Appendix A SI Units and Gaussian Units A.1 Conversion of Amounts All factors of 3 (apart from exponents) should, for accurate work, be replaced by 2.99792456, arising from the numerical value of the velocity of light. [43] Physical quantity Symbol SI(MKSA) Gaussian Length l 1 meter (m) 102 centimeters (cm) Mass m 1 kilogram (kg) 103 grams (gm or g) Time t 1 second (sec or s) 1 second (sec or s) Frequency f 1 hertz (Hz) 1 hertz (Hz) Force F 1 newton (N) 105 dynes Work, Energy W; U 1 joule (J) 107 ergs Power P 1 watt (W) 107 ergs/s Charge q 1 coulomb (C) 3 £ 109 statcoulombs Charge density % 1 C/m3 3 £ 103 statcoul/cm3 Current I 1 ampere (A) 3 £ 109 statamperes Current density J 1 A/m2 3 £ 105 statamp/cm2 Potential ' 1 volt (V) 10¡2=3 statvolt Electric ¯eld E 1 V/m 10¡4=3 statvolt/cm Electric induction D 1 C/m2 12¼ £ 105 statvolt/cm Polarization P 1 C/m2 3 £ 105 moment/cm3 Magnetic flux © 1 weber (Wb) 108 maxwell (Mx) Magnetic induction B 1 tesla (T) 104 gauss (Gs) Magnetic ¯eld H 1 A/m 4¼ £ 10¡3 oersted (Oe) Magnetization M 1 A/m 10¡3 moment/cm3 Conductance G 1 siemens (S) 9 £ 1011 cm/s Conductivity σ 1 S/m 9 £ 109 1/s Resistance R 1 ohm (­) 10¡11=9 s/cm Capacitance C 1 farad (F) 9 £ 1011 cm Inductance L 1 henry (H) 109 cm 674 Appendix A A.2 Formulas in SI (MKSA) Units and Gaussian Units Name of formula SI (MKSA) Gaussian r £ E = ¡@ B r £ E = ¡1 @ B @ t c @ t Maxwell r £ H = @ D + J r £ H = 1 @ D + 4¼ J @ t c @ t c equations r ¢ D = % r ¢ D = 4¼% r ¢ B = 0 r ¢ B = 0 1 Lorentz force F = q(E + v £ B) F = q(E + c v £ B) Constitutional D = ²0E + P = ²ED = E + 4¼P = ²E equations B = ¹0(H + M) = ¹HB = H + 4¼M = ¹H J = γE J = γE Constitutional ² = ²0(1 + Âe) = ²0²r ² = 1 + 4¼Âe = ²r parameters ¹ = ¹0(1 + Âm) = ¹0¹r ¹ = 1 + 4¼Âm = ¹r n £ (E2 ¡ E1) = 0 n £ (E2 ¡ E1) = 0 4¼ Boundary n £ (H2 ¡ H1) = J s n £ (H2 ¡ H1) = c J s equations n ¢ (D2 ¡ D1) = %s n ¢ (D2 ¡ D1) = 4¼%s n ¢ (B2 ¡ B1) = 0 n ¢ (B2 ¡ B1) = 0 Z Z 1 % 0 1 % 0 Coulomb's law E = 4¼² 2 r^dV E = ² 2 r^dV V r V r Z Z 1 % 0 1 % 0 ' = 4¼² r dV ' = ² r dV V V Z Z ¹ J £ r^ 0 ¹ J £ r^ 0 Biot-Savart B = 4¼ 2 dV B = c 2 dV V r V r law Z Z ¹ J 0 ¹ J 0 A = 4¼ r dV A = c r dV V V 2 % 2 % Poison r ' = ¡ ² r ' = ¡4¼ ² equations 2 2 4¼ r A = ¡¹J r A = ¡ c ¹J SI Units and Gaussian Units 675 Name of formula SI (MKSA) Gaussian 2 ²¹ 2 Wave r2E ¡ ²¹@ E = 0 r2E ¡ @ E = 0 @ t2 c2 @ t2 equations 2 ²¹ 2 r2H ¡ ²¹@ H = 0 r2H ¡ @ H = 0 @ t2 c2 @ t2 Dynamic B = r £ A B = r £ A potentials E = ¡r' ¡ @ A E = ¡r' ¡ 1 @ A @ t c @ t @' ²¹ @' Lorentz gauge r ¢ A + ²¹ = 0 r ¢ A + = 0 @ t c @ t @2' % ²¹ @2' % D'Alembert r2' ¡ ²¹ = ¡ r2' ¡ = ¡4¼ @ t2 ² c2 @ t2 ² equations 2 ²¹ 2 r2A¡²¹@ A =¡¹J r2A¡ @ A =¡4¼ ¹J @ t2 c2 @ t2 c Z Z 1 %(t ¡ r=c) 0 1 %(t ¡ r=c) 0 Retarding '= 4¼² r dV '= ² r dV V V potentials Z Z ¹ J(t ¡ r=c) 0 ¹ J(t ¡ r=c) 0 A= 4¼ r dV A= c r dV V V 1 1 Energy density w = 2(E ¢ D + H ¢ B) w = 8¼ (E ¢ D + H ¢ B) c Poynting vector P = E £ H P = 4¼ E £ H 676 Appendix A A.3 Pre¯xes and Symbols for Multiples Multiple Pre¯x Symbol 10¡18 atto a 10¡15 femto f 10¡12 pico p 10¡9 nano n 10¡6 micro ¹ 10¡3 milli m 10¡2 centi c 10¡1 deci d 10 deka da 102 hecto h 103 kilo k 106 mega M 109 giga G 1012 tera T 1015 peta P 1018 exa E Appendix B Vector Analysis B.1 Vector Di®erential Operations B.1.1 General Orthogonal Coordinates r ³ @ x ´2 ³ @ y ´2 ³ @ z ´2 u1; u2; u3; h1; h2; h3; hi = + + ; i = 1; 2; 3 @ ui @ ui @ ui A = u^1A1 + u^2A2 + u^3A3 X3 1 @ ' 1 @ ' 1 @ ' 1 @ ' r' = u^ = u^ + u^ + u^ (B.1) i h @ u 1 h @ u 2 h @ u 3 h @ u i=1 i i 1 1 2 2 3 3 1 X3 @ r ¢ A = (hjhkAi) h1h2h3 @ ui · i=1 ¸ 1 @ @ @ = (h2h3A1) + (h3h1A2) + (h1h2A3) (B.2) h1h2h3 @ u1 @ u2 @ u3 · ¸ X3 1 @ @ r £ A = u^i (hkAk) ¡ (hjAj) hjhk @ uj @ uk i=1 · ¸ 1 @ @ = u^ (h A ) ¡ (h A ) 1 h h @ u 3 3 @ u 2 2 2 3 · 2 3 ¸ 1 @ @ + u^ (h A ) ¡ (h A ) 2 h h @ u 1 1 @ u 3 3 3 1 · 3 1 ¸ 1 @ @ + u^3 (h2A2) ¡ (h1A1) (B.3) h1h2 @ u1 @ u2 678 Appendix B 1 X3 @ ³h h @' ´ r2' = j k h1h2h3 @ ui hi @ ui · i=1 ¸ 1 @ ³h h @' ´ @ ³h h @' ´ @ ³h h @' ´ = 2 3 + 3 1 + 1 2 (B.4) h1h2h3 @ u1 h1 @ u1 @ u2 h2 @ u2 @ u3 h3 @ u3 r2A = r(r ¢ A) ¡ r £ r £ A · ³ ´¸ 1 @F0 1 @F3 @F2 = u^1 ¡ ¡ h1 @ u1 h2h3 @ u2 @ u3 · ³ ´¸ 1 @F0 1 @F1 @F3 + u^2 ¡ ¡ h2 @ u2 h3h1 @ u3 @ u1 · ³ ´¸ 1 @F0 1 @F2 @F1 + u^3 ¡ ¡ (B.5) h3 @ u3 h1h2 @ u1 @ u3 where F0 = r ¢ A · ¸ h1 @ @ F1 = h1(r £ A)1 = (h3A3) ¡ (h2A2) h2h3 @ u2 @ u3 · ¸ h2 @ @ F2 = h2(r £ A)2 = (h1A1) ¡ (h3A3) h3h1 @ u3 @ u1 · ¸ h3 @ @ F3 = h3(r £ A)3 = (h2A2) ¡ (h1A1) h1h2 @ u1 @ u2 B.1.2 General Cylindrical Coordinates @ h @ h u ; u ; z; h = 1; 1 = 0; 2 = 0 1 2 3 @ z @ z 1 @' 1 @' @' r' = u^1 + u^2 + z^ (B.6) h1 @ u1 h2 @ u2 @ z · ¸ 1 @ @ @Az r ¢ A = (h2A1) + (h1A2) + (B.7) h1h2 @ u1 @ u2 @ z · ¸ 1 @A @ r £ A = u^ z ¡ (h A ) 1 h @ u @ z 2 2 2 · 2 ¸ 1 @ @A + u^ (h A ) ¡ z 2 h @ z 1 1 @ u 1 · 1 ¸ 1 @ @ + u^3 (h2A2) ¡ (h1A1) (B.8) h1h2 @ u1 @ u2 · ³ ´ ³ ´¸ 2 2 1 @ h2 @' @ h1 @' @ ' r ' = + + 2 (B.9) h1h2 @ u1 h1 @ u1 @ u2 h2 @ u2 @ z Vector Analysis 679 2 2 2 r A = r AT + z^r Az (B.10) where Az is the longitudinal component and AT is the transverse 2- dimensional vector of A A = AT + z^Az; AT = u^1A1 + u^2A2 · ³ ´ ³ ´¸ 2 2 1 @ h2 @Az @ h1 @Az @ Az r Az = + + 2 h1h2 @ u1 h1 @ u1 @ u2 h2 @ u2 @ z µ ¶ 1 @F 1 @F @2A 2 ^ 0 z 1 r AT = u1 ¡ + 2 h1 @ u1 h2 @ u2 @ z µ ¶ 1 @F 1 @F @2A ^ 0 z 2 + u2 + + 2 (B.11) h2 @ u2 h1 @ u1 @ z where · ¸ 1 @ @ F0 = r ¢ AT = (h2A1) + (h1A2) h1h2 @ u1 @ u2 · ¸ 1 @ @ Fz = jr £ AT j = (h2A2) ¡ (h1A1) h1h2 @ u1 @ u2 B.1.3 Rectangular Coordinates x; y; z; h1 = 1; h2 = 1; h3 = 1 @' @' @' r' = x^ + y^ + z^ (B.12) @ x @ y @ z @A @A @A r ¢ A = x + y + z (B.13) @ x @ y @ z µ ¶ µ ¶ µ ¶ @A @A @A @A @A @A r£A = x^ z ¡ y +y^ x ¡ z +z^ y ¡ x (B.14) @ y @ z @ z @ x @ x @ y @2' @2' @2' r2' = + + (B.15) @ x2 @ y2 @ z2 2 2 2 2 r A = x^r Ax + y^r Ay + z^r Az (B.16) B.1.4 Circular Cylindrical Coordinates ½; Á; z; h1 = 1; h2 = r; h3 = 1 @' 1 @' @' r' = ½^ + Á^ + z^ (B.17) @ ½ ½ @Á @ z 1 @ 1 @A @A r ¢ A = (½A ) + Á + z (B.18) ½ @ ½ ½ ½ @Á @ z 680 Appendix B · ¸ · ¸ · ¸ 1 @A @A @A @A 1 @ @A r £ A=½^ z ¡ Á + Á^ ½ ¡ z + z^ (½A ) ¡ ½ (B.19) ½ @Á @ z @ z @ ½ ½ @ ½ Á @Á µ ¶ 1 @ @' 1 @2' @2' r2' = ½ + + (B.20) ½ @ ½ @ ½ ½2 @Á2 @ z2 µ ¶ µ ¶ 2 @A A 2 @A A r2A=½^ r2A ¡ Á ¡ ½ +Á^ r2A + ½ ¡ Á +z^r2A (B.21) ½ ½2 @Á ½2 Á ½2 @Á ½2 z B.1.5 Spherical Coordinates r; θ; Á; h1 = 1; h2 = r; h3 = r sin θ @' 1 @' 1 @' r' = r^ + θ^ + Á^ (B.22) @ r r @ θ r sin θ @Á 1 @ 1 @ 1 @A r ¢ A = (r2A ) + (sin θA ) + Á (B.23) r2 @ r r r sin θ @ θ θ r sin θ @Á · ¸ 1 @ @A r £ A = r^ (sin θA ) ¡ θ r sin θ @ θ Á @Á · ¸ · ¸ 1 1 @A @ 1 @ @A + θ^ r ¡ (rA ) + Á^ (rA ) ¡ r (B.24) r sin θ @Á @ r Á r @ r θ @ θ µ ¶ 1 @ @' 1 @ ³ @'´ 1 @2' r2' = r2 + sin θ + (B.25) r2 @ r @ r r2 sin θ @ θ @ θ r2 sin2 θ @Á2 · ¸ 2 ³ @A @A ´ r2A = r^ r2A ¡ A + cot θA + csc θ Á + θ r r2 r θ @Á @ θ · ¸ 1 ³ @A @A ´ + θ^ r2A ¡ csc2 θA ¡2 r +2 cot θ csc θ Á θ r2 θ @ θ @Á · ¸ 1 ³ @A @A ´ + Á^ r2A ¡ csc2 θA ¡2 csc θ r ¡2 cot θ csc θ θ (B.26) Á r2 Á @Á @Á B.2 Vector Formulas B.2.1 Vector Algebraic Formulas A ¢ B = B ¢ A (B.27) A £ B = ¡B £ A (B.28) A ¢ (B £ C) = B ¢ (C £ A) = C ¢ (A £ B) (B.29) A £ (B £ C) = (A ¢ C)B ¡ (A ¢ B)C (B.30) (A £ B) ¢ (C £ D) = (A ¢ C)(B ¢ D) ¡ (A ¢ D)(B ¢ C) (B.31) (A £ B) £ (C £ D) = (A £ B ¢ D)C ¡ (A £ B ¢ C)D (B.32) Vector Analysis 681 B.2.2 Vector Di®erential Formulas r(' + Ã) = r' + rà (B.33) r('Ã) = 'rà + Ãr' (B.34) r(A ¢ B) = (A ¢ r)B + (B ¢ r)A + A £ (r £ B) + B £ (r £ A) (B.35) r ¢ (A + B) = r ¢ A + r ¢ B (B.36) r ¢ ('A) = A ¢ r' + 'r ¢ A (B.37) r ¢ (A £ B) = B ¢ (r £ A) ¡ A ¢ (r £ B) (B.38) r £ (A + B) = r £ A + r £ B (B.39) r £ ('A) = r' £ A + 'r £ A (B.40) r £ (A £ B) = A(r ¢ B) ¡ B(r ¢ A) + (B ¢ r)A ¡ (A ¢ r)B (B.41) r ¢ r' = r2' (B.42) r £ r' = 0 (B.43) r ¢ r £ A = 0 (B.44) r £ r £ A = r(r ¢ A) ¡ r2A (B.45) B.2.3 Vector Integral Formulas Volume V is bounded by closed surface S.