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. Turner in this series.
Appendix 5 129 Appendix 6 Exercises
Chapter 2 1 Use the Lorentz transformations for the components of a force, of a velocity and of the coordinates, given in Appendix 5, to show that the Coulomb force F' on a moving charge in a mo.ving frame, equation [2.14], transforms into the Lorentz force F, equation [2.15], in the laboratory frame. 2 A narrow beam of electrons of energy 7 GeV is circulating in a large storage ring. What are the maximum instantaneous values of the electric and magnetic fields due to each electron at a distance of 5 mm from the beam and in which direction is each vector relative to the beam? 3 Show that the vector potential at it point P, distance r from-the centre of a thin, circular wire carrying a persistent current I, is: poIfell A(r) = 41T R where R is the distance from the circuit element ell to P and r> a, the radius ofthe circle. Hence show that: Po (m X r) A(r)= 4m3
where m is the magnetic dipole moment of the circular current L 4 Using the definitions of A and tP given in equations [2.26] and [2.29] , show that Maxwell's equations div D = PI and curl H = jf + aD/at lead to the following inhomogeneous wave equations for a linear, isotropic medium of permittivity € and permeability p:
130 Electromagnetic waves a2 A " 2 A = - €Il ~ = -Ilj,
a2 1/> ~ " 2 I/> = - €Il at 2 = - € •
S Explain why the electric field due to a charge is not given by the force per unit charge, nor the gradient of its electric potential, when the charge is moving.
Chapter 3 1 Show that the displacement current density in a linearly polarised plane wave E = Eo exp i (w - k.r) is i€owE and calculate itsroot mean square value when Eo = 5.1 mY m- 1 and w/2n = 1 GHz. 2 A stationary observer is looking at a mirror that is travelling away from him at a speed v < c. Show that if he shines a laser beam of frequency v at the moving mirror he will see a reflected frequency: v' = v { (1 -vlc)/(1 + vic) } Vz.
3 Solar energy falls on the earth's surface at about 1 kW m-2 • Estimate the r .m.s. electric and magnetic fields in the sunshine received. 4 Assuming that the earth's magnetic field is due to a dipole at its centre which produces a field of 15 mT at the North Pole, estimate the total magnetic energy in the earth's external magnetic field. S A radio antenna at the surface of the earth is emitting its radiation radially. If its average power is 20 kW, what is the energy flux at a domestic receiver 50 km away? What are the r.m.s. values of the E and H fields at the receiver?
Chapter 4 1 A charge q, mass m, oscillating with amplitude a at an angular frequency Wo radiates energy at a rate (p.oq2 Wo 4 a2)/(12nc). By solving the equation of motion of a harmonically bound charge, subject to a damping force proportional to its speed and driven by a force F exp (iwt), find the mean energy of the oscillator and hence show that the damping constant which
Appendix 6 131 simulates the radiation is '1 = (21rP-Ocq2 )/(3m"Ao 2 ), where "Ao = 21r c/wo. Estimate the natural width due to this·cause of a line in an atomic emission spectrum. 2 Show, from Maxwell's equations, that the wave equations for a polarisable, magnetisable dielectric of relative permittivity €r and relative permeability IJr are: "iI 2 E - 1. a2E "iI2 B _ 1.. a2B - 1./2 af ' - v2 af
where 1.1 = C (p,.€r r% . 3 The electric vector of an electromagnetic wave in a dielectric is:
Ex = Eo exp (-Ilz) exp iw (t - nRz/c) where Il is the absorption coefficient. Show that the magnetic vector is perpendicular to Ex and find the phase difference between the vectors.
Chapter 5 I Estimate the reflectance and transmittance at normal incidence for (a) light.and (b) radio waves from air into water. How can you explain this microscopically? 2 Show that the reflection roefficient for radiation at normal incidence from free space on to a plane surface of material of refractive index (nR - inIl is: (nR -1)2 + nI2 R = (nR + 1)2 + nI2 .
For a metal at low frequencies (w/21r) and conductivity 0 (nR - inI)2 = - iO/(WEo). Show thatR = 1 -(SwEo/O)% when o>WEo. 3 A transparent dielectric of refractive index n has a plane bound• ary, forming the (y. z) plane, with free space. A linearly polarised plane wave is incident on the boundary from the medium. The magnetic field B in the incident (I), reflected (R) and trans• mitted (T) beams is parallel to the positive axis and has components:
132 Electromagnetic waves BJ = A exp [iw {t - n (x cos a + y sin a)/c} J
BR =Bexp [iw {t-n(-xcosa'+ysina')/c}]
BT = C exp [iw {t - (x cos ~ + y sin ~)/c } ]
Draw a diagram showing the directions of the three beams and of their electric fields, indicating the angles a, a' and ~. By considering the y-dependence of B at the surface, show that a = a' and sin ~ = n sin a. Show that a solution of the equation sin ~ = n sin a can be obtained when n sin a > 1 by putting ~ = rr/2 + ili, where cosh li = n sin a. If n = 1.4, a = 800 and the radiation has wavelength of 400 nm in free space, find how far from the surface the magnetic field has dropped to 10% of its value at the surface. [Hint: sin ('y + ili) = sin 1 cosh li + i cos 1 sinh li; cos (1 + ro) = cos 1 cosh li - i sin 1 sinh li.J 4 A laser beam, having ap?wer of 100 MW and a diameter of 1 mm, passes through a glass window of refractive index 1.59. Find the peak values of the electric and magnetic fields of the laser beam (a) in the air, (b) in the glass. 5 Show that the wave impedance Z of a medium of permeability Jl and permittivity € is (Jl/E)v" . 6 A uniform plane wave is normally incident froQl a medium I into a parallel slab of thickness 1 of medium 2 and emerges into medium 3 after two partial reflections. Show that there are no reflections: (a) when media I and 3 are the same and kzl = mrr, where m is an integer; (b) when kzl = rr/2 andZ2 = (ZtZ3)v".
Chapter 6 1 Show that an electromagnetic wave with complex E and H fields given by E = (ER + iEV exp iwt and H = (HR + zHJ) exp iwt, has an average Poynting vector given by
Appendix 6 133 pressure Pr = 2Ui, where Ui is the energy density of the incident wave. 3 Electromagnetic waves of frequency 1 MHz are incident normally on a sheet of pure copper at O°C. (a) Calculate the depth in the copper at which the amplitude of the wave has been reduced to half its value at the surface, if the conductivity of copper at 0° C = 6.4 X 107 S m-I • (b) Explain how you could calculate this depth at higher frequencies and at lower temperatures. 4 Discuss the possibility of using radio waves to communicate with a submarine submerged in seawater of conductivity 4.0 S m-1 • S A spacecraft returning to earth produces a cloud of ionised atoms of density 1015 m-3 • Find (a) the plasma frequency of this cloud, (b) the cut-off wavelength for cormnunication with the ground. 6 Show that the energy dissipated by a current 1 flowing in a long, straight wire of conductivity a and radius a can be described as flowing into it radially from its surroundings. Hence show that the power dissipated per unit length is P /( frO'; ).
Chapter 7 1 Using the Lorentz condition for the vector potential A show that the equation:
aAz=( Jl.o/)~{/o cos w (t- ;IC)] az 4fr az r can be solved to fmd the scalar potential:
t/)= 4!EO{ ~ qo sin w (t-rlc) + CO;8 10 cosw(t-clr) )
where I, r, z, (J are given in Fig. 7.l(b) and 10 = Wllo .. i For the radiation field of a Hertzian dipole the vector potentialis:
A(r,t)=IJ;!O (cosw(t-rlc)) (cos8i-sin88)
and. the scalar potential is given in exercise 1. Show that the electric vector is E = Ei), where E9 = - Eorl sin w (t - ric) and Eo = (wUo sin (J)/(4frEoC2). 3 The Poynting vector for the raaiation field of a Hertzian dipole is:
134 Electro1Tlllgnetic waves E 2 fI= _0_ sin2 w (t - r/c)i II-ocr Show that the average Poynting vector over one cycle is:
A#. lI-oCP 2. 2 k 2 _ 1 ~>= 3')_1r 2 0 sm 9 2r r where 10 is given in exercise 1 and k = w/c. 4 Show that the radiation resistance of a current-loop antenna of radius a is 201r2(kat. where k is the wave number of the radiation. Estimate- the radiation resistance for a loop with the Bohr radius ao emitting red light. S Show that the power radiated by a linear quadrupole antenna (Fig. 7.3(a» is given by (p.oW 6 Q0 2)J(2401rc3 ) and that its radi• ation resistance is 4 (kIt. Hence show that the ratio of the power radiated from a quadrupole antenna to that from a Hertzian dipole of the same length is (kl)2/20. 6 The radiation field of a Hertzian dipole of moment P (t) = Po exp ;wt has vector potential:
A (r. t) = 411-0 [p] (cos 9i - sin 9th 1rr where [p] is the time derivative at the reduced time (t. - ric) of p. Show that the radiation fields are: sin 9 11-0 sin 9 .. E - f;;] B [ ] 8 - 41r€oc 2 r IJ'. I/J 41rcr p
and that the power radiated is W = [P] 2/(61r€oc 3 ).
ChapterS
1 For the TE lO mode in a lossless, rectangular waveguide (Fig. 8.3(a». obtain expressions for (a) the average electrical energy, (b) the average magnetic energy. per unit length of guide. and hence show that the total electromagnetic energy per unit length is Eo 2 €oab/4, where Eo is the peak amplitude of the electric vector. 2 Calculate the cut-off frequency of the following modes in a rec• tangular waveguide of internal dimensions 30 mm X 10 mm:
Appendix 6 135 TEo I, TE 10, TM I l, TM2 l' Hence show that this waveguide will only propagate the TEl 0 mode of 10 GHz radiation. 3 A rectangular cavity has internal dimensions (in mm) of 30 X 15 X 45. Find the three lowest resonant modes and calculate their frequencies. 4 A microwave receiver is connected by 30 m of waveguide of internal cross-section 23 mm X 10 mm to an antenna. Find the ratio of (a) the phase velocity and (b) the signal velocity in the waveguide to that in free space, for reception at 12 GHz. 5 A rectangular cavity made from waveguide of aspect ratio 2.25: 1 resonates at 8.252,9.067 and 9.967 GHz.lfthese frequencies are those of adjacent TElOI modes find the length of the cavity, assuming that the cut-off frequency of the TE IO mode of the waveguide is 6.56 GHz.
136 Electromagnetic waves Appendix 7 Answers to exercises
Chapter 2
2 E = 0.78 V m-1 , radial; B = 2.6 X 10-9 T, azimuthal.
Chapter 3
1 200p.A m-2 • 3 600 V m-1 ; 2 p.T. 4 1019 J. 5 1.27 p.W m-2 ; 22 mV m-1 ; 58 p.A m-1
Chapter 4 1 1.2 X 10-14 m. 3 tan 1) = nI/nR.
Chapter 5 1 (a) R = 2%; T = 98%; (b) R = 64%, T = 36%; interference between radiation induced and incident. 3 153 nm. 4 (a) 312 MV m-1 , 1.04 T; (b) 248 MV m-1 , 1.31 T.
Chapter 6 3 (a) 44 p.m. 4 At 100 Hz, skin depth = 25 m. 5 (a) 280 MHz; (b) 1.1 m.
Appendix 7 137 Chapter 7
4 10-11 il.
Chapter 8 1 (a) Eo2eoab/8; (b) same. 2 15,5,15.8,18.0 GHz. 3 101,102,201; 6.0, 8.3,10.5 GHz. 4 (a) 1.19, (b) 0.84. 5 119.8 mm.
138 Electromagnetic waves INDEX Index
absorption coefficient, (j, 54, 121 Compton scattering, 103 acceleration, due to gravity,g, conductance, Qf dielectric, G, 55 124 conduction current density, if, admittance, of dielectric, Y, 55 4,78,80 ampere, definition, 21 conductivity, electrical, 0, 78, Ampere's law, 6, 30 120 anisotropic media, 1, 8 conductors: density of charge antennas, 95-9 carriers, 77; wave parameters, 79 Avogadro number, NA, 124 conservation: of charge, 42; of energy flow, 43 Biot-Savart law, 25, 92 continuity, equation of, 42 Bohr magneton, JIB, 124 Coulomb force, F, 13; of moving Bohr radius, a 0, 124 charge, 16-18 Boltzmann constant, kB, 51, critical angle, 8c, 75 59-60, 119, 124 cross-section, scattering, 0, 101 boundary relations, 62-7 current density i, 1,4,12,25-6, Brewster angle, 70-1 30-2,85-6,91,120;of conduction current, if, 4; of Cartesian differential operators, magnetisation current, im, 125 5-7; of polarisation current, cavities: coupling to, 116-18; jp,4 helical, 118 ; rectangular, current, electric, I, 19-20, 28-30, 115-16;re-entrant,118 91,97,120 cavity radiation, 118-19 currentloop,5,95-6,130 charge density p, I, 120; offree cylindrical polar differential charges Pf. 3; of polarisation operators, 126 charges Pp, 3 classical electron radius, TO, 101 D'Alembertian operator, 0,24,31 classical limit, of quantum theory, Debye equations, of complex vii, 103, 119 permittivity, 61 Clausius-Mossotti equation, 50-1, density, of charge carriers in 77 conductors, 77
Index 141 dielectric constant, see per- electrostatic units, 122-3 mittivity, relative Er energy: quantised, viii; relativistic, dielectric relaxation, 59-61 11 differential operators, 125-6 energy density, U, 28,31,42-5, dilatation, of time, 10 121; at a boundary, 71-2; dimensions, of electromagnetic of cavity radiation, 119; in a quantities, 120-1 conductor, 83-5 dipole moment, electric, p, equation of continuity, 42 48,90,100-1,120; of polar equipartition of energy, 119 molecules, Po. 51 evanescent wave, 75-6 dipole moment, magnetic, m, 95-6,121,130 Faraday, M., vii dispersion: anomalous, 57; Faraday's law, 29 normal, 57 Feynman, R.P., 62,104 displacement current, 78, 80 FitzGerald contraction, 10 divergence theorem, Gauss's, 4, four-vector: of current density jv, 28,30,43,127 12; of position rv, 12; of . potential, Av, 24; of retarded Einstein, A., vii, 9, 119 potential, Av, 32 electric charge, q, 120 Fresnel's equations, 67-70 electric constant, Eo. 1, 124 electric current, I, 19-20, 28-30, Galilean transformation, 9-10 91,97,120 gas discharge, 77,88 electric dipole, 47-8,90, 100-1 Gaussian units, viii, 122-3 electric displacement, D, 2, 7-8, Gauss's divergence theorem, 4, 62-5,72,120 28,30,43, 1·27 electric field, E, 1,21,22,120; of Gauss's law for polarised electric quadrupole, 96; Of dielectrics, 3-4 halfwave antenna, 97-8; of gravitational constant, G, 124 Hertzian dipole, 92-3; of group velocity, u, 109 magnetic dipole, 95-6; of guided waves, 104-18 moving charge, 14-16 electric potential, see potential, half-wave antennas, 97-9 scalar Hertzian dipole, 89-93 electric quadrupole, 96 horn, waveguide, 99-100 electromagnetic energy, flux of, f/, 42-5, 71-3 induction field, 92 electromagnetic equations, in inhomogeneous wave equations, Gaussian units, 123 31-2,89,130-1 electromagnetic units, 122-3 intensity of wave: <.9>, 44; electromotive force, &, 28 in a conductor, 84; in a electron, classical radius, ro, 101 dielectric, 73; in a laser beam, electron volt, 124 85 electronic charge, e, 124 invariance: of electric charge, 10; electrostatic energy, U, 27-8,120 of physical laws, 9
142 Index ionosphere, 77, 88 millimetre waves, 86 isotropic media, 1, 8, 52, 72, 78, molar volume, at S.T.P., 124 130-1 momentum: flux, 84; relativistic, 10 Laplacian operator '\7 2, 125-6 laser, viii, 133 natural units, 122 linear charge density, A, 120 natural width, of absorption linear media, 1, 8, 52, 72, 78, 130 line, 56-7 Lissajous figures, 110 Newton's laws of motion, 9 Lorentz force law, 13, 18 non-homogeneous media, 1, 8 Lorentz gauge, 23-5, 91 non-linear media, 1, 8, 52 Lorentz transformations, 9-12, 128-9; of charge density, optical fibre, 114-15 11-12; of current density, 12; of electric field, 129; of mag• parabolic reflector, 99-100 netic field, 129 penetration depth, in a super• Lorentzian shape, of absorption conductor, 79 line, 57 permeability,JI, 62,130-1,133 Lorentz-Lorenz equation, 53, 55 permeability, relative JIr, 8, 72, loss tangent, tan 1),54-5 78-9,121, .132; differential, 8 permittivity, E, 62, 130-1,133 magnetic constant JIo 1, 124 penpittivity, relative Er, 8,120; magnetic dipoles,S complex;54, 60-1; of air, 56; magnetic field, Bo 1, 120; of of free space, see electric current, in a wire, 20,25-6; constant; of plasmas, 86-7; of earth, 131 ; of electric quad• of water, 56 rupole, 96-7; of halfwave phase velocity, v, 53; in a con• antenna, 98; of Hertzian ductor, 83; in a dielectric, dipole, 92-3; of magnetic 65; in a waveguide, 109 dipole, 95-6; of moving photon, viii, 119 charge, 13-18 physical constants, 124 magnetic vector potential, see Planck constant, h, viii, 124 potential, vector A plane waves, 36 -40 magnetisation, M, 2, 5-7, 121 plasma, definition, 85; frequency, magnetisation current density, wp, 87-8,121 jm,4-7 Poisson's equation, 25, 27, 31 magnetising field, H, 2, 7-8, 121 polarisability, a, 1, 20; of non• magnetostatic energy, U, 28-31 polar dielectric, 49-51; of non• 121 polar gas, 47-8 mass: of electron, 124; of proton, polarisation, P; 2-4, 120; of 124; relativistic, 10 dielectrics, 49-51; rotational, Maxwell's equations, vii, 1,8, 59-60 24-5; in dielectrics, 51-2, polarisation of waves: 63; in free space, 2, 33 circular, 39; direction of microwaves, 61, 75-6, 79-83 rotation, 39, 40; elliptical, 39;
Index 143 polarisation of waves (cont.) 75-6,114-15 linear, 38; by reflection, refraction, law of, 67 70-1; transverse electric, refractive index, n, 53, 121; 68-9,73; transverse magnetic, complex, 53,56-7; of gas, 69-70,74 57-8; at high frequencies, 58 polarisation current density, relativistic energy, 10; momentum, jp,4-5 10 potential, scalar cp, 21, 23-4, 120; relaxation peak, 61 retarded, 32, 89-91 relaxation time, T, 59 potential, vector A, 21-6,121; resistance, at radio frequencies, of current, 25-6; of moving Rrf. 81; radiation, Rr, 95-7, charge, 21-2; retarded, 32, 135 89-91,96 resonant scattering, 102 potential energy, of charge, 27 rest mass: of electron, 124; of power, radiated, W, 94-5, 97-9, proton, 124 110-11,121 retarded potentials, 32, 89-91, Poynting vector. f/, 43-5,71-3, 105 83-5,93-4, 121; in a con• ductor, 83-5; in a cylinder, scalarpotential,cp, 21-4,120; 113; for electric quadrupole, retarded, 32, 89-91 96-7; for Hertzian dipole, scattering, 99-103 93-4; for magnetic dipole, semiconductor, 77, 88 95-6; time average, 44-5, SI units, viii, 120-3; oflength, 73,83-4,93-4,96-7,101; 35; of time, 35 in a waveguide, 110 skindepth,i>, 79-84,121 Snell, W., 67 quadrupole, electric, 96-7 solar corona, 102 quadrupole moment, electric, Q, speed of light, c, 9,15,32,34-6, 120 38,44-5,53,122-3,124 spherical polar, coordinates, 90 -1 ; radiant energy, 92-3 differential operators, 126 radiated power, W, 94-5, 97, spherical waves, 40-1, 91-3 110-11,121, 135 Stokes's theorem, 29, 127 radiation field, 92-3, 96-7 superconductor, 79 radiation pressure,Pr, 84-5,121 surface current density, i, 6, radiation resistance, R r , 95-7, 120;.of magnetised matter, 135 im ,6 radio waves, 44,59-60,81,83, susceptance, of dielectric, 55 84,88 susceptibility, electric Xe, 47, Rayleigh scattering, 101 120 Rayleigh-Jeans law, 119 susceptibility, magnetic, Xm, 121 reflectance,R, 73, 82-3,121 reflection: coefficient, see re- temperature, of photons, viii flectance; diffuse, 85; laws of, tesla, unit of magnetic field, 120, 67; specular, 84; total internal, 123
144 Index Thomson scattering, 102-3 wave equations: in conductors, time, dilation, 10 78; in dielectrics, 65; general total internal reflection, 75-6, solution, 34-5; inhomo• 114-15 geneous, 3 I, 131; in space, transmission coefficient, see trans• 34; in waveguides, 107-9 mittance wave impedance, Z, 45-6, 69-70, transmittance, T, 73, 121 81-3,121 transverse electric and magnetic wave intensity, <11'>,44-5,73, modes, TEM, 113-14 83-4,94,96,101 transverse electric modes, TEmn, wave number k, 36,65-7, 121; 107-12 complex, 79 transverse magnetic modes TMmn, wave parameters: in conductors, 113 78-81;in dielectrics, 65-7; transverse waves, 38 in plasmas, 86-7 waveguide: equation, 106; horn, ultraviolet catastrophe, 119 99-100; modes, 107-14; units, viii, 120-3 rectangular, 106-11 waveguide, A, 36; cut-off, Xc, 106; vector identities, 125 in free space, AI), 105; in vector potential, A, 21-6, 121; of waveguide, A~, 105 current loop, 130; retarded, waves: plane, 36-40; spherical, 32,89-91,96 40-1,92-3 velocity of light, c, 9, 15, 32, 34-6,38,44-5,53,122-3,124
Index 145 STUDENT PHYSICS SERIES
RELATIVITY PHYSICS
Relativity Physics covers all the material required for a first course in relativity. Beginning with an examination of the paradoxes that arose in applying the principle of relativity to the two great pillars of nineteenth-century physics-classical mechanics and electromagnetism - Dr Turner shows how Einstein resolved these problems in a spectacular and brilliantly intuitive way. The implications of Einstein's postulates are then discussed and the book concludes with a discussion of the charged particle in the electromagnetic field. The text incorporates details of the most recent experiments and includes applications to high-energy physics, astronomy, and solid state physics. Exercises with answers are included for the student.
R.E. Turner Dr Roy Turner is Reader in Theoretical Physics at the University of Sussex.
ISBN 0-7102'()OOI-3 About 128 pp., 198mm x 129mm, diagrams, April 1984 STUDENT PHYSICS SERIES
CLASSICAL MECHANICS
A course in classical mechanics is an essential requirement of any first degree course in physics. In this volume Dr Brian Cowan provides a clear, concise and self-contained introduction to the subject and covers all the material needed by a student taking such a course. The author treats the material from a modern viewpoint, culminating in a fmal chapter showing how the Lagrangian and Hamiltonian formulations lend themselves particularly well to the more 'modern' areas of physics such as quantum mechanics. Worked examples are included in the text and there are exercises, with answers, for the student.
B. P. Cowan Dr Brian Cowan is in the Department of Physics, Bedford College, University of London
ISBN 0-7102-0280-6 About 128 pp., diagrams, 129mm x 198mm, April 1984 STUDENT PHYSICS SERIES
ELECTRICITY AND MAGNETISM
Electromagnetism is basic to our understanding of the properties of matter and yet is often regarded a difficult part of a first degree course. In this book Professor Dobbs provides a concise and elegant account of the subject, covering all the material required by a student taking such a course. Although concentrating on the essentials of the subject, interesting applications are discussed in the text. Vector operators are introduced at the appropriate points and exercises, with answers, are included for the student.
E.R.Dobbs Professor Roland Dobbs is Hildred Carlile Professor of Physics at the University of London.
ISBN 0-7102'()lS7-S About 128 pp., 198mm x 129mm, diagrams, April 1984