a) 1.56098 x 10-5 Q.

(b) 1.4303 X 10-5 Q. 11. Rmin = 17.98 Q,

Rmax = 29.3 Q 12. (a) 0' = 4b1017ra2V. 13. 0' = O'o(b2 - a2) [ (~ - a2)(L - p) + pb2 ]

a2 9pb2 - (L - p)(b2 - a2) 14. (a) VBA = 2.944VI(O'1faR + 2.944)V. 15. V = 1MO'I7rrf + 1h/0'27rr; + 1M0'37rrf. 16. R = O'(O'od + ad3 13)lb2, IVI =

I -I( fa ) b2.jiiCiO tan V -;;~/ . 17. (a) VAB =

2.944JaI0'. 18. 15.9 Q. 19. (a) 1.628 mm for series connection, 7.54 rom for parallel connection. (b) 16.65 kg for series connection, 357.2 kg for parallel connection. 20. (a) 2.76 x 10-6 Q

(b) 8.354 X 1010 W. 21. (a) Jiron = VO'la, Jcopper = 5VO'la. (b) Piron = V20'bela, P copper = V2 50'bel a. 22. (a) 1.894 x 10-5 Q.

(b) 5.904 X 10-4 Q. (c) 8.085 W. 23. (a) 16 A. 24. (b) I = 1.2e-IOOt Cis. 25. QI = -8IV0'27rb2/(dI0'2 + d20'1), Q2 = V7rb2(810'2 - 820'1)/(dI0'2 + d20'1), Q3 = 82 VO'I7rb2 I(dl 0'2 +d20'1). 26. (a) Ecopper = 0.0223 Vim, Ealuminum = 0.03536 Vim. (b) wcopper = 14,220 W/m3, Waluminum = 22,515 W/m3• 27. (a) V = 8.944 mY, 1= 3512 A. (b) 1.755 X 107 W/m3•

CHAPTER 8

1. B = 9/-Loll27ra. 2. B = +12 x 1O-711L. 3. H = zPsfJJ(b - a)l2.

4. BI = -Y;:a(l + Jz), B2 = ~ /-L1 ~ 2...ti/-Lo1a2

-y 47rd· S. B = z7r(4h2 +a2)(2h2 +a2)112·

ANSWERS 1207

1208 ANSWERS

(b) 4> = JLola. 24. Ijt"" = 7.958 X 103 A. 25. H = -yl,OOO Aim.

CHAPTER 9

Leq = 8JLNfe2/(a + e). 23. JLoM212 ]1m3• 24. AW = n2 l27rb2(JLo - JL)I2. 25. AW = -JLoNa2l 2/d. 26. (a) 4> = JLle In(d/b)I27r. (b) 4> = JLJLole(d - b)

JLo7r(d + b) + Ig(JL - JLo)

28 L _ 7ra2 JLJLoN • 12 - ~

(27rro - Ig)JLO + IgJL 7ra2JLJLoN

27rroJLo + IgJL . 29. H =

JLIJL2 [Nlll + N2h] JLOJL2[d + a - 2b] + JLOJLild + a - 2b] + JLIJL2/g'

30. B = 2JLOJLIJL2Ni . JLo [7r(rl + r2) - 4d ](JL2 + JLI) + 8JLIJL2d

31. (a) W = (Nlll +N2l2)2JLoJLIJL2be

33. Fm = YO.005 N (up-

CHAPTER 10

1. (a) 3.056 x 10-3 A. (b) l(t) = (V2 + vI)Bod

2r[P - (V2 + VI)t] 2. (a) v = VIBod. (c) D = VlIr. 3. (a) F = -Xwbd2 m cos(wt) sin(wt)IR. (b) zero. 4. V = -voJLoJr:I2R. 5. Vab = Bovd.J2 cos(wt)l2. 6. V = 2 07rrl dB sin 20m. 7. emfnnmd/emfsquare = 7r. 8. emf = rifBed sin(2nft). 9. (a) emf = wBo/212. (b) emf = wBo{1 - (I + l)e-I ).

10. (a) V = -wd2JLoI06 cos(wt)/8. 11. emfAB = -0.17r(sin(8007rt +

800Jrcos(800m»). 12. 157,080 A. 13. 0.001 N· m. 14. (a) 11 V, 13.64 A. (b) 1.364 A. (c) Zp = 80.7 n, Zs = 0.807 n, ZplZs = 100. 15. V = 1.019 X 10-2 sinwt. 16. (a) NI/N2 = 0.024, II = 40,000 A, 12 = 960 A. (b) NI/N2 = 0.024. (c) 12 = 950 A, <1>= 0.25 Wb. 17. (a) 17.68t (b) 5.3 t. 18. (a) WII = 4.11sin(314t), W21 = 0.41 sin(314t). (b) W31 = 49.32 sin(314t). 19. (a) emf l = 2000Jrcos(2Jr x 105t), emf2 = 200Jrcos(2Jr x 105t). (b) emf l = 1990Jrcos(2Jr x 105t), emf2 = 100cos(2Jr x 105t).

J-Lowclo a +b 20. emf = ---In--coswt.

2Jr a N J-LocIow a + b

21. emf = 2 In -- coswt. Jr a

22. (a) 40 V. (b) 0.02 n.

CHAPTER 11

1. Jd = i(0.51J-L)(cos lOOt) sin 5z. 2. Id = 4Jr8oabwVo cos wtl(b - a). 3. (a) Jd = rW8r80VOcoswtlrln(bla). (b) Id = 2Jr8r80WLVo coswtl In(bla). 5. 7.992 GHz. 6. llconJIIlldis:rip I = UIW8. 7. t, = 35.416 X 10-9 s. 8. Jdmax = 0.885 Alm2. 9. A = 0.3183, (3 = 0.04189.

20 x 1O-4ei(104t+lo-4z)

15. (a) E = Y 9 104 x J-L08o 20 X 10-4 ei(104t+10-4z)

(b) D = Y 104 J-Lo ' H =

5 X 107 ei(104t+1O-4z)

x 16. H(t) = Jr

y1250cos(106t - 50z), B(t) = y5 x 1O-4Jrcos(106t - 50z). 17. V2J, -

a2j. aJe 8 2 0 J-L8atf = J-Lu-at. 1. V1/!= .

19.(a) -V2A+V(V.A)= J-LJ. (b) E=-aAiat. (c) V2A = -J-LJ. 20. (a) H = -V1/!.

2 alF a (b) -v F + '11('11. F) = -8f,L[jfi - 8J-LFtV1/I.

(c) V· F = -J-L~ '111/1. 21. (b) V2F = J-LJ + at

a2F J-L8[jfi. 22. lEI I = 5 Vim, IDII = 1080 Clm2,

IE21 = 3.81 Vim, ID21 = 11.45680 C/m2•

23. B2 = 4Jr X 10-7 (i24 + y5 - zlO) T. 24. Hit = H2t = 0, H2t = O. 25. (a) J =

ANSWERS 1209

x106/4Jr + Y106/4Jr. (b) B2 = ZO.l T. 26. (a) HI = x105 +Y2 X 105 + z104 Aim, H2 = x105 + Y2 X 105 + z104 Aim. (b) HI = X(105 + 1001 .J2)+Y(2 x 105 -1001 JI)+z104 Aim, H2 =x(105 -100IJI)+y(2 X 105 + 1001JI) + z104 Aim. 27. (a) H2 = x(105 -X200IJI) + y(2 x 105 - 2001.J2) + z104 Aim. 28.(a) A·A = a2-b2+j2a.b. (b) A·A* = a2+b2. (c) A • B = (a • c - b • d) + j(b . c + a . d). (d) A . B* = (a. c + b . d) + j(b • c - a . d). (e) A x A = 0 (f) A x A* = -j2(a x b). (g) A x B = (a x c - b x d) + j(b x c + a x d). (h) A x B* = (a x c + b x d) + j(b x c - a x d). 29. H(z, t) = y5 cos(wt- (3z). 30. H(x,y, z) =

Ho sin m;x cos n;y [cos(kz) - j sin(kz)], . mJrx nJrY

H(x,y,z) = Ho sm ---;;- cos TL - kz,

H( ) lJ· mJrx nJrY -jkz x,y,z = nosm---;;-cosTe .

31. E(z) = EIL(-kz + 1/1) + E2L(kz + 1/1), E(z) = EI,!(-kzH ) + E2,!(kzH ).

32. E(x,z,t) = Eocos(wt - (30(xsin8; + z cos 8;» 33. (a) Ex(z) = Eoe-j(kz-¢). (b) Eiz) = jwEoe-j(kz-¢). 34. (a) H = -x1250 cos(106t - 50z) + y1250cos(106t - 50z). (b) H = -x12501!"e-j50z + Y12501!"e-j50z . 35. (a) H(x,y, z, t) = (Xhx + yhy + zhz) cos(wt + (3z + cf» - (Xgx + ygy + zgz)sin(wt + (3z + cf». (b) H(x,y,z) = IXhx + yhy + zhzIL({3z + cf» + IXgx + ygy + zgz I L({3z + cf> + JrI2).

CHAPTER 12

1. (a) V2E - v(!!.) - J-L aJ = O. (b) V2E-8 at

a2E a2B 8f,L-2 = O. 2. (a) V2B - J-L8-a 2 = O. & t 3. V2A+W2J-L8A = O. 4. V2D+W2J-L8D = O. 5. V2B + w2 J-L8B = o. 6. (a) E = w2 J-L8~ -Vv. (b) V2Tt+w2J-L8~=0,Gauge:V.~=-V. 7. E(20) = z5 cos(109Jrt - 200Jr/3). 8. (b) vp = c = 3 X 108 m/s,j = 1.05 X 1010 Hz. 9. 663.13 W. 10. 'Pav = -rIJ/4Jr2R3u. 11. (a) E = 1027 Vim, H = 2.72 Aim. (b) E = 275 Vim, H = 0.73 Aim. 13. (b) 17(lOOHz) ~ 9.943 x 10-3 +j9.943 X 10-3 n, 17(100 MHz) = 9.943 + j9.943 n, 17(10 GHz) = 74.47 + jlO.92 n. 14. E(x = 1m) = -u.87 x 10-17 e-j38.95 Vim,

1 21 0 ANSWERS

15. (a) Pt = 2097.1 W (b) Pt = 0.012 W 16. (b) P = fl(E5/17)e-2ay cos2(wt - Py). (c) IHI = 0.057 Aim. 17. (a) 0.266 W 18. (a) PI = 1.0 X 10-3 W; P2 = 1.026 X 10-11 W (b) Vpl = 2.268 X 108 mis, Al = 604.8 run, 171 = 2.84.98 n, Vp2 = 1.896 X 108 mis, A2 = 506 nm, 172 = 238.4 n. (c) ¢2 - <PI = 2.04 X 1010 rad. 19. (a) 17 = 367.9 n, f3 = 21r.j1.05f/c rad/m, vp = 2.927 X 108 mls. (b) 170 = 377 n, f30 = 2;if1c rad/m, VpO = c = 3 X lOs mls. 20. a = 1.21 x 10-6 Np/m. 21. (a) 17 = 2.63 X 10-3 + j2.63 X 10-3 n. (b) 17 ::::: 2.63 x 10-3 + j2.63 X 10-3 n. 22. EI = 1.035 X 10-2 Vim. 23. (a) z I /z2 = 10-4 • (b) vpllvp2 = 10-4• 24. 0" = 3.36x 107

S/m, a = 230.26 Np/m. 25. (a) 2.198 m. (b) 695.295 m. 26. (a) 8", = 6.61 X 10-7 m. (b) 8bg = 5.03 X 10-6 m. 27. (a) In copper: 1760Hz = 2.04 X 10-6 + j2.04 X 10-6 n, 1710GHz = 2.63 X 10-2 + j2.63 X 10-2 n. In iron: 1760Hz = 1.54 X 10-4 + j1.54 X 10-4 n, I7IOGHz = 0.198 + j0.198 n. 30. 0.0987 m. 31. 8 = 2.98 X 10-9 m. 32. (a) f « 8.88 x lOs Hz. (b) f » 8.88 x lOs Hz. 33. J = 0.000062 Alm2• 34. (a) 790 A. (b) 2317 A. (c) 12,566 A. 35. (a) d", = 2.9123 X 10-4 m, dill = 3.665 X 10-4 m, d",u = 3.1 X 10-5 m, dpo = 69.54 m. (b) cost: aluminum. weight: mumetal. volume:

1 mumetal. 36 (b) v - -::::---;;;::===::;;:::;:::::::o:

• P - ..fji8Jl - w;lw2' ';"""'w2"-_-W--'2

Vg = c. (c) vp = wlf3 = 00, Vg = O. w..fji8

.;r-w-:-'2 J.L-e---n'""""'2-=-1 a w 37. (b) Vg = ,vp / 2 2/. •

wJ.Le v w J.Le - n a 39. (b) vi! = 100) = 0.316 mis, vi! = lOs) = 104 mis, vi! = 100) = 0.632 mis, vi! = 106) = 2 X 104 mls. 40. (a) vp = 1.498 X lOs mls. (b) Vg = 1.503 X lOs mls. (c) v. = 1.4979 X lOs mls. 41. (a) E =

mAo mAo Ao sin Wc t+ - cos(wc-ws)t- -2- cos(wc+w,)t.

2 mA (b) E(z, t) = Ao sin(wct - f3cZO) + T cos«wc -

w,)t-{3/ZO)-~o cos«wc+W,)t-{3bZO). (c) vp =

1

CHAPTER 13

1. E,.(z) = _XO.8E;12.0S9z , Et(z) = XO.2E;e-2.1 x 1O-8zIIS.73z. 2. (a) EI""", = 11.716 Vim. (b) HI""", = 0.031 Aim. 3 E""", = 1 + 1(172 - 170)/(172 + 170)1.

• Emin 1 - 1(172 - 170)/(172 + 170)1 4. (a) Pcrmverttd = 343.24 W/m2, eff = 24.5%. (b) P crmverttd = 350 W/m2, eff = 25%. 5. (a) Egkus = 5.244 Vim, Hgkus = 0.01866 Aim, (b) Pgkus = 7.686 X 10-10 W 6. 735.78 W 7. P = 84.9 W; W = 509.4 W· h. 8. (a) 3.25A. (b) 192.8 Vim. 9. EIH = -jl7o tan {3z. 10. (a) J, = -flE;ll7o. 11. (a) Elt = E2t , Hit - H2t = 1s.

1 1 (b) 1/2 = (1 + })-. -, 82 = ~'

C1ZUZ V "Jf.L2U2

Vp2 = J 2w , Y2 = (1 + })JrifI1-20"2' 11-20"2

12. a = 45°. 13. (a) H; = 0.265(ycosa - zsina)e-jPo<ysina+zcosa). (b) E,. = -XEre-jPo<Y sin a-z cos a) ,

H,. H (~ + ~. ) -jPo<ysina-zcosa) = r ycosa zsma e .

( ) J <:::t\ 530 -jfJoysina C :r = Xv. cos ae . ~EJI cos 8; ( 2

14. (a) PIIV = Y 2 1 + r.L + 170

) , ... .Eft sin 8; 2) r.L2 cos(2{3oz cos 8;) + z 2170 (1 - r.L .

EJITI~ ~. 8) (b) P I1V = --,zcos8t + ysm t· 2172

15. (a) 8; = 63°26', 8t = 46°55' (c) III = 0.11, 111 = 0.726. (d) Etx = 7.26, Ely = 3.63 Vim, Err = 1.1, Ery = 0.55 Vim. 16. (a) Elt = Ezt , Hit - H2t = 1s. 17. (a) E; = 37700( -ycosa +zsina)e-jPo<Ysina+zCOSa).

(b) H,. = XHre-jPo<ysina-zcosa), H,. = ilOOe-j2094.4(ysina-zcosa). (c) Js = -flOO Aim. 18. (a) EI (x,y, z) = f.j9794.46 sin(173 .2z)e-jlOOy,

'IOOy HI (x,y, z) = i30 cos 173.2ze-J •

(b) Eptak = ±9794.46 Vim at z = mnl246.4,

m = 1,3,5,7, ... , Hpeak = ±30 Aim at z = m1l1173.2, m = 0,1,2,3,4, .... (c) 'P",,(x,z) = YS4825eos2(173.2z) W/m2•

19. (a) vpr = cI sin {;lj. (b) vpr = 3 X 108 mls.

20. d = (d2I2d1)JO.0016 + 3df m. 21. (a) J = YO.0459 Aim. (b) J = XO.053 Aim. 22. (a) CPTJ. =

»0~1J-2 cos {;l. cos {;It - '!!!!:l. eos2 {;l. (

., 2(72' 2(72 ' ) tan-1

3'!!!!:l. eos2 {;l. + 2»0 ~ eos{;l·eos{;lt • 2(72 /./ V 2(72 /

2 r]o ~21J- cos {;lj cos (;It -1( V 2~ ) (b) CPT = tan . 1. 2 '!!!!:l. eos2 {;l. - »2 eos2 {;l

2(72 / '/0 t

24 _ eos{;lj - ~h.1 - sin2 {;lj . rJ. - , eos{;lj + .12.1 - sin2 {;lj

.12.1 - sin2 {;lj - 2.1 eos{;lj 111 = .

.12.1 - sin2 {;lj + 2.1 eos{;lj 25. 111 = (r]2 - r]1)/(r]2 + r]1), 111 = 2r]2 / (r]2 + r]1), {;lj = 0, 111 = 1, 111 = 0, {;lj = 90°. 26. (a) {;lb = 83°39'. (b) {;lb = 63°26'. (c) {;lb = 56°18'. 27. 82 = 3.53780. 28. (a) {;le = 6°23'. (b) {;le = 41°25'. (c) (;le = 41°49'. 29. 8r = 2.894. 30. (a) 111 = -0.12388, Til = 2.2478. (b) 8e =30°. 31.(a) 8r =2. (b) 8e = 70°32'. (c) 8e = 45°. 34. (a) d = nAI2. 35. (a) d = tan-l (r]2 e-.ftr12 )Ir]IW J/.t2 8 2' 36. d = 1..12. 37. 3.14 mm. 38. (a) r]1 = 377 n, r]2 = 188.37 - jO.42 n. (b) ZI =

(i(~ij+~~) tan(P2 -ja2)d+2~O~2e-j/lo'-j(~ij-~~) tan(/32 -ja2)de+j/lo') r]1 (2~o~2+j[1fo+~~ltan(fJ2-ja2)de JPO'+j[1fo-~~ltan(fJ2-ja2)d) , Z = q2[~o+~21e("2+jll2)d+~2[qO-~21e-("2+jP2)d

2 r]2 2~2~O cos(a2+j/32)d+j2~~ sin(a2+j/32) ,

Z3 = r]o. 39. El = X[le-ftrz + (-0.46693 - j0.12261)r], z < 0, HI = n1e-ftrz + (0.46693 + j0.12261)irz]I377, z < 0, E2 = X[(0.75552 - jO.020435)e-j1.SlrZ -(0.22246 + jO.10218)ei1.5lrZ], 0 < z < 0.1 m, H2 = Y[(0.75552 -jO.020435)e-j1.SlrZ+(0.22246+ j0.10218)ei1.SlrZ]1251.34, 0 < z < 0.1 m, E3 = X[(0.65441 + jO.055284)e-j2lrZ -(0.16564 + jO.14313)ei2lrZ ], 0.1 m < z < 0.2 m, H3 = Y[(0.65441 + jO.055284)e-jbrz + (0.16564 + j0.14313)ei2lrZ]/188.5, 0.1 m < z < 0.2 m, E4 = X[(0.74918 - jO.45302)e-j1rZ], z > 0.2 m, ~ = n(0.74918 - jO.45302)e-ftrz ]/377, z > 0.2 m. (b) !'slab = -0.46693 - j0.12261, Tslab = 0.74918 - jO.45302. 40. 4.93 W/m2. 41. (a) !'slab = -0.9226 + j0.3858. (b) d = 0.0745n, n = 0, 1,2, ....

ANSWERS 1 211

CHAPTER 14

1. (a) C1 = 80.26 pF/m, Ll = 0.1386 /.tRIm, C2 = 722.34 pF/m, L2 = 0.1386 /.tRIm. (b) ZI = 41.56 n, Vpl = 2.9979 X 108 mis, Z2 = 13.8534 n, Vp2 = 9.993 X 107 mls 2.(b) C = 5.6x 10-12 F/m,L = 1.98 x 10-6 RIm. (c) L = 0.2634 /.tRIm, C = 42.24 pF/m. 3. C = 66.67 pF/m, L = 0.375 /.tRIm. 4. (a) Zo = 145.6 n, vp = 1 X 107 mls. 5. vp = c, Zo = 3.77 n. 6. (a) 299.79 m (b) L = 0.497 /.tH, C = 22.388 pF. 7. (a) 14 mm. (b) Zo ~ 21.4 n, a = 0.00374 Np/m (c) 307.8 m . 8. (a) d = 0.0117 m. (b) Zo ~ 441 n, a = 3.77 x 10-3 Np/m. (c) 2,289 m . 9. (a) y = 4.86 X 10-5 + j4.86 X 10-5

(b) Zo = 8.85 + j8.85 n, y = 4.86 X 10-5 + j4.86 X 10-5•

10. (a) ZL = Zo = 635.68 - j19.9 n. (b) 0.0104 w. 11. (a) C = 16.67 X 10-12

F/m, L = 1.499 X 10-6 RIm, R = 0.0628 n/m. (b) y = 1.047 X 10-4 + j3.14 X 10-3. 12. (a) E = r(24.15Ir)e-j/3z, H = ;;;0. 127Ire-j/3z. (b) P = 24 W. 13. (a) 282 kW. (b) 902 kW. (c) Hmox = 0.796 Aim, E11/IIX = 300 Vim. 14. (a) d = nAI2, n = 0,1,2, .... (b) Same. 15. d = nAl4, n = 1,3,5, .... 16 . ZI(1 + ne-j2/32d) - Zz(1 + ne-j2 /32d) -jZ/3ja

. nn ZI(1 + ne-j2/32d) + Z2(1 + nrj2/32d) e . 17. ZjnCr) =

e(0.01OHjZO.96)r + 0.598e-(0.0105+jZO.96)r 18.84 e(0.0105+jZO.96)r _ 0.598e-(0.0105+j20.96)r . 18. Z = 45.23 + j14.69 n. 19. (b) Zinl = 9Z0/1O. 20. Zin50 + j153.9 n. 21. ZLl = 200 n, ZL2 = 12.5 n. 22. (a) d = 0.1762 + 0.5n, n = 0,1,2, .... m. (b) d = 0.8976 + O.5n, n = 0,1,2, .... m. (c) d = 0.8238 + 0.5n, n = 0,1,2, .... m.

ZL(1 + e-j2/32 /l) + ZI (1 - e-j2 /32 /l) 23. Zjn = ZI '2 I '2/3 I • ZL(1- e-} /32 l) + ZI(1 + e-} 2 l) 24. d = vpO/4lfz - fi). 25. (b) dmax = vpO(2fi - ji)/4fdi· 26. (a) Zo = 100 n, a = 1.02 x 10-4 Np/m, f3 = 5.88 X 10-5 rad/m. (b) 0.435 W. 27. (a) Zadd. = 83.69 n. (b) z",jn = 31.25 n, Zmox = 80 n. Minima at: z = 0, 1..12, A, 1.51.., ... , Maxima at: z = 1../4, 3A14, 51../4, . . . . 29. (a) 10 n (b) 300 V and -200 V. (c) 6 A and 4 A. 30. (a) 4.4.

1 21 2 ANSWERS

(b) V""", = 100 V, Vmin = 22.73 V. (c) V""", at z = n')../2, n = 0,1,2, ... , Vmin at z = (2n + 1)')../4, n = 0,1,2, .... 31. (a) 2.0. (b) Vmin = 100 V, V""", = 200 V. (c) Vmin at z = n')../2, n = 0,1,2, ... , V""", at z = (2n + 1)')../4, n = 0,1,2, ....

32. (a) Z(z) = '50 -1 + tan(20nz) . J 1 + tan(20nz)

33. (a) Z(z) = '50 1 + tan(20nz) . J 1 - tan(20nz)

34. (a) r(z) = 0.3e-j (0.042z-1.267) (b) 1.857. (c) z = 109.98 m. (d) SWR = I +(25 x 1O-12.,2+j500x 10-6",)1(1002+25 x 10-12.,2) 1-(25xI0 l2()+j500xI0-6",)I(I()()2+25xI0 lZ()'

35. (a) r(z) = 0.3e-j(0.042z+J.263) (b) 1.857. (c) z = 44.69 m.

2 5ai + 1013 + 1013 - j5w X 106 (d) SWR = . .

2.5w2 + 1013 - 1013 + j5w X 106

36. (a) Zin = Zo ZL + Zo tanh«a + jf3)1) . Zo + ZL tanh«a + jf3)1)

(b) n = (ZL - Zo)/(Zo + ZL).

(c) 'P1fl) = ~Re{(V+V+OIZ~)(I-nri+n-rD}. 37. (a) VL = 9.97 L - 76.8° V. (b) 0.0026 w.

CHAPTER 15

1. (a) r = 0.6L22°. (b) r = 0.6L - 100°. (c) 4.0. (d) 42 - j75 n. (e) V""", at 0.03').., V min at 0.28')... 2. (a) 80.44 V. (b) 30.56 V. (c) Minima: 0.088').. + n(O.5')..), n = 0,1,2, ... , maxima: 0.338').. + n(0.5')..), n = 0,1,2, ... . 3. 50 + j50 n. 4. (a) SWR = ) 4Z6 + (XL + ZI tan(f3Id)2 + IXL + ZI tan(f3Id)1

)4Z6 + (XL +ZI tan(f3Id)2 -IXL +ZI tan(f3Id)1

(b) d = (>../2n) tan-I (-XLIZI)' S. (a) 210 - j648 n (b) 9. (c) 17.5 W. 6. (a) 71 - j21 n. (b) Maxima: 0.45').., 0.95').., 1.45').., 1.95').., 2.45').., 2.95').., 3.45').., 3.95').., 4.45').., 4.95').., 5.45').., 5.95').., 6.45').., 6.95').., 7.45').., 7.95').., and 8.45')... Minima: 0.2').., o.n, 1.2').., I.n, 2.2').., 2.n, 3.2').., 3.n, 4.2').., 4.n, 5.2').., 5.n, 6.2').., 6.n, 7.2').., 7.n, and 8.2')... (c) n = 0.244L - 36°, SWR = 1.75. 7. (a) 1 m. 8. (a) ZL = 97.5 - j78 n. (b) r = 0.431L - 50°. 10. (dlQ = 0.462').., d2Q = 0.102')..) and (dlb = 0.12').., d2b = 0.39n). 11. Position: 0.224').. (1.12m), length: 0.153')..

(0.765m) or: position 0.419').. (2.085m), length: 0.34n (1.735m). 12. dlQ = 0.029').., d2Q = 0.105').. or: dlb = 0.189').., d2b = 0.396')... 13. (a) r = l/3L - 180°. (b) 2.0. (c) location: 0.403').. (0.959m), length: 0.148').. (0.352m) or: location: 0.098').. (0.233m), length: 0.352').. (0.838m). (d) n = 0.53L - 96°. 14. dB = 0.375').., d21 = 0.177').. or: d12 = 0.125').., d22 = 0.094')... 15. dB = 0.264').., d21 = O.D4lA or: d12 = 0.144').., d22 = 0.439')... 16. (b) Zt = ;Z;Z;. 17. d = ')../4, Z, = ZoJ(l - InO/(l + InO, if ZL < Zo, Z, = ZoJ(l + InO/(l - InO, if ZL > Zoo 18. d = 0.1611').., Zt = 114 nor: d = 0.411 lA, Zt = 38.46 n. 19. (a) dmin = 2.6316')... (b) Zt = 367.42 n. (c) 1.0 and 1.5.

CHAPTER 16

4. (a) zero. (b) > 3 f.LS. 6. (a) 9.26 V. (b) 10 V. 8. (a) 33.34 V. (b) 0.667 A (c) 0.1 ms. (d) 0.2 ms. 9. (a) V(IOf.Ls) = -0.02235 V, I(lOtLs) = -0.57125 A 10. V(lOf.Ls = -3.3976V,I(lOtLs) = -0.06745 A. 11. (a) Vg = 7.11 V, VL = 6.912 V, v,,- = 7.373 V, v,,+ = 7.373 V, Vb- = 7.465 V, Vb+ = 7.465 V. (b) Vg = 7.1424 V, VL = 6.912 V, v,,- = 7.1424 V, v,,+ = 7.1424 V, Vb- = 7.189 V, Vb+ = 7.1885 V. 12. Vg = 7.1424 V, VL = 0 v, v,,- = 7.1424 V, v,,+ = 7.1424 V, Vb- = 1.659 V, Vb+ = 1.659 V. 13. (a) VL(t) = 100(1 _ e-(t-IO-6)150xlO-6 ),

t ::: llt. (b) h(t) = 2e-(t-lO-6)150xlO-6 ,

t ::: llt. 14. (a) VL(t) = 100e-(t-lO-4)150x 10-6 ,

t ::: llt, h(t) = 2(l _ e-(t-l0-4)150xlO-6 ),

t ::: 10-4 s. (b) V(t) = 100e-(t-IO-4)150xlO-6 V, I(t) = 1 - e-(t-IO-4)150xlO-6 A 17. (a) 3000 m. 18. (a) 1500 m. (b) Z = 16.7 n. 19. (a) 3000 m. (b) Z = 150 n. 20. (a) 1500 m.

CHAPTER 17

1. (a) Vg = 5.43 X 108 mis, ')..g = 0.003 m, Z1M = 208.4 n. (b) E(x,z) = x1.1134cos(1000nx)e-j2094.4z +

ij1.67 sin(1000nx)e-j2094.4z. (c) H(x,z) = YO.0053 cos(1000nx)e-j2094.4z. (d) 'P = ZO.0059cos2(1000nx)e-j4188.8z -~0.0088 sin(1000nx) cos(1000nx)e-j4148.& W/m2•

2. (a) 160 GHz. (b) 160 GHz. (c) ZTE = 232.7 0, ZTM = 174.5 0, ZTEM = 201.5 O. 3. (a) 31.84 x 10-12 W. (b) 31.84x 10-12 w. (c) PWTEM = 106 x 10-12 W. 4. 3 GHz. S. (a) r = -0.29, T = 0.71. (b) SWR = 1.817 (left), SWR = 1 (right). 6. E = X(rJ>J)"z)2Ho cos(nxla) cos(wt) sin(2nZ/)..z) -z(,,,)../)..,)2Ho sin(1rxl a) sin(wt) sin(2:n'z1)..z), H = YZHo cos(1rxl a) sin(wt) cos(2:n'z1)..z). 7. E = X(17)../)"z)Ho cos(nxla)[sin(wt + 2:n'z1)..z) + r sin(wt - 2:n'z1)"z)] + Z(I7)"I)..z)Ho sin(nxla) [cos(wt+2:n'z1l..z)+ r cos(wt-2:n'z1)..z)], H = YHo cos(nxla)[sin(wt + 2ru:J)..z) -r sin(wt - 21rZ/)"z)]. 8. E(z) = yEoe-)OJ,fiiOiOz, H(z) = -X(Eo/l7o)e-jw.jiiOiOz. 9. E(x, z) = fj2Eo sin(mnxlti)E-j27rZ1l.g , •

H(x, z) = -~(2Eo/I70)()../)..g) sin(mnxlti)e-J27rZ1l.g -

z(2Eo/I70)()../)..",,) cos(mnxlti)e-j27rZ1l.g ••

10. E = i2Hol7o()../l..g) cos(mnxlti)e-J27rZ1l.g + ij2H0I70()"II..",,) sin(mnxlti).e-j27rZ1l.g ,

H = Y2Ho cos(mrrxhJ)e-J27rZ1l.g •

11. (a) b = 1.5 m. (b) 8. 12. (a) 2.0793 GHz. (b) 1.1865 GHz. 13. TEIO (5.263 GHz), TEoI (11.886 GHz), TEn = 1Mn (12.999 GHz), TE20 (10.526 GHz), TE21 = TE21 (14.024 GHz) 14. (a) E = 1.755 X 106 VIm, H = 3052.73 Aim. 15. (a) > 30 MHz. (b) 30 MHz.

16. (a) Ie = ~J (21 r + (21bt (b) t = BrlP,rl a

d[~p,oBoJl- (7)2 +~P,IB)I- (7)1 18. (a) TEIO (15 GHz). (b) 0.0298 w. (c) 0.0199 W. 19. 2.326 x 106 W. 20. (b) 5.3 X 1013 Hz (infared). (c) 1.62 x 10-3

W. 26. (a) adTElo = 0.0179 Np/m. (b) 3%. 27. (a) 2.98 mW/m. (b) a", = 0.00472 Np/m. 28. (a) a = 0.0716 Nplm. (b) 0.0298 W/m. 29. (a) 4.5 GHz. (b) TEol1 (5.59 GHz), TE201 (7.9 GHz), TEI03 (8.38 GHz). 30. TElol (2.1 GHz), TEI02 (2.164GHz), TEI03 (2.266GHz), TEI04 (2.4 GHz), TElOS (2.564 GHz), TEI06 (2.75 GHz), TEI07 (2.955 GHz), TEI08 (3.175 GHz), TEI09 (3.408 GHz), TEIO,IO (3.65 GHz). 31. (a) d = 0.03867 m. (b) TEIOI (5 GHz). 32. f = nl2d~P,IBI'

ANSWERS 121 3

CHAPTER 18

2. (a) E ~ R(1.53ljR3) cos 9 + 8(0.763ljR3) sin 9, H ~ +(6.37 x 1O-3IR2) sin 9. (b) E = 8{j7.54IR)singe-j>rR, H = +(j0.02lR)singe-j>rR. (c) Rrllll = 0.316 0, PnuI = 0.632 W. (d) 27.5 lan. 3. (a) 'Pw = R(Vl12170R2)sin29. (b) PnuI = 4nVl/3170. 4. (a) 5.12 mW. (b) 44.92 mW. S. (a) a = J /)'l'Irr{3. 6. (a) Prllll = 102w2 p,2 f32 na4151 12 17. (b) ~ = 102w2p,2f32rra4/617. . 7. (a) E = 8{jf31710U8rrR)singe-iPR, H = +JifJloU8rrR) sin ge-jPR , 'Pw = R(fJ21715L2/128rr2R2)sin29. (b) R.,.I111 = 2 On2(U)..)2 . 8. P W = 7.54 X 10-10 W/m2• 9. (a) ~ = 73.08 0, efJ = 99.43%. (b) 99.86%. 10. (c) 96.8%. 11. (a) 1.4615. (b) 1.4857. (c) 159. 12. (a) l(z) = 10 cos(nz'IL). (b) l(z) = 10 sin(2rrlz'IIL). (c) l(z) = -10 cos(3d IL). (d) l(z) = 10 cos(rr/4 - 3nlz'II2L). (e) l(z) = -10 sin(4nlz'IIL). (f) l(z) = -10 sin(n/4 - 5nlz'II2L). 13. (a) [(.(9)1 = l(cos«3rr/4) cos 9) + Ilv'l)1 sin 91. (b) [f.(9) 1 = 1 (cos«n) cos 9) + 1)1 sin 91. (c) [f.(9) 1 = 1 cos«3rrl2) cos 9)1 sin 91. (d) [f.(9)1 = 1 (cos«2n) cos 9) - 1)1 sin 91. (e) [f.(9) 1 = 1 cos«5rrl2) cos 9)1 sin 91. 14. (a) 3 m. (b) 99.54 0 (c) 49.7715 W. (d) D = 4.822. 15. (a) 2.433 A. (b) 23.2 lan. 16. (a) The thicker an-tenna. (b) 99.46% and 99.94%. (c) 0.995. 17. (a) 0.00439 O. (b) 0.0021915 w. (c) d = 1.5. 18. (a) 3 m. (b) 36.54 O. (c) 18.2715 W. (d) 1.642. 19. (a) E = ~81710e-jPR cos«nl2) cos 9) . (f3h . II ",)

-""'---- Sin sm u cos." , nR sin 9 •~loe-jPR cos«nl2) cos 9) . (lIh . II ",)

H - sm p smucos.". - nR sin 9

(b) [f.(9,1/»1 = 1 (cos«nl2) cos 9)1 sin 9)1 sin 9) sin(f3h sin 9 cos 1/»1, o 5 9 5 1r12, 0 5 I/> 5 21r. 20. (a) 1{.(9) 1 = 1 (cos(2rr cos 9) - 1)/sin91. (b) 64.9 w. (c) [f.(9) 1 = l(cos(3n cos 9) + 1)1 sin 91, 73.9 W 21. (a) 1(.(9,I/>,qI)1 = 1 cos«nl2) cos 9)1 sin 91. (b) 1(.(9,I/>,qI)1 = 1 cos«rrl2) cos 9) cos«rrl2) sin 9 cos 1/»1 sin 91. (c) 1(.(9,1/>, qI)1 =

1214 ANSWERS

1 - cos«rrl2) cos 8) sin«rrl2) sin 8 cos</»/ sin 81. (d) [f,,(8, </>, q;)1 = 1 cos«rrl2) cos 8) cos«rrl2) sin 8 cos </> + rr/4)/ sin 81. (e) [f,,(8, </>, q;)1 = 1 cos«rrl2) cos 8) cos«rr/2) sin 8 cos </» + rr/8)/ sin 81· (f) [f,,(8, </>, q;)1 = 1 cos«rrl2) cos 8) cos(rrsin 8 cos </»/ sin 81. (g) [f,,(8, </>, q;)1 = 1 - cos«rrl2) cos 8) sin«rrsin 8 cos </»/ sin81. (h) [f,,(8, </>, q;)1 = 1 cos«rrl2) cos 8) cos(rr sin 8 cos </> + rr/4)/ sin 81. (i) [f,,(8, </>, q;)1 = 1 cos«rrl2) cos 8) cos(rr sin 8 cos </> + rr/8)/ sin 8 I. (j) [f,,( 8, </>, q;) 1 = 1 cos«rrl2) cos 8) cos«3rrl2) sin 8 cos </»/ sin81. (k) [f,,(8, </>, q;)1 = 1 - cos«rrl2) cos 8) sin«3rrl2) sin 8 cos </»/ sin 81. (1) [f,,(8, </>, q;)1 = 1 cos«rrl2) cos 8) cos«3rrl2) sin 8 cos </> + rr/4)/ sin 81. (m) [f,,(8, </>, q;)1 = 1 cos«rrl2) cos 8) cos«3rr/2) sin 8 cos </> + rr/8)/sin81. 26. (a) [f,,(8,</>,q;)1 =

I Cos«rrl2) cos 8) II sin(3rr(sin8cos</>-I)I2) I

sin 8 5 sin(0.6rr(sin8cos</> - 1)12) . (b) [f,,(8, </>, q;)1 = I cos«rr?) cos 8) II sin(6rr(0.6 sin 8 cos </> - 1)12) I

sm8 6 sin(rr(0.6sin 8 cos </> -1)12) . (c) [fi8, </>, q;)1 = I cos«rr~2) cos 8) II sin(1Orr(0.7 sin 8 cos </> - 0.75)12) I

sm8 1Osin(rr(0.7 sin 8 cos </> - 0.75)12) . (d) [fa(8, </>, q;)1 = I cos«rrl2) cos 8) II sin(5rr(0.7sin8cos</>+ 1)12) I

sin 8 5 sin(rr(0.7 sin 8 cos </> + 1)12) . 27. (a) [(.(8, </>, 0)1 = 11 sin«5rr/4) sin 8 cos </» I . "5 sin«rr/4)sin8cos</» 1 sm81·

28. (a) [f,,(8, </»1 =

I cos«:~~~cos 8) IIcos2 (rr sin :cos</» I.

29. (a) [fa(8, </»1 =

I cos«rr~2) cos 8) II s~[(nrrl2) sin 8 cos </>] I sm8 nsm[(rrl2)sin8cos</>] ,

o ::: 8 ::: rrl2. (b) E = 9 x jTJ10 e-jPR ..i(n-l)("I2) sin o cos <j>

2rrR ~ x cos«rrl2) cos 8) sin[(nrrl2) sin 8 cos </>]

sin 8 sin[(rrl2) sin 8 cos </>] ,

o ::: 8 ::: rrl2, H = .f!~e-jPR x

i(n-1X"I2) sin Ocos <j> cos«rrl2) cos 8) sin[(nrrl2) sin 8 cos </>] sin 8 sin[(rrl2) sin 8 cos </>] ,

o ::: 8 ::: rrl2. 30. (a) E =

• wl-Lf3Irrd2 /4 e-jPR sin 8i(n-l)>/I12 sin(m/ll2) 4rrR sin( ""12) ,

H ~9wl-Lf3Irrd2/4 -jPR . 8..i(n-l)·'·12 sin(n""l2) =- e Sln~~· . 4rrTJR sin( ""12)

(b) F (8,J.. ) = sin(0.08rrnsin8cos</» n , ,/" q; . (0 08' , sm . rrsm 8 cos </»

[f,,(8,</>,q;)1 = I! S~(0.08rrn~in8cos</»I· n sm(0.08rrsm8cos</»

31. At 520 kHz: A. = 39726.7 sin2 8, at 1.6 MHz: A. = 4196.5 sin2 8. 32. (a) P rtceivtslJ/(11UIX) = 3c2 P132rr2f2 R2.

3c3 /p (b) Ip•ak = 8J3Ra2rr3V ~. 33. (a) I = 2.469 X 10-4 A (b) I = 1.633 X 10-5 A 34. (a) 125.66 (b) 10m2• 35. 7955m. 36. 198.44m2•

Index

Airless sprayer, 604 alternator, 645 Ampere's force law, 597 Ampere's law, 486, 496, 630 Ampere, Andre Marie, 109,420,472 ampere, unit of current, 420

experimental definition of, 601 amplitude modulation (AM), 781 anechoic chamber, 866 angle of incidence, 808

TM, in parallel plate waveguide, 1067

anodic reaction, 455 antenna, 1121

aperture, 1193 arrays, 1165 broadside, 1177 dipole, 1151 down cables, 896 high gain, 1191 hom, 1193 linear, 1151 linear, of arbitrary length, 1151 loop, 1151 monopole, 1151 parabolic, 864 patch,1193 properties of, 113 3 quasi static solution, 1128 radome, 861 receiving, 1180 reflector, 1193 resistance of, 1182

antenna array adaptive, 1167 broadside, 1171 end-fire, 1177 large baseline, 1166

antenna radiation pattern, normalized, 1170

approximation process, 355 array factor, normalized, 1170, 1177 attenuation

below cutoff, 1093 in lossy dielectrics, 796 in rectangular waveguide, 1091 in the atmosphere, 801 in waveguide, 1093

attenuation constant, 767, 814, 824, 883, 892,896

due to wall losses in waveguide, 1092 for evanescent waves, 1048 in low loss dielectrics, 771 low loss, TE, 1053 low loss, TEM, 1053 low loss, TM, 1053 TE propagation, 1049

average power density electric dipole antenna, 1147 magnetic dipole antenna, 1147

average radiation intensity, 1140 electric dipole antenna, 1147 magnetic dipole antenna, 1147

azymuthal angle, 34, 1169

B(H) curve (see magnetization curve), 550

1215

1216 INDEX

BAG-GAB rule, 22 back emf, 636 bandwidth (in waveguide), 1086 battery, 447 beamwidth,1139 Bell, Alexander Graham, 456 binomial expansion, 140,205,771 Biot,Jean-Baptiste, 476 Biot-Savart law, 476,500,501,596,1124 boundary conditions, 287, 399 boundary value problems, 283,353

analytic methods, 283 breakdown

in air, 265 Brewster's angle, 850

for parallel polarization, 850 for perpendicular polarization, 851

Cable coaxial, 897 television (CA1V), 897

capacitance, 231 calculation of, 325 in CMOS devices, 381 of cables, 237 of electrode, 378 of infinite structures, 236 of overhead line, 326 of the globe, 235 perunitlength,236,326,886,998 shunt, 882

capacitive fuel gauge, 242 loading on transmission lines, 1023

capacitor, 231 high voltage, 235 multilayer, 243 parallel plate, 233 two layer, 245

capacitors in parallel, 239 in series, 239

carrier frequency, 781 cathode ray tube, 159 cathodic protection, 455 cathodic reaction, 455 cavity resonator, 1097

detection of materials, 1108 parallel plate, 1117

characteristic impedance, 895, 910, 925 in parallel plate waveguide, 1063

in transmission line, 893 charge, 122

bound,212 distributions, 284 induction of, 163 line distribution, 143 on electrode, 378 point, 123 surface distribution, 147 volume distribution, 152

charge density, 122,424 at surface of conductor, 209 line, 123 nonuniform, 124 surtace, 123,224,704 time dependent, 439 uniform, 124 volume, 124

charged layer, 183 charging

by friction, 265 of conductors, 164

circulation, 60 of vector field, 62 total, 105

clamping ampere meter, 660 coercive field intensity, 550,610 communication

in fiber optic cables, 801 in sea water, 803 underwater, 802 with satellites, 774 with spacecraft, 793 with submarines, 776

complex permittivity, 765 complex plane, 711 computer programs, 354 conductance

per unit length, 886 shunt, 882

conduction microphone, 455 conductivity, 162, 206,425 conductor, 206

bimetal,467 perfect, 207, 453,556

conservation of charge, 418, 437, 688, 691,1123

constants of integration, 287 constitutive relation, 696

for current density, 427

for electric flux density, 696 for magnetic flux density, 475, 696

contact potential, 447 continuity equation, 437,691 contour integral, closed, 60 convergence, 367 coordinate system, 29

Cartesian, 5, 30 cylindrical, 34 orthogonal,29 rectangular, 4 spherical, 40

coordinate transformation Cartesian to cylindrical, 36, 80 cylindrical to Cartesian, 36 cylindrical to spherical, 42 spherical to Cartesian, 41 spherical to cylindrical, 43

core memory, 568 corona, 162 cosine formula, 15 Coulomb

force, 474 gauge, 701, 702 torsional balance, 158

Coulomb's gauge, 502 Coulomb's law, 126, 164,257 Coulomb, Charles Augustine de, 123,

126 coulomb, unit of charge, 123 Coulomb-Lorentz force equation, 592 coupled charge device (CCD), 412 coupling coefficient, 656 critical angle, 852 critically coupled cavity, 1108 cross product (see vector product), 17 Curie temperature, 552, 614 curl,97

associative property, 102 distributive property, 102 in Cartesian coordinates, 100 in cylindrical coordinates, 101 in spherical coordinates, 101 of a vector field, 96 of the electric field intensity, 175

current, 418 conduction, 418, 425 convection,418,419,462 convection due to rain, 422 definition of, 438

INDEX 1217

in photovoltaic tube, 423 induced, 661, 673 steady, 439 surface, 883 transformer, 659 transient, 441

current density, 418, 421,595 as a field, 443 at surface of perfect conductor, 558 conduction, 425 convection, 419 equivalent, 541 in a fuse, 430 surface,554,555,704,883 time dependent, 439

current wave backward propagating, 1024 forward propagating, 893, 910, 1024

cutoff frequency in parallel plate waveguide, 1061 lossless, TE, 1053 TE propagation, 1048 TEM,1053 TEM propagation in parallel plate

waveguide, 1072 TM in rectangular waveguide, 1077 TM propagation, 1053

cutoff wavelength in parallel plate waveguide, 1061 TE propagation, 1048 TM in rectangular waveguide, 1077

cutoff wavenumber, 1100 in parallel plate waveguide, 1061 TE propagation, 1048 TM in rectangular waveguide, 1077

Deep space network, 1193 degenerate mode, 1085, 1089, 1103 del operator, 76

in Cartesian coordinates, 76 in cylindrical coordinates, 81 in spherical coordinates, 83

demagnetization, 550 demagnetizing field, 551 Descartes, Rene, 30 diamagnetic materials, 546 dielectric, 217

breakdown, 220 breakdown in semiconductors, 221 coatings, 862 losses in waveguide, 1091

1218 INDEX

dielectric (cont.) lossy, 453, 764 materials, 212 perfect, 212, 453, 764 strength,220 waveguide, 1107 window, 861, 863

dielectric constant, 126 differential oflength

in Cartesian coordinates, 31 in cylindrical coordinates, 35 in spherical coordinates, 41

differential of surface in Cartesian coordinates, 31 in cylindrical coordinates, 35 in spherical coordinates, 41

differential of volume in Cartesian coordinates, 31 in cylindrical coordinates, 35 in spherical coordinates, 41

dipole, 205 dipole antenna

full wavelength, 1157 half wavelength, 1156, 1164, 1168 Hertzian, 1129 three halves wavelength, 1158

directional derivative, 73, 75 directivity, 1141, 1184

of arbitrarily long antenna, 1155 of electric dipole antenna, 1147 of half wavelength antenna, 1157 of Hertzian dipole antenna, 1141 of magnetic dipole antenna, 1147 of quarter wavelength monopole

antenna, 1164 of receiving antenna, 1181, 1190

Dirichlet boundary condition, 362,400 discontinuities in waveguides, 1070 discretization, 395 dispersion, 780, 783

anomalous, 783 in optical fibers, 1107 in the atmosphere, 784 normal,783 relation, 783

dispersive medium, 783 displacement

current, 688 density, 688, 691 in a capacitor, 692

distributed parameters, 882 divergence

in Cartesian coordinates, 87 in cylindrical coordinates, 91 in spherical coordinates, 92 of a vector field, 85 of current density, 440 of the electric field intensity, 175 of the magnetic vector potential, 700

divergence theorem, 93, 94, 174,214, 252,438,751

domain wall, 549 dominant mode, 1085 dot convention, 655 dot product (see scalar product), 14 drift velocity, 425, 427

E-plane pattern, 1136 eddy current, 661

levitation, 674 testing of materials, 670

Edison, Thomas Alva, 456 effective aperture, 1182, 1183

of small loop antenna, 1186 effective area, 1182, 1183 effective length, 1181

vector, 1181 electric

charge, 121 field intensity, 132 field lines, 13 5 gun,263 motor, 607

electric dipole, 139,212, 531, 1122, 1123

electric dipole antennas table of properties, 1147

electric dipole moment, 141, 212, 1123 electric energy

stored, 750, 752, 1128 in cavity resonator, 1105

electric energy density stored, time averaged, 763

electric field at surface of conductor, 209 lines of, 444 normal component, 208 of point charges, 134 tangential component, 208 time dependent, 417,630

electric field intensity

calculation from potential, 203 far field of antenna, 1128 near field of antenna, 1128 near field of loop antenna, 1148 obtained from potential, 295 of antenna, 1125 of arbitrarily long antenna, 1153 of half wavelength antenna, 1156 ofloop antenna, 1144, 1146 time dependent, 439

electric flux density, 155, 178 electric length, 894 electric potential, 173, 190,699

due to distributed charges, 195 due to point charges, 191

electric scalar potential, 284, 698 electric susceptibility, 217 electrical length, 917 electrical welding, 456 electrocardiography (ECG), 458 electrokinetic momentum, 499 electromagnetic

field,687 power density, 750 propulsion of ships and submarines,

672 radiation, 1119 radiation safety, 799 shielding, 804 spectrum, 1120 wave equation, 729 waves

propagation in water, 741 electromagnetic spectrum, 748 electrometer, 166 electromotive force, 447,586,631,673

induced in loop antenna, 1186 motional, 634, 636 transformer action, 643

electron gun, 159 electrostatic

bed,263 clamping, 263 cleaner, 159 deflection, 159, 160 discharge (ESD), 222, 260 energy, 265 energy density, 253 energy stored in a capacitor, 574 extrusion, 160

INDEX 1219

field, 121, 174,284,285,629 force, 130,474 headphones, 258 lens, 161 levitation, 169 pistol,263 precipitator, 159, 338 sandpaper deposition, 261 scrubber, 159 separator, 159 shield,260 speaker, 258 spinning, 160 spraying, 158

electrostatic field of antenna, 1128

emf (see electromotive force), 447 energy

density, 69, 583 dissipated,750 in a cloud, 250 in a gap, 601 in a system of point charges, 249 in the electrostatic field, 245, 252 relations in cavity resonator, 1105 stored,69

in a capacitor, 248 in an inductor, 573 in the magnetic field, 581

stored in an inductor, 676 energy function, 395, 396

for Laplace's equation, 395 for Poisson's equation, 395

equipotential surface, 294 error criterion, 367 evanescent wave, 1048, 1093 explicit method of solution, 361, 365

Far field, 1127, 1128 of Hertzian dipole, 1127 of magnetic dipole, 1148

farad, unit of capacitance, 127 Faraday's cage, 260 Faraday's disk, 667 Faraday's law, 629, 630, 747

in differential form, 632 in integral form, 633

Faraday, Michael, 232, 631 ferrimagnetic materials, 553 ferromagnetic materials, 548

1220 INDEX

field conservative, 60, 106, 108,284 curl-free, 108 divergence free, 109 electrostatic, 245 irrotational, 108 nonconservative, 60, 64, 106 nonrestoring, 60 of force, 473 restoring, 60 time harmonic, 711

field point, 144,476 field radiation pattern, 1138

normalized electric dipole antenna, 1147 magnetic dipole antenna, 1147 of quarter wavelength monopole

antenna, 1164 relative

of arbitrarily long antenna, 1155 of full wavelength antenna, 1158 of half wavelength antenna, 1157

finite difference approximation, 362 backward difference formula, 357 central difference formula, 357 first order derivatives, 357 for, 3d Laplace's equation, 368 forward difference formula, 357 second order derivatives, 357

finite difference grid, 362 finite difference method, 356 finite element, 391

mesh,396 one dimensional, 413 quadrilateral, 414 triangular, 391

finite element method, 389 implementation, 394

flux density, 217 due to square loop, 506 leakage, 656 lines, 494 linkage,561 through a loop, 495

force, 126 between overhead transmission lines,

598 centrifugal, 131 distribution, 59

field, 59, 132 in a magnetic circuit, 604 in a uniform magnetic field, 602 in the electrostatic field, 255 in the geomagnetic field, 597 in the magnetic field, 592 lines, 135

Foucault currents, 662 Foucault,]ean Bernard Leon, 662 Fourier series, 332, 335 Franklin, Benjamin, 122 Fraunhoffer zone, 1127 frequency

measurement in cavity resonator, 1109

modulation (FM), 781 Fresnel zone, 1127 Friis transmission formula, 1185, 1190

Gain of antenna, 1142 of half wavelength antenna, 1157

Galvani, Luigi Alosio, 191 Gauss'law, 173, 178, 179, 187, 198,886 Gauss,]ohann Carl Friedrich, 178 Gauss-Seidel,364 Gaussian elimination, 364,401 Gaussian surface, 180, 198, 886 generator

ac,641,643,648 acyclic (homopolar), 667 dc,634 linear, 636, 638

geomagnetic field, 511, 609 Gibbs,]osiah Willard, 1,90 Gilbert, William, 527 gradient, 73, 76, 203

in Cartesian coordinates, 76 in cylindrical coordinates, 79 in spherical coordinates, 83, 205 of electric potential, 295

Green's function, 376 ground electrode, 466 group velocity, 780, 782 guided waves, 835, 1055 guiding direction, 1057

H-plane pattern, 1137 head to tail method, 9 Heaviside, Oliver, 1,689 Helmholtz equation, 736, 765

for electric flux density, 798 for magnetic flux density, 798

Helmholtz theorem, 108, 110,443,496, 499,698

Helmholtz, Hermann Ludwig Ferdinand von, 109

hematite, 511, 526 Henry,Joseph, 306, 545 henry, unit of inductance, 561 Hertz electric potential, 798 Hertz, Heinrich Rudolph, 690, 731 Hertzian dipole, 1123, 1129, 1150,

1181,1183 high loss materials, 769 homogeneity, 219, 696 hysteresis cwve (see magnetization

cwve),550

Impedance normalized, 954 on Smith chart, 957

impedance matching, 654, 953 conjugate, 911 double stub, 973, 976 in transmission lines, 911 network, 967 parallel stub, 969 quarter wavelength transformer, 984 series stub, 973 single stub, 969 stub,969 using the Smith chart, 967

implicit method of solution, 361 index of refraction, 841 induced charge, 208 inductance, 559

between wire and loop (mutual), 570 between wire and toroidal coil

(mutual), 568 external, 563, 567 external per unit length, 885 in multiple coils (mutual), 570 in multiple coils (self), 570 internal, 563, 567 internal per unit length, 885 mutual,560 oflong solenoid (self), 564 of toroidal coil (self), 563 per unit length, 564, 885, 998 per unit length, of coaxial cable, 565 self,560

INDEX 1221

series, 882 series connection, 576, 578 total,585

induction melting, 664 surface hardening, 664

inductive loading, 1025 rone, 1127

inductor, 561 initial condition on line, 1029 initially charged line, 1029 input line impedance, 913 insulator, 426

perfect, 764 interface conditions, 223, 295

between dielectric and conductor, 228 between two dielectrics, 224 for current density, 449 for electric field intensity, 449 for normal components of B, 555 for normal components ofH, 554 for static electric field, 709 for static magnetic field, 709 for static magnetic fields, 553 for tangential components of B, 555 for tangential components ofH, 554 for the electric field, 704 for the electric flux density, 225 for the electromagnetic field, 703, 708 for the magnetic field, 706 in poor conductors, 453

intrinsic impedance, 743, 747, 768, 810, 1046,1133

in free space, 743 in good conductor, 824

ion beam, 161

etching, 161 milling, 161

implantation, 161 lithography, 161 propulsion, 264 thruster, 264, 468

isotropic antenna, 794, 1140 isotropy, 219, 696 iteration procedure, 366

Joule's law, 433, 459, 758 in differential (point) form, 434 in integral form, 434

1222 INDEX

Kelvin, Lord, 293 Kirchhoff's

current law, 437, 439 for magnetic circuits, 591

laws, 891 for magnetic circuits, 591

voltage law, 448, 654 for magnetic circuits, 591

L'Hospital rule, 1176 Langmuir-Child law, 424 Laplace operator, 112 Laplace's equation, 328

direct integration, 286 in Cartesian coordinates, 328 in cylindrical coordinates, 336 in three dimensions, 330 in three dimensions, finite difference

approxnnation, 371 solution to, 357 two dimensional, finite difference

approxnnation,360 Laplace, Pierre Simon, 285 Laplacian

in Cartesian coordinates, 112 in cylindrical coordinates, 112 in spherical coordinates, 112 three dimensional, finite difference

approxnnation,359 two dimensional, finite difference

approxnnation,359 vector, 112

laser printer, 163 law of sines, 54 Lentz's law, 633, 651, 653, 666 Lentz, Emil Khristianovich, 633 lightning, 220,262

cloud to cloud, 262 cloud to earth, 262 earth to cloud, 262 on Jupiter, 263 protection, 489

line impedance, 912, 913, 926 line integral, 58 linear array

antenna, 1166 11 element, 1175

linearity, 219, 696 lodestone, 525 Lodge, Sir Oliver Joseph, 1042 London relation, 794

longitudinal component in cavity resonator, 1103 in parallel plate waveguide, 1063 TE in rectangular waveguide, 1084 TM in rectangular waveguide, 1076 TM, in parallel plate waveguide, 1067

Lorentz force, 474, 493, 714 Lorentz force equation, 592 Lorentz gauge (condition), 701 Lorentz, HendrikAntoon, 474 Lorentz-Coulomd force, 475 loss tangent, 765 losses

in conductors, 662 in rectangular waveguide, 1091

low loss condition, 773 dielectrics, 771 materials, 769

Magellan spacecraft, 1192 magnetic

bearings, 553 brake,666 circuit, 586, 649 damper, 667 disks,610 energy, 583 energy stored in inductor, 574 fluids, 553 flux, 586 flux linkage, 634 force, 474, 512, 595,602,676 force density, 474 needle, 609 particle testing, 610 path, 586 pole, 497 properties of materials, 534 prospecting, 511 recording, 609,610, 614 reluctance, 587 retarder, 667 seals, 553 shielding, 613 shielding of MRI equipment, 611 surveying, 511 tapes, 610

magnetic dipole, 528, 531, 536,1143 magnetic dipole antenna

table of properties, 1147

magnetic dipole moment, 531 in atom, 532 of square loop, 532

magnetic domain, 548 magnetic energy

stored, 583, 750, 752, 1128 in cavity resonator, 1105

magnetic energy density stored, time averaged, 763

magnetic field due to overhead transmission line, 488 lines of, 474 of distributed currents, 483 of ground plane, 485 static, 471 time dependent, 630

magnetic field intensity, 472, 475 due to infinite sheet, 489 due to long, thick: line, 488 due to thin wire, 488 far field of antenna, 1128 in long solenoid, 492 in toroidal coil, 490 near field of antenna, 1127 near field of loop antenna, 1148 of antenna, 1125 of arbitrarily long antenna, 1153 of half wavelength antenna, 1156 ofloop antenna, 1144, 1146

magnetic flux, 493, 494,555 magnetic flux density, 472, 493

of electromagnet, 541 oflong magnet, 540

magnetic materials, 525 hard,552 hard, table of, 552 soft, 551 soft, table of, 552

magnetic moment, 1143 magnetic monopole, 497 magnetic path, 658 magnetic permeability, 475,545

of free space, 475 relative, 545

magnetic resonance imaging (MRI), 511,608

magnetic scalar potential, 509, 698, 699, 702

of a magnet, 509 magnetic susceptibility, 544

INDEX 1223

magnetic vector potential, 499,581, 699,1124

ofloop antenna, 1144 retarded, 1124

magnetite, 511, 526,611 magnetization, 534,613

surface current density, 539 volume current density, 538

magnetization curve, 549, 610 initial, 550

magneto-optic recording, 609 magnetohydrodynamic (MHD)

channel,672 generator, 672 pump, 672

magnetometer, 511 magnetomotive force, 586, 590 magnetostatic

energy in terms of fields, 580 field,629

magnetostrictive material, 795 material constant, 795

magnetotactic bacteria, 611 magnetron, 754,864 main lobe, 1175, 1177 maximum directivity, 1141

of electric dipole antenna, 1147 of half wavelength antenna, 1157 of half wavelength dipole, 1184 of Hertzian dipole antenna, 1142,

1184 of magnetic dipole antenna, 1147 of quarter wavelength monopole

antenna, 1164 maximum effective aperture

of half wavelength dipole, 1184 ofHertzian dipole antenna, 1184 of loop antenna; 1186

maximum effective area, 1184 Maxwell's equations, 687, 725

differential form, 690 integral form, 694 time dependent, source free form, 715 time harmonic form, 711, 714 time harmonic, source free form, 716

Maxwell, James Clerk, 689 Maxwell-Lorentz equations, 696 method of images, 286, 292

analogy, 293

1224 INDEX

method (cont.) calculation of capacitance, 325 charged line over a plane, 305 in curved geometries, 316 in planar geometries, 293 multiple planes and charges, 312 overhead transmission line, 308 point and line charges, 294 point charge inside conducting hollow

sphere, 321 point charge outside conducting

sphere, 319 underground cable, 324

method of moments, 373, 1193 hand computation, 383

methods of solution analytic, 353 for second order PDEs, 286

microstrip line, 943, 946 microwave

absorbing materials, 867 antennas for therapy, 1194 applicators, 1194 cooking, 829,863,867 freeze drying, 863 hyperthermia treatment, 1194 integrated circuits (MIC), 1111 mapping, 865 monolythic integrated circuits

(MMIC), 1111 oven, 754 power density exposure, 799 radiation of tumors, 1194 reflector, 864 remote sensing, 865 testing of materials, 1149

Millikan's experiment, 169 mmf (see magnetomotive force), 631 mode

dominant, 1072 index, 1093 of propagation, 1043, 1061 separation (in waveguide), 1086 TEl 0, 1061 TEmO, 1061 TM, in rectangular waveguide, 1074

monopole antenna, 1163 quarter wavelength, 1164

motional emf, 643 motor

acyclic (homopolar), 668 efect, 636 linear, 639

Nabla, 76 near field, 1127

of Hertzian dipole, 1127 of magnetic dipole, 1148

Neumann boundary condition, 400 Newton's third law, 128, 597 nondestructive testing, 610

of materials (electrostatic), 261 numerical methods, 354

Oersted, Hans Christian, 472 Ohm's law, 418, 426, 428, 887 Ohm, Georg Simon, 428 optical

fiber, 856, 1043, 1107 connector, 814 magnetometer, 794 multimode, graded index, 1107 single mode, 1107

resonator, 1108 waveguide, 856, 1093, 1107, 1115

Parallel polarization, 831 oblique incidence on conducting

interface, 837 oblique incidence on dielectric

interface, 839, 844 parallelogram rule, 10 paramagnetic materials, 547 permanent magnet, 473,525,551,613

levitation, 676 permeability,493 permittivity, 126,218

of free space, 127 relative, 21 7

perpendicular polarization, 831 oblique incidence on conducting

interface, 832 oblique incidence on dielectric

interface, 841 phase angle

of the intrinsic impedance, 77 5 phase constant, 767, 814, 824, 883, 892,

896 guide, 1093

in parallel plate waveguide, 1062 in guide direction, 1059

in transverse direction, 1059 lossless, TE, 1053 lossless, TEM, 1053 lossless, TM, 1053 low loss, TE, 1053 low loss, TEM, 1053 low loss, TM, 1053 TE propagation, 1049 TM, in parallel plate waveguide, 1067

phase velocity, 732, 739, 768, 894, 896, 998, 1046

guide in parallel plate waveguide, 1062 TM in rectangular waveguide, 1078

in free space, 740 in guide direction, 1057 in low loss dielectrics, 772 in space, 1124 in transverse direction, 1057 TE propagation, 1053 TEM propagation, 1053 TM propagation, 1053

phased array, 1167 phasors, 711

exponentialform, 712 polarform,712 rectangular form, 712

photoconductor, 162 photon energy, 1120 Pioneer, 10,688,793,1193 Planck constant, 1120 plane of incidence, 831 plane wave, 737, 1046

in conductors, 774 in low loss dielectrics, 771 nonuniform nonuniform, 835 uniform, 737

Pliny (Caius Plinius Secundus), 526 Poisson's equation

direct integration, 286 for electrostatic field, 284 in Cartesian coordinates, 285 in curved geometries, 316 in cylindrical coordinates, 285 in spherical coordinates, 285 in three dimensions, 368 in two dimensions, 370 solution to, 357, 367 two dimensional, finite difference

approximation, 360

INDEX 1225

Poisson, Simeon Denis, 284 polarization, 212

circular, 786, 787 elliptical, 786, 787 linear, 786 losses, 765 of charges, 764 of plane waves, 786, 796 surface charge density, 215 vector, 213, 217

in coaxial cable, 218 volume charge density, 214, 215

polarized materials, 212 polarizing angle, 851 position vector, 47 postulates, 284

for current density, 444 for electric and magnetic fields, 629 for time dependent electromagnetic

field,633 integral form, 176 of the electrostatic field, 176 of the static magnetic field, 496, 498

potential, 191,203,284 absolute, 191 difference, 191 distribution in coaxial line, 291 energy, 190 of a dipole, 205

potential drop method of nondestructive testing, 454

potential energy, 61, 245, 395,602 potential function, 499

time dependent, 697 power, 433

dissipated in a fuse, 434 dissipated in lightning strike, 436 dissipation, 433 flow, 752 flow into a conductor, 825 lines, 260 loss, 1092 reactive, 763 real,762 received, 1185 scattered, 1189 time averaged, 755, 761, 762,1057

power density, 752 dissipated,434 electric, 751

1226 INDEX

power (cont.) in a fuse, 434 in semiconductors, 435 in waveguide, 1085 instantaneous, 848 magnetic, 751 of half wavelength antenna, 1156 time averaged, 755, 761

in cavity resonator, 1105 in far field of magnetic dipole

antenna, 1148 in parallel plate waveguide, 1063 in sinusoidal fields, 759 in the far field of antenna, 1133 of receiving antenna, 1182

power radiation pattern, 1137, 1138 normalized, 113 7

electric dipole antenna, 1147 magnetic dipole antenna, 1147 of quarter wavelength monopole

antenna, 1164 rectangular, 1138 relative

of arbitrarily long antenna, 1155 of full wavelength antenna, 1158 of half wavelength antenna, 1157

Poynting theorem, 750, 753, 755,1046, 1092

complex, 762 Poynting vector, 752, 755, 810, 906

complex, 760 for transmitted wave, 825 instantaneous, 848 time averaged, 755, 759, 782, 830,

1056 of arbitrarily long antenna, 1154 TM waves in parallel plate

waveguide, 1068 Poynting, John Henry, 752 Priesdey, Joseph, 126 propagation

in lossy dielectrics, 796 in water, 796 of narrow pulses on distortionless

transmission lines, 1002 of narrow pulses on lossless

transmission lines, 998 propagation constant, 766, 810, 892,

1044 in lossy medium, 1046

lossless, TE, 1053 lossless, TEM, 1053 lossless, TM, 1053 low loss, TE, 1053 low loss, TEM, 1053 low loss, TM, 1053 TE propagation, 1047 TM in rectangular waveguide, 1077 TM, in parallel plate waveguide, 1067

Quadrupole, 167 quality factor, 1106 quasi-static field equations, 715

Radar, 749, 801, 1188 a-static, 1189 absorbing materials, 866 automobile, 1194 autonomous landing, 1194 bi-static, 1189 collision avoidance, 1194 Doppler, 1191, 1194 ground penetrating, 1191 look down, 1191 sensing of the environment, 817 space-born, 1192 synthetic aperture (SAR), 1191

radar cross section, 1189 reduced, 1191

radar equation, 1190 radiated power, 1133

electric dipole antenna, 1147 magnetic dipole antenna, 1147 of arbitrarily long antenna, 1154 of half wavelength antenna, 1156 of quarter wavelength monopole

antenna, 1164 radiation, 1120

exposure, 870 ionizing, 1120 non ionizing, 1120 quantum of, 1120 safety, 1120

radiation efficiency, 1142 electric dipole antenna, 1147 magnetic dipole antenna, 1147

radiation field, 1128 radiation intensity, 1140

electric dipole antenna, 1147 magnetic dipole antenna, 1147

radiation pattern

absolute E-field, 1136 E-field of magnetic dipole, 1149 H -field of magnetic dipole, 1149 nonnalized, in the E-plane, 1136 of element antenna, 1170 of receiving antenna, 1181 three dimensional, 1138

radiation resistance, 1134 electric dipole antenna, 1147 electric dipole antenna, in air, 1147 magnetic dipole antenna, 1147 magnetic dipole antenna, in air, 1147 of arbitrarily long antenna, 1154 of half wavelength antenna, 1156 of half wavelength antenna, free space,

1156 of quarter wavelength monopole

antenna, 1164 of receiving antenna, 1182

radio tagging of animals, 1193 telescope, 1167, 1194

radome, 819, 863 rail launcher (rail gun), 669 Rayleigh, Lord, 1042 reciprocity, 1180 reciprocity theorem, 1180 rectangular waveguide, 1087, 1115 reflection

angle, 808 at air-conductor interface, nonnal

incidence, 821 at air-Iossless dielectric interface,

nonnal incidence, 819 at air-lossy dielectric interface, nonnal

incidence, 816 at an interface, oblique incidence, 831 at general dielectric interface

nonnal incidence, 809 by the ionosphere, 835 for conducting slab, nonnal incidence,

860 for lossless dielectric slab backed by a

conductor, nonnal incidence, 860 for lossless dielectric slab, nonnal

incidence, 859 for lossy dielectric slab, nonnal

incidence, 857 in layered materials, nonnal incidence,

856

INDEX 1227

of plane waves, 807 reflection coefficient, 810, 811, 999

for perpendicular polarization, 842 generalized, 912, 915 load, 910, 911, 915 of dielectric slab, 859 parallel polarization, 845, 850 perpendicular polarization, 850 phase angle of, 911, 958 plane, 953 polar fonn, 954 rectangularfonn, 954

reflection diagram, 1011 reflections from discontinuities, 1018 reflectometty, 826

frequency domain, 936 time domain, 1006, 1033

refraction of electric field intensity, 227 of magnetic field, 557 optical index of, 228

relative radiation pattern, 1136 relaxation, 441 relaxation time, 442 reluctance, 658 re1uctivity, 587 remnant magnetic flux density, 550,610 remote sensing of the environment,

1192 resistance, 425, 428, 459

ac, in conductors, 778 contact, 466 ground,466 in parallel, 432 in series, 432 of fuse, 430 per unit length, 882, 884 series, 882 surface, 883, 1106

resistive position sensor, 432 resistivity,430 resistor, 454 resonant

frequencies, 1102, 1103 modes, 1102

resonant circuit parallel, 932 series, 933

right hand rule for magnetic field, 473 Rowland, Henry A, 516

1228 INDEX

Saturation point, 550 Savart, Felix, 476 scalar

component, 4 electric potential, 191 quantity,2 static, 2

scalar field, 26 time dependent, 26

scalar potential, 360, 698 scalar product, 13

commutative property, 14 distributive property, 14

scalar triple product, 23 scatterer, 1189 scattering

cross section, 1189 of waves, 865 tropospheric, 865

second order operators, 111 separation of variables, 286,328,1074

in Cartesian coordinates, 328 in cylindrical coordinates, 336

shape function, 392 derivatives of, 392

shielded enclosure, 77 5 shielding

of transmission, 795 structures, 796

SIunits, 7 sidelobe, 1175, 1177

index, 1177 siemens, unit of conductance, 425 Siemens, Werner von, 426 simple material, 220 skin depth, 774, 802, 824, 829, 1095 slotline, 1106 Smith chart, 953

admittance point, 958 as an admittance chart, 964 as impedance chart, 973 open circuit point, 957 short circuit point, 957

Snell's law, 839, 853 for reflected wave, 841 for refracted wave, 841

solar wind, 419, 462 source point, 144,476 speed of light, 740 spot welding, 456

standing wave, 814, 821, 967,1057 complete, 823, 918 incomplete, 823 nodes of, 1059

standing wave pattern, 872 standing wave ratio (SWR), 823, 918,

960 steady current, 417 steady state solution

current on transmission lines, 1010 voltage on transmission lines, 1010

stealth aircraft, 866 steradian, 1140 Stokes' theorem, 104,504,581,632 Stokes, Sir George Gabriel, 104 stored energy, time averaged, 763, 782 strain gauge, 457 stream lines, 444 stripline, 1070 stud sensor, 289 superconducting

magnetic energy storage (SMES), 576 materials, 546 power line, 903 power transmission, 794

superparamagnetic materials, 553 surface integral, 64

closed,67 switching power supply, 240, 265 system of equations, 364

Taylor series expansi~n, 88, 99 TE modes, 1084

in cavity resonator, 1102 in rectangular waveguide, 1083

TE propagation condition for, 1048

TEwaves in cavity resonator, 1098 table of properties, 1053

telephone circuit, 456 TEM waves, 743

in parallel plate waveguide, 1072 table of properties, 1053

Tesla, Nicola, 494 tesla, unit of flux density, 475 Thales ofMiletus, 122,526 time constant, 441 TMmodes

in cavity resonator, 1099, 1103 TM propagation

in microstrip waveguide, 1068 in parallel plate waveguide, 1066

TMwaves in cavity resonator, 1098 table of properties, 1053

torque, 606 on a square coil, 607 on an electron, 608

total differential, 75 total reflection, 852, 866

internal, 11 07 transfonner, 610, 641, 652, 675, 968

ideal, 653 ratio, 654 real, finite penneability, 654 real, finite penneability and flux

leakage, 656 toroidal, 657

transfonner action emf, 632, 641 transients

on transmission lines, 997 finite length pulses, 1015 long pulses, 1007 ~threactiveloading, 1023

transmission at air-conductor interface, nonnal

incidence, 821 at air-lossless dielectric interface,

nonnal incidence, 819 at air-lossy dielectric interface, nonnal

incidence, 816 at an interface, oblique incidence, 831 at general dielectric interface

nonnal incidence, 809 for conducting slab, nonnal incidence,

860 for lossless dielectric slab, nonnal

incidence, 859 for lossy dielectric slab, nonnal

incidence, 857 in layered materials, nonnal incidence,

856 of plane waves, 807 of power into solar cells, 869

transmission angle, 808 transmission coefficient, 811, 999

for perpendicular polarization, 842, 844

of dielectric slab, 859 parallel polarization, 845

INDEX 1229

transmission line, 185,488,881 circuits, 953 coaxial, 888, 941 distortionless, 895, 900 equations, 890 finite, 909 general, 892 high voltage dc (HVDC), 1030 infinitely long, 895 long, 898 lossless, 895, 898

resistively loaded, 925 terminated, 916

lossy, 898 low loss, 898 low resistance, 902 matched, 922 method of oil recovery, 938 open, 923 parallel plate, 888, 907 parameters, 882

(table of), 888 power relations, 928 resonant, 932, 938 sensors, 952 shorted, 922 simulated, 939, 1033 two~re, 888

transparent materials, 820 transverse component

in cavity resonator, 1100, 1103 in parallel plate waveguide, 1063 TE in rectangular waveguide, 1084 TE propagation, 1047 TM in rectangular waveguide, 1076,

1078 TM, in parallel plate waveguide, 1067

transverse electric wave (TE), 1043 transverse electromagnetic propagation

(TEM),881 transverse electromagnetic wave, 905 transverse electromagnetic wave (TEM),

1043 transverse electromagnetic waves, 1046 transverse electromagnetic waves

(TEM), 743 transverse magnetic (TM) wave, 1067 transverse magnetic wave (TM), 1043 two-element array, 1167

Uniqueness theorem, 287, 292

1230 INDEX

unit vector, 3, 5 transformation of, 36

Vector addition, 8 algebra, 1 calculus, 1, 57 component, 3,4 field, 27, 59 identities, 113 Laplacian, 502 magnitude of, 3 null identities, 108 Poisson equation, 502 potential, 698 product, 17 quantity,2 scaling, 12 space dependent, 2 subtraction, 10 time dependent, 2 triple product, 22 velocity,7

vector field circulation of, 96 nonsolenoidal, 110 nonsolenoidal, irrotational, 110 nonsolenoidal, rotational, 110 rotational, 110 solenoidal, irrotational, 111 solenoidal, rotational, 111 static, 27

vector integral, 70 vector potential

current, 702 electric, 701, 702

vector product associative property, 18 commutative property, 17 cyclic property, 19 distributive property, 18

vector scaling associative property, 12 commutative property, 12 distributive property, 12

vector sum associative property, 11 commutative property, 11 distributive property, 11, 13 of electric fields, 134

vector transformation

Cartesian to cylindrical, 37 Cartesian to spherical, 42 cylindrical to Cartesian, 37 cylindrical to spherical, 43 spherical to Cartesian, 42 spherical to cylindrical, 43

velocity of energy transport, 740, 780, 782

virtual displacement, 256, 602 virtual work, 256, 601 volt, unit of potential, 191 Volta pile, 447 Volta, Alessandro Giuseppe Antonio

Anastasio, 191 voltage

drop, 448 induced in receiving antenna, 1183

voltage wave backward propagating, 1024 forward propagating, 893, 910, 999,

1024 voltaic cell, 455 volume integral, 70 Voyager 1, 263

Wall losses, 1091 watthourmeter, 668 wave

backward propagating, 738, 810, 856, 910,929,1009,1044,1100

backward propagating, in rectangular waveguide, 1075

eartquake,726 elastic, 72 6 equation, 892 forward propagating, 729, 738, 856,

909,929,1009,1044,1100 forward propagating, in rectangular

waveguide, 1075 left circularly polarized, 789 left elliptically polarized, 789 linearly polarized, 790 monochromatic, 780 nonmonochromatic, 780 nonpropagating,814 propagation in lossy dielectrics, 764 propagation in materials, 764 retarded,1124 right circularly polarized, 789 right elliptically polarized, 792 sound,727

transmitted, 810 wave equation, 1124

D'Alembert solution to, 728 electromagnetic, 731 for current, 892 for longitudinal component of H,

1047 for voltage, 892 homogeneous, 732 homogeneous, for A, 732 homogeneous,forH,734 homogeneous, for V, 733 lossy, source free, 765 nonhomogeneous, 732 nonhomogeneous, for A, 732 nonhomogeneous, for H, 734 nonhomogeneous, for V, 733 scalar, 728 source free, 735,905 source free, lossless, 736 time dependent, 73 5 time harmonic, 732, 734 time harmonic, source free, lossless

(for B), 736 time harmonic, source free, lossless

(for 11),736 time harmonic, source free, lossless,

(for V), 736 wave impedance, 732, 743, 768, 1046

in parallel plate waveguide, 1063

INDEX 1231

TE propagation, 1051, 1053 TEM propagation, 1053 TM in rectangular waveguide, 1078 TM propagation, 1052, 1053 TM, in parallel plate waveguide, 1068

waveguide, 1042 as high pass filter, 1096 below cutoff, 1096 cylindrical, 1097 elliptical, 1097 integrated circuit, 1068 parallel plate, 1055 rectangular, 1073

wavelength, 740, 768, 894, 896 guide

in parallel plate waveguide, 1061 TM in rectangular waveguide, 1078

in low loss dielectrics, 772 TE propagation, 1049, 1053 TEM propagation, 1053 TM propagation, 1053 TM, in parallel plate waveguide,

1067 wavemeter, 1109 wavenumber, 740

free space, 740, 747 Weiss, Pierre, 548 work, 190, 245

Xerography, 161

Gradient, divergence, curl, and the Laplacian in various coordinates:

Cartesian coordinates:

VU ..... au ..... au ..... au =xa; +y~ +za;'

aAx aAy aAz V ·A= a; + &y + 8;'

..... (aAz aAy ) ..... (aAx aAz ) ..... (aAy aAx ) VxA=x &y -8; +y az - ax- +Z a;- &y ,

V2U _ a2u a2u a2u - ax-2 + &y2 + az2

Spherical coordinates:

..... au ..... 1 au ..... 1 au Vu = R aR +Ozroo- +.RsinO &p'

1 a 2 1 a. 1 aAq, V·A= --(RAR)+-.--(AesmO)+-.---, R2 aR RsmO ao RsmO arp

V x A = i-~ _ (a(Aq, sin 0) _ aAe) + i! (~ aAR _ a(RAq,») RsmO ao aq, R smO arp aR

...... ! (a(RAe) _ aAR ) + R aR ao'

2 1 a ( 2 au) 1 a (. au) 1 a2 u V u = R2 aR R aR + R2 sinO ao sm0-oo + R2 sin2 0 arp2

The divergence theorem:

l(V .A)dv = fA. ds

Stokes's theorem:

i(V x A) • ds = i A • dl

Some useful vector identities:

v x (VU) = 0,

V .(V x A) = 0,

V2 A = V(V • A) - V x (V x A),

V(UQ) = U(VQ) + Q(VU),

V· (UA) = U(V .A) + (VU) .A, V • (A x B) = -A • (V x B) + (V x A) • B,

V x (UA) = U(V x A) + (VU) x A,

V· VU = V2u,

V2A =XV2A,r +yv2Ay + ZV2Az

Some physical constants:

Charge of the electron (q.) Rest mass of the electron (m.) Speed of light in vacuum (c) Permittivity of free space (eo) Penneability of free space (/-Lo) Plank's constant (h) Intrinsic impedance in free space (110)

-1.602129 X 10-19 C 9.103534 X 10-31 kg 2.997992 X 108 mls 8.854187 X 10-12 F/m 41r x 10-7 Him 6.62620 x 10-34 J . s 376.7304 Q