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Appendix 1 Electromagnetic Quantities

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Appendix 1 Electromagnetic Quantities Appendix 1 Electromagnetic quantities Quantity Symbol Units Dimensions Equations Electric current I A A SI unit Electric charge q C AT [7.1] Electric dipole moment p Cm ALT [ 4.7] Electric quadrupole moment Q Cm2 AL2T [7.19] Electric field E Vm-l A-1MLT-3 [2.10] Electric potential tP V A-I ML2 T-3 [2.24] Electrostatic energy U J ML2T""2 [2.46] Electric polarisation P C m""2 AL""2T ~ P/vol Polarisability a Fm2 A2M-1T4 [4.3] Electric susceptibility Xe none [ 4.1] Dielectric constant (relative [1.25], permittivity) E, none [4.32] Electric displacement D C m""2 AL2T [1-.9] Electric charge density p Cm-3 AC3T q/vol Linear charge density A Cm-l ACIT q/length Electric current density A m""2 AL""2 q/area/s Surface current density Am-l ACI [ 1.18] 120 Electromagnetic waves Quantity Symbol Units Dimensions Equations Electrical conductivity a S m-I A-2M-IC 3T3 [ 6.1] Magnetic field B T A-IMT-2 [2.10] Magnetic dipole moment m Am2 AL2 [7.14] Magnetisation M Am-I ACI ~ m/vol Magnetising field H Am-I ACI [1.10] Magnetic vector potential A Wb m-1 A- IMLT-2 [2.41] Magnetic susceptibility Xm none M/H Relative permeability f.l.r none [ 1.24] Magnetostatic energy U J ML2T-2 [ 2.51] Electromagnetic [3.32], energy density u J m-3 MCIT-2 [5.40] Poynting vector f/ Wm-2 MT-3 [3.31], [5.39] Wave impedance Z n A-2ML 2T-3 [3.38] Refractive index n none [4.26], [ 4.29] Wave number k m-I C l [3.10], [6.10] Absorption coefficient {j m-I C l [ 4.31] Skin depth 8 m L [ 6.13] Reflection coefficient R none [ 5.45] Transmission coefficient T none [5.46] Plasma frequency wp S-I T-I [6.35] Radiation pressure Pr Pa (J m-3) MCI T-2 [6.27] Radiated power W W ML2 T-3 [7.12] Appendix 1 121 Appendix 2 Gaussian units The Systeme International d'Unites (SI) used in this book is that adopted by the General Conferences of Weights and Measures (CGPM) and endorsed by the International Organisation for Standardisation (ISO) for use by engineers and scientists through­ out the world. It is based on six fundamental units: metre :(m), kilogramme (kg), second (s), ampere (A), kelvin (K) and candela (cd). Such a system is, of course, arbitrary and its chief merit is that it is agreed internationally. For convenience in theoretical physics two other systems of units are often chosen: natural and Gaussian units. In the system of natural units the universal constants (h, kB, c) are chosen to be dimensionless and of unit size, which is useful in the theo.ry of elementary particles. In the Gaussian system the older metric units of centimetre, gram and second (c.g.s.) are retained with an electrostatic unit (e.s.u.) for electric charge and an electromagnetic unit (e.m.u.) for electric current. The ratio of current (in e.s.u.) to current (in e.m.u.) has the dimensions of a velocity and is the velocity of light in vacuo, c (in c.g.s. units). The net results are that eo and #Lo are dimensionless and of unit size in this system, leading to the replacement of (eo#Lor~ by c, and that the absence of 1/47T in Coulomb's law of force leads to the presence of 47T in terms involving charges and currents. The main advantages of this system are that E and B have the same dimensions and are of equal magnitude for electromagnetic waves in free space. How­ ever, in media D = eE and B = #LH and some of this simplicity is lost. Some of the important equations in electromagnetism are given 122 Electromagnetic waves in Table A2.1 in Gaussian units, th~ equation number being that for SI units in the text. Conversions of some Gaussian units to SI units are given in Table A2.2, assuming c = 3 X 108 m S-I. Table A2.1 Electromagnetic equations in Gaussian units Maxwell I div E = 41Tp/e [ 1.1] ; div D = 41TP [1.14] MaxwelllI div B = 0 [ 1.2] 1 aB curlE=--­ Maxwell III cat [1.3 ] 1 aE 1 aD[ Maxwell IV curlB=-J+--41T. [1.4]' curIH=-JIf+--1.23]41T. c c2 at' c c at Lorentz force F=q (E+U;B) [2.10] Electric displacement D = E + 41TP [ 1.9] Magnetising field H= B-41TM [1.10] 1 Electric susceptibility Xe = 41T (e-1) [4.1 ] 1 Magnetic susceptibility Xm = 41T (p - 1) Energy density u = (l/81T)(E.D+ B.H) [5.40] Table A2.2 Conversion of Gaussian units to SI units Electric charge q 3 X 109 e.s.u. = I coulomb Electric current I 1 e.m.u. = 10 ampere Electric potential cp 1 stat volt = 300 volt Electric field E 3 X 104 stat volt cm-1 = 1 volt metre-1 Electric displacement D 121T X 105 e.s.u. cm-2 = 1 coulomb metre-2 Energy density u 1013 erg cm-3 = 1 joule metre-3 Radiated power W 10 7 erg second-1 = 1 watt Resistance R 1 stat ohm = 9 X 1011 ohm Capacitance C 9 X 1011 cm = 1 farad Inductance L 109 e.m. u. = 1 henry Magnetic field B 104 gauss = 1 tesla Magnetising field H 41T X 10-3 oersted = I ampere metre-1 Appendix 2 123 Appendix 3 Physical constants Constant Symbol Value Electric constant Eo = 1/(p.oc2) 8.85 X 10-12 F m-1 Magnetic constant Ilo 41T X 10-7 H m-1 Speed of light c 3.00 X 108 m S-l Electronic charge e 1.60 X 10-19 C Rest mass of electron me 9.11 X 10-31 kg Rest mass of ptoton mp 1.67 X W-27 kg Planck constant h 6.63 X W-34 J s II = h/21T 1.05 X 10-34 J s Boltzmann constant kB 1.38 X 10-23 J J(""1 Avogadro number NA 6.02 X 1023 mor1 Gravitation constant G 6.67 X 10-,11 N m2 kg-2 41TEo1l2 Bohr radius ao=~ 5.29 X Hr11 m me Bohr magneton IlB=A 9.27 X 10-24 J rl 2me Electron volt eV 1.60 X 10-19 J Mo18r volume at S.T.P. Vm 2.24 X 10-2 m3 morl Acceleration due to gravity g 9.81 m S-2 124 Electromagnetic waves Appendix 4 Vector calculus For general vectors A, B and a scalar n. Identities div (nA) = n div. A + A.grad n [A4.1] div (A X B) = B.curl A - A.curl B [A4.2] curl (nA) = n curl A + grad n X A [A4.3] curl grad n = 0 [A4.4] div curl A =·0 [A4.5] curl curl A = grad div A - 'il 2 A [A4.6] Cartesian differential operators an. anA an A grad n = 'il n = ax 1+ ay j + az- k [A4.7] aAx aAy aAz divA='il.A=-+-+- [A4.8] ax ay az curl A = 'ilX A·= i j k [A4.9] a/ax a/ay a/az Ax Ay A z . 2 a2 a2 a2 dlV (grad) = 'il = ax2 + ay2 + az2 [A4.10] Appendix 4 125 Spherical polar differential operaton an . 1 an . 1 an· grad n = a,:- r + -;:ao- 8 + ,sin 8 a", '" [A4.l1 ] div A = ~ :, (yl Ar) + 'S~ 8 a~ (Ao sin 8) + _1_aAIjI ,sin 8 a", [A4.l2] 1 ·r cur I A .2. 8 riJ ,sin8~ [A4.13] = r. SIO alar a/a8 a/a", Ar rAB r sin 8AIjI Cylindrical polar differential operaton d n an. + an 9" + an . gra =a,-r ,ao ~z [A4.lS] 1 a aA B aA z div A = r ar (rAr) + ra8 + az [A4.l6] curl A = ! f r8 [A4.l7] r z alar a/a8 a/az [A4.l8] 126 Electromagnetic waves Theorems For a smoothly varying vector field A. Gauss's divergence theorem ~ A.dS = Iv div A dr [A4.l9] where the surface S encloses the volume V, dS = odS is a vector of magnitude dS along the outward normal 0 to the surface dS and dr is an element of the volume V. Stokes's theorem Ie A.ds = Is (curl A).dS [A4.20] where the closed loop C bounds the surface S and ds is a vector element of the loop C. Appendix 4 127 Appendix 5 Lorentz transformations The transformation of physical quantities from an inertial frame S (the laboratory frame) to a frame S' moving with respect to S at a speed u in the positive x direction is given in. Cartesian coordinates, where (j = u/c and 'Y = (1 - (j2rlh. For the inverse transforms replace u by - u. (See also Chapter 2 and Fig. 2.1.) Coordinates x' = 'Y (x - ut), y' = y, Z' = z, t' = 'Y { t - ( ~ ) x } [AS.1] Velocities I Vx - U I Vy I Vz V = V = V =---:---=---:----:: X {I -(J3/c}vx }' y 'Y{1-(J3/c)vx}"z 'Y{1-W/c)vx} [AS.2] Components of a force I W/c) (vyFy + vzFz) Fx =Fx- {l-W/c)vx} [AS.3] I Fy [AS.4] Fy = 'Y{1 - W/c)vx} [AS.S1 128 Electromagnetic waves Electric field Ex' =Ex' Ey' = 'Y(Ey -(kBz), Ez' = 'Y(Ez + ~cBy) [AS.6] Magnetic field Bx' = Bx,By' = 'Y{By + (~/c)Ez} ,Bz' = 'Y{Bz - W/c)Ey} [AS.7] For further information, see Reliltivity Physics by R.E.
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