ELECTROMAGNETIC FIELD THEORY EXERCISES

Tobia Carozzi Anders Eriksson Bengt Lundborg Bo Thidé Mattias Waldenvik E LECTROMAGNETIC F IELD T HEORY E XERCISES Draft version released 9th December 1999 at 19:47 Downloaded from http://www.plasma.uu.se/CED/Exercises Companion volume to E LECTROMAGNETIC F IELD T HEORY by Bo Thidé E LECTROMAGNETIC F IELD T HEORY Exercises Tobia Carozzi Anders Eriksson Bengt Lundborg Bo Thidé Mattias Waldenvik Department of Space and Plasma Physics Uppsala University and Swedish Institute of Space Physics Uppsala Division Sweden Σ Ipsum This book was typeset in LATEX 2ε on an HP9000/700 series workstation and printed on an HP LaserJet 5000GN printer   Copyright c 1998 by Bo Thidé Uppsala, Sweden All rights reserved Electromagnetic Field Theory Exercises ISBN X-XXX-XXXXX-X C ONTENTS Preface ix Maxwell’s Equations 1.1 1.2 1.3 Coverage Formulae used Solved examples Example 1.1 Macroscopic Maxwell equations Solution Example 1.2 Maxwell’s equations in component form Solution Example 1.3 The charge continuity equation Solution Electromagnetic Potentials and Waves 2.1 2.2 2.3 Coverage Formulae used Solved examples Example 2.1 The Aharonov-Bohm effect Solution Example 2.2 Invent your own gauge Solution Example 2.3 Fourier transform of Maxwell’s equations Solution Example 2.4 Simple dispersion relation Solution Relativistic Electrodynamics 3.1 Coverage Draft version released 9th December 1999 at 19:47 1 1 4 5 9 9 10 11 11 13 13 15 15 17 17 i ii 3.2 3.3 Formulae used Solved examples Example 3.1 Covariance of Maxwell’s equations Solution 17 18 18 18 Example 3.2 Invariant quantities constructed from the field tensor 20 Solution 20 Example 3.3 Covariant formulation of common electrodynamics formulas Solution Example 3.4 Fields from uniformly moving charge via Lorentz transformation Solution Lagrangian and Hamiltonian Electrodynamics 4.1 4.2 4.3 Coverage Formulae used Solved examples Example 4.1 Canonical quantities for a particle in an EM field Solution Example 4.2 Gauge invariance of the Lagrangian density Solution Coverage Formulae used Solved examples Example 5.1 EM quantities potpourri Solution Example 5.2 Classical electron radius Solution Example 5.3 Solar sailing Solution 27 27 28 28 28 29 29 31 Example 5.4 Magnetic pressure on the earth Solution Radiation from Extended Sources 6.1 6.2 6.3 23 23 27 Electromagnetic Energy, Momentum and Stress 5.1 5.2 5.3 21 21 Coverage Formulae used Solved examples 31 31 32 32 32 35 35 37 37 39 39 41 41 41 42 Example 6.1 Instantaneous current in an infinitely long conductor 42 Draft version released 9th December 1999 at 19:47 iii Solution Example 6.2 Multiple half-wave antenna Solution Example 6.3 Travelling wave antenna Solution Example 6.4 Microwave link design Solution Multipole Radiation 7.1 7.2 7.3 Coverage Formulae used Solved examples Example 7.1 Rotating Electric Dipole Solution Example 7.2 Rotating multipole Solution Example 7.3 Atomic radiation Solution Example 7.4 Classical Positronium Solution 53 Radiation from Moving Point Charges 8.1 8.2 8.3 Coverage Formulae used Solved examples Solution Example 8.2 Synchrotron radiation perpendicular to the acceleration Solution Example 8.3 The Larmor formula Solution ˇ Example 8.4 Vavilov-Cerenkov emission Solution Radiation from Accelerated Particles Coverage Formulae used Solved examples Draft version released 9th December 1999 at 19:47 53 53 54 54 54 56 56 58 58 59 59 63 Example 8.1 Poynting vector from a charge in uniform motion 9.1 9.2 9.3 42 47 47 50 50 51 51 63 63 64 64 64 66 66 67 67 69 69 71 71 71 72 iv Example 9.1 Motion of charged particles in homogeneous static EM fields Solution Example 9.2 Solution Example 9.3 Solution Example 9.4 Solution 72 72 Radiative reaction force from conservation of energy 74 74 Radiation and particle energy in a synchrotron 77 77 Radiation loss of an accelerated charged particle 79 79 F Formulae F.1 F.2 F.3 F.4 83 The Electromagnetic Field F.1.1 Maxwell’s equations Constitutive relations F.1.2 Fields and potentials Vector and scalar potentials Lorentz’ gauge condition in vacuum F.1.3 Force and energy Poynting’s vector Maxwell’s stress tensor Electromagnetic Radiation F.2.1 Relationship between the field vectors in a plane wave F.2.2 The far fields from an extended source distribution F.2.3 The far fields from an electric dipole F.2.4 The far fields from a magnetic dipole F.2.5 The far fields from an electric quadrupole F.2.6 The fields from a point charge in arbitrary motion F.2.7 The fields from a point charge in uniform motion Special Relativity F.3.1 Metric tensor F.3.2 Covariant and contravariant four-vectors F.3.3 Lorentz transformation of a four-vector F.3.4 Invariant line element F.3.5 Four-velocity F.3.6 Four-momentum F.3.7 Four-current density F.3.8 Four-potential F.3.9 Field tensor Vector Relations F.4.1 Spherical polar coordinates Draft version released 9th December 1999 at 19:47 83 83 83 84 84 84 84 84 84 84 84 84 85 85 85 85 86 86 86 86 86 87 87 87 87 87 87 87 88 F.4.2 Base vectors Directed line element Solid angle element Directed area element Volume element Vector formulae General relations Special relations Integral relations Draft version released 9th December 1999 at 19:47 88 88 88 88 88 89 89 90 91 v vi Draft version released 9th December 1999 at 19:47 78 L ESSON R ADIATION FROM ACCELERATED PARTICLES tion ❚ v˙ ❚ To this end we use what is left of (9.23), that is m0 γ v˙ ✓ ✑ qv B (9.29) The acceleration is found by taking the norm of this last equation and since v have is the scalar equation m0 γ ❚ v˙ ❚❯✑➉❚ q ❚ vB0 ❙ B all we (9.30) This equation is easily solved to give ✬ ❚ q ❚ B0 ✑ v˙ m0 γ v (9.31) ☎ ☎ qB The factor m γ0 ✫ ωc is known as the synchrotron angular frequency and is the relativistic value of the angular frequency for gyro-harmonic motion or cyclotron angular frequency Now we can insert this expression into the relativistic generalisation of the Larmor formula µ q2 v˙2 γ µ q2 ω γ dU ✑ ✑ c v2 dt ✻ 6π c 6π c To find the energy per revolution, we need the period of revolution which is T so dU 2π µ0 q2 ωc2 γ ✑ v dt ✻ ωc 6π c µ0 ❚ q ❚ γ B0 v2 3cm0 ✑ T Urev ✑ ✑ µ0 q2 ωc2 γ v ✑ 6π c (9.32) ✑ 2π ✒ ωc , (9.33) (9.34) (b) The task here is to equate the total energy and the radiated energy and solve for velocity and then see what total energy that velocity is associated with The total energy is E ✑ m0 γ c2 and the radiated energy is Urev which when equated gives c2 v2 γ µ0 q3 B0 3cm20 ✑ ✬ c2 v2 ✕ 1✑ ✬ v2 c2 ✑ (9.35) µ0 q3 B0 3cm20 1✖ ✑ µ0 ❚ q ❚ γ v2 B0 ✒ ✙ 3cm0 ✚ , (9.36) (9.37) µ0 q3 B0 3cm20 So, after a little algebra, γ ✑ ➔ 1✖ 2cm20 µ0 ❚ q ❚ B0 (9.38) Thus the particle energy for which the radiated energy is equal to the total particle energy is Draft version released 9th December 1999 at 19:47 9.3 S OLVED E 79 EXAMPLES ✑ m0 c2 ➔ ✖ 2cm20 µ0 ❚ q ❚ B0 ✑ ✮ 16 TeV (9.39) This is obviously the upper limit for which radiation effect can be neglected in the treatment of synchrotron motion of the particle E ND OF EXAMPLE 9.3 ✦ ✌ R ADIATION LOSS OF AN ACCELERATED CHARGED PARTICLE E XAMPLE 9.4 A charged particle, initially at rest, is accelerated by a homogeneous electric field E0 Show that the radiation loss is negligible, even at relativistic speeds, compared to the particle’s own energy gain Assume that under all circumstances E0 ✶ 108 V/m Solution The relativistic equation of motion is dp dt ✻ ✛ d m0 γ v ✚✇✑ m0 γ v˙ ✖ ✙ dt ✻ ✑ ✎ γ2 v ✙ v˙ v ✚➢✢ c ✑ qE0 (9.40) ❙ v˙ ❙ E0 then Since v m0 γ v˙ ✙ ✖ β γ ✚✩✑ ✑ m0 γ v˙ ✬ m0 γ v˙ ✬ v˙ ✑ ✛ ✛ β2 ✢ ✕ β2 m0 γ v˙ ✖ ✢ ✕ β2 ✑ m0 γ v˙γ ✑➉❚ q ❚ E0 (9.41) ✑➉❚ q ❚ E0 (9.42) ❚ q ❚ E0 m0 γ (9.43) Having found v˙ we try to derive the radiation field generated by this motion; we have that ✓ µ0 q ✓ r ✙ rv v˙ ✚ 4π s3 µ0 q r 4π r3 ✙ ✕ β cos θ ✚ ✑ Erad ✑ ✑ ✑ ✑ ✑ ✓ v ▼Õ◗ r ✕ r ❘ c ✓ µ0 q r 3 4π r ✙ ✕ β cos θ ✚ ❦❤ r ✓ ✓ µ0 q 4π r3 ✙ ✕ µ0 q 4π r3 ✙ ✕ µ0 qv˙ 4π r ✙ ✕ β cos θ ✚ β cos θ ✚ sin θ β cos θ ✚ ✙r ✓ v˙ ✕ ✓ v˙ ❖ ❧ r ✓ ♥ ❋v ●■❍ v❏ ˙ c ✧ v˙ ✚ r r2 v˙ sin θθˆ θˆ Draft version released 9th December 1999 at 19:47 (9.44) 80 L ESSON R ADIATION FROM ACCELERATED PARTICLES where have used the particular geometry of the vectors involved Then we may determine the Poynting vector, which is S Erad rˆ µ0 c ï ï ✑ ï ï ✑ Sr rˆ (9.45) The radiated energy per unit area per unit time corresponding to this is ïï E ∂U ✑ Sr r2 ✑ ∂t rad ï r2 ï µ0 c ïï (9.46) but the energy radiated per unit area per unit time at the source point is ∂U ✑ ∂ t✻ ∂U ∂ t ✑ ∂t ∂t✻ ∂U s ✑ ∂t r rs Erad µ0 c ï ï ✕ β cos θ ✚ µ02 q2 v˙2 ✑ r2 ✙ ✑ µ0 q2 v˙2 sin2 θ 16π c ✙ ✕ β cos θ ✚ µ0 c 16π ï ï sin2 θ r2 ✙ ✕ β cos θ ✚ (9.47) The total radiated energy per unit time is π µ0 q2 v˙2 sin3 θ ✳ π dθ 16π c ✙ ✕ β cos θ ✚ µ q2 v˙2 1 ✕ x2 x ✑ ✕ cos θ 2π ✳ dx ✑✗❹ ❺ ✑ dx dθ ✑ sin θ 8π c ❃ ✙ ✖ β x ✚ µ q2 v˙2 ✛ x2 ✑ 2π ✳ ❃ ✕ ✢ dx 8π c ✙ ✖ β x✚ ✙ ✖ β x✚ µ q2 v˙2 µ0 q2 v˙2 µ0 q4 E02 ✑ 2π ✑ ✑ π γ 8π c ✙ ✕ β ✚ 8π c 6π m20 c ∂ U˜ ✑ ∂ t✻ ✳ ∂U dΩ ✑ ∂ t✻ We compare this expression for the radiated energy with the total energy E dE dt ✻ ✑ m0 c dγ dt ✻ (9.48) ✑ m0 γ c2 , so (9.49) From the equation of motion we find that d dγ m γ v ✚✇✑ m0 γ v˙ ✖ m0 v dt ✻ ✙ dt ✻ and, from the expression for v, ˙ m0 γ v˙ ✑ ✑➉❚ q ❚ E0 ❚ q ❚ E0 (9.50) (9.51) γ2 Hence Draft version released 9th December 1999 at 19:47 9.3 S OLVED m0 v 81 EXAMPLES ✛ ✑➉❚ q ❚ E0 ✕ ✢ ❋ ●■❍ γ ❏ dγ dt ✻ (9.52) v2 c2 dγ dt ✻ ✬ ❚ q ❚ E0 v ✑ (9.53) m0 c and dE dt ✻ ✑ m0 c Finally, for E0 dU˜ dt ✻ ✆ ❚ q ❚ E0 v m0 c ✑➉❚ q ❚ E0 v (9.54) ✑ 108 V/m, and an electron for which ❚ q ❚❯✑ e, we find that µ0 q4 E02 ✑ 6π cm20 ❚ q ❚ E0 v dE dt ✻ ✑ ✑ ✓ ❃ µ0 ❚ e ❚ E0 6π cm20 v ✎ ✓ ❃ E0 ✙ 4π 10 ✚ ✙ ✮ 10 19 ✚ ✎ ✓ ✎ ✓ v 6π ✮ 998 108 ✙ ✮ 11 10 ❃ 31 ✚ ✑ 1✮ Which of course is a very small ratio for a relativistic electron (v ✝ even for E ✑ 108 V/m ✓ 10 c ❃ (9.55) v ✑ ✮ 998 E ND Draft version released 9th December 1999 at 19:47 12 E0 ✓ 108 m/s) OF EXAMPLE 9.4 ✦ Draft version released 9th December 1999 at 19:47 82 A PPENDIX F Formulae F.1 The Electromagnetic Field F.1.1 Maxwell’s equations ☎✝✆ D✂ ρ ☎✝✆ B✂ ☎✵✟ ∂ E ✂☛✡ B ☎✠✟ ∂t H✂ j☞ ∂ D ∂t (F.1) (F.2) (F.3) (F.4) Constitutive relations D✂ εE B H✂ µ j✂ σE P ✂ ε0 χ E Draft version released 9th December 1999 at 19:47 (F.5) (F.6) (F.7) (F.8) 83 84 A PPENDIX F F ORMULAE F.1.2 Fields and potentials Vector and scalar potentials ☎✵✟ B✂ E ✂☛✡ ☎ A (F.9) ∂ A ∂t φ✡ (F.10) Lorentz’ gauge condition in vacuum ☎➄✆ ∂ A☞ φ ✂ (F.11) c ∂t F.1.3 Force and energy Poynting’s vector ✟ S✂ E H (F.12) Maxwell’s stress tensor Ti j ✂ F.2 Ei D j ☞ Hi B j ✡ δ E D ☞ Hk Bk ❾ ij ❽ k k (F.13) Electromagnetic Radiation F.2.1 Relationship between the field vectors in a plane wave B✂ ✟ kˆ E c (F.14) F.2.2 The far fields from an extended source distribution ③ Brad ω ② x ✂ ✡ iµ0 eik ➙ x ➙ ↕ ↕ ③ Erad ω ② x ✂ i eik ➙ x ➙ ✟ ↕ ↕ xˆ d3 x ✁ e ➛ ➣ V❩ 4πε0 c x 4π x ➣ V❩ d3 x ✁ e ➛ ik ✞ x❩ jω ✟ ik ✞ x❩ k (F.15) ✟ jω k Draft version released 9th December 1999 at 19:47 (F.16) 85 F.2 E LECTROMAGNETIC R ADIATION F.2.3 The far fields from an electric dipole ω µ0 eik ➙ x ➙ ✟ rad ③ ↕ ↕ pω k Bω x ✂ ✘ ✡ ② 4π x (F.17) ✟ ✟ eik ➙ x ➙ ↕ ↕ pω k ③ k 4πε0 x ② ③ ✂ ✡ Erad ω ② x ✘ (F.18) F.2.4 The far fields from a magnetic dipole µ0 eik ➙ x ➙ ✟ ✟ rad ③ ↕ ↕ ② mω k ③ k Bω x ✂ ✘ ✡ ② 4π x (F.19) ✟ k eik ➙ x ➙ ↕ ↕ mω k 4πε0 c x ③ Erad ω ② x ✂ (F.20) F.2.5 The far fields from an electric quadrupole rad ③ Bω ②x ✂ iµ0 ω eik ➙ x ➙ ✆ ✟ ↕ ↕ k Qω ③ k 8π x ② (F.21) ③ Erad ω ② x ✂ ✆ ✟ ✟ i eik ➙ x ➙ ↕ ↕ ê ② k Qω ③ kë k 8πε0 x (F.22) F.2.6 The fields from a point charge in arbitrary motion E ② t → x③ ✂ q Rv ø ✡ 4πε0 s3 ÷ B ② t → x③ ✂ ③ ② x ✡ x✁ s✂ ↕ v2 ☞ c2 ù ✟ ✟ Rv v˙ ③ ② x ✡ x✁ c2 ú E ② t → x③ ↕ ↕ c x ✡ x✁ ✟ ↕ ✆v x ✡ x✁ ✡ ② x ✡ x✁ ③ c Rv ✂ ∂t✁ ø ∂t ù ③ ② x ✡ x✁ ✡ ✂ x ↕ ↕ x ✡ x✁ x ✡ x✁ s ↕v c (F.23) (F.24) (F.25) (F.26) ↕ (F.27) Draft version released 9th December 1999 at 19:47 86 A PPENDIX F F ORMULAE F.2.7 The fields from a point charge in uniform motion E ② t → x③ ✂ q ø 1✡ 4πε0 s3 B ② t → x③ ✂ v ↕ ↕2 s✂ R0 ✂ F.3 R0 ✟ v2 R0 c2 ù E ② t → x③ c2 ✡ ø R0 (F.28) (F.29) ✟ v c (F.30) ù x ✡ x0 (F.31) Special Relativity F.3.1 Metric tensor 0 ✡ ❭❬ ❪❭ 0 ✡ 0 ✡ gµν ✂ ❫❴ ❴ 0 ❵ (F.32) F.3.2 Covariant and contravariant four-vectors gµν vν vµ ✂ (F.33) F.3.3 Lorentz transformation of a four-vector x✁ µ ✂ Λµν xν γ ✡ γβ ❭❬ ❪❭ 0 Λµν ✂ γ✂ β ✂ (F.34) ❫ ❴ ✡ γβ 0 ❴ γ 0 0 0❵ 1 ✟ (F.35) (F.36) ✡ β2 u c (F.37) Draft version released 9th December 1999 at 19:47 87 F.4 V ECTOR R ELATIONS F.3.4 Invariant line element ds ✂ c dt ✂ γ c dτ (F.38) F.3.5 Four-velocity dx µ ✂ ds uµ ✂ v c è γ→ γ é (F.39) F.3.6 Four-momentum pµ ✂ ③ ② E → cp m0 c2 uµ ✂ (F.40) F.3.7 Four-current density jµ ✂ v c ρ0 u µ ✂ è ρ→ ρ é (F.41) F.3.8 Four-potential ③ ② φ → cA Aµ ✂ (F.42) F.3.9 Field tensor ∂ Aν ∂ A µ ✡ ✂ ∂ x µ ∂ xν F µν ✂ F.4 Ex ❭❬ ❪❭ Ey Ez ✡ Ex cBz ✡ cBy ✡ Ey ❫ ❴ ✡ Ez ❴ cBx cBy ✡ cBx ❵ ✡ cBz (F.43) Vector Relations Let x be the radius vector ↕ (coordinate vector), from the origin to the point ③ ✠ x → y→ z ③ and let x ↕ denote the magnitude (“length”) of x Let further x → x → x ② ② α ② x ③ → β ② x ③ →☛✡☞✡☞✡ be arbitrary scalar fields and a ② x ③ → b ② x ③ → c ② x ③ → d ② x ③ →☞✡☛✡☞✡ arbitrary vector fields ☎ The differential vector operator is in Cartesian coordinates given by ☎ ✠ ∑ i✌ where xˆ i , i ✂ def ✠ xˆ i ∂ def ✠ ∂ ∂ xi → → is the ith unit vector and xˆ ✠ (F.44) x, ˆ xˆ ✠ Draft version released 9th December 1999 at 19:47 yˆ , and xˆ ✠ zˆ In 88 A PPENDIX F F ORMULAE component (tensor) notation ∇i ✂ ∂i ✂ ∂ ☎ ∂ can be written ∂ ø ∂x → ∂x → ∂x ù ✂ ∂ ∂ ∂ ø ∂x → ∂y → ∂zù (F.45) F.4.1 Spherical polar coordinates Base vectors rˆ ✂ sin θ cos ϕ xˆ ☞ sin θ sin ϕ yˆ ☞ cos θ zˆ (F.46) θˆ ✂ cos θ cos ϕ xˆ ☞ cos θ sin ϕ yˆ ✡ sin θ zˆ (F.47) ϕˆ ✂☛✡ sin ϕ xˆ ☞ cos ϕ yˆ (F.48) Directed line element dx xˆ ✂ dl ✂ dr rˆ ☞ r dθθˆ ☞ r2 sin θ dϕ ϕˆ (F.49) Solid angle element dΩ ✂ sin θ dθ dϕ (F.50) Directed area element d2xnˆ ✂ dS ✂ dS rˆ ✂ r dΩ rˆ (F.51) drdS ✂ r2 dr dΩ (F.52) Volume element d3x ✂ dV ✂ Draft version released 9th December 1999 at 19:47 89 F.4 V ECTOR R ELATIONS F.4.2 Vector formulae General relations ✆ ✆ a b ✂ b a ✂ δi j b j ✂ ab cos θ ✟ ✟ a b ✂☛✡ b a ✂ ✆ a ②b ✟ a a ✟ ✟ c③ ✂ (F.53) εi jk a j bk xˆ i (F.54) ✟ ③ ✆ ②a b c (F.55) ✟ ③ ✆ ③ ✆ ③ ② b c ✂ b② a c ✡ c② a b (F.56) ✟ ③ ✟ ✟ ③ ✟ ✟ ③ ②b c ☞ b ②c a ☞ c ②a b ✂ (F.57) ✟ ③✆ ✟ ③ ✆ ✟ ✟ ③ ② a b ② c d ✂ a êb ② c d ë ✂ ☎ ✟ ③ ✟ ✟ ③ ②a b ②c d ✂ ✆ ③ ✆ ③ ✆ ③ ✆ ③ ②ac ②bd ✡ ②ad ②bc ✟ ✆ ③ ✟ ✆ ③ ② a b d c✡ ② a b c d ☎ ☎ ③ ② αβ ✂ α β ☞ β α ☎✘✆ ☎✔✟ ☎✿✆ (F.59) (F.60) ✆☎ ☎✘✆ ③ ② αa ✂ a α ☞ α a (F.61) ☎✗✟ ✟ ☎ ③ a✡ a α ② αa ✂ α ✟ ③ ✆ ☎❼✟ ③ ✆ ☎ a ✡ a ② ②a b ✂ b ② (F.58) ✟ (F.62) b③ Draft version released 9th December 1999 at 19:47 (F.63) 90 A PPENDIX F F ORMULAE ☎ ✟ ✟ ③ ☎✲✆ ③ ☎➉✆ ③ ✆↔☎ ③ ✆↔☎ a✡ ② a ③ b ② a b ✂ a② b ✡ b② a ☞ ② b (F.64) ☎ ✆ ③ ✟ ☎☛✟ ③ ✟ ☎✴✟ ③ ✆↔☎ ✆☎ b ☞ b ② a ☞ ② b ③ a☞ ② a ③ b ②ab ✂ a ② (F.65) ☎ ✆Õ☎ α ✂ ∇2 α (F.66) α ✂ (F.67) ☎✝✟⑦☎ ☎ ✆ t ☎ ✟ ③ a ✂ ② ☎✗✟ ☎ ② ✟ a③ ✂ ☎ (F.68) ② ☎☛✆ ③ a ✡ ∇2 a (F.69) Special relations In the following k is an arbitrary constant vector ☎ ✆ ☎ x✂ ✟ x✂ ☎ ↕ ↕ (F.71) x x (F.72) ↕ ↕ x ✂ ☎ (F.70) x ø ↕ x ↕ ù ✂☛✡ ↕ x ↕ ☎✴✆ (F.73) ③ ø ↕ x ↕ ù ✂☛✡ ∇ ø ↕ x ↕ ù ✂ 4πδ ② x x Draft version released 9th December 1999 at 19:47 (F.74) 91 F.4 V ECTOR R ELATIONS ☎ ☎ ✟ ÷k ✟ ✟ k x ø ↕ x ↕ ù ú ✂☛✡ ↕ x ↕ ☎ x ø ↕ x ↕ ù ú ✂☛✡ ✆ k x ø ↕ x↕ ù (F.75) ↕ ↕ ✍ if x ✂ (F.76) ✂ k∇2 ø ↕ ↕ ù ✂✘✡ 4π kδ ② x ③ x (F.77) ✟ ③ ☎➠✆ ③ ✟ ☎✏✟ ③ ☎ ✆ ③ a ✡ ② k a ✂ k② a ☞ k ② ②k a (F.78) k ∇2 ø ↕ ↕ ù x ☎ ✆ ✆ ☎ ✂ k ÷ k ø ↕ x↕ ù Integral relations Let V ② S ③ be the volume bounded by the closed surface S ② V ③ Denote the 3-dimensional volume element by d3x ② ✠ dV ③ and the surface element, directed along the outward pointing surface normal unit vector n, ˆ by dS ② ✠ d 2x nˆ ③ ☎✏✆ ③ ✆ a d x ✂ ✎ dS a ➣ V② ➣ V② (F.79) S ☎ α ③ d3x ✂ ✎ S dS α (F.80) ☎✐✟ ③ ✟ a d x ✂ ✎ dS a (F.81) ➣ V② S If S ② C ③ is an open surface bounded by the contour C ② S ③ , whose line element is dl, then ✎ ✎ C α dl ✂ ✆ C a dl ✂ ➣ dS ✟ ☎ ➣ S α ✆ ☎✔✟ dS ② S (F.82) a③ Draft version released 9th December 1999 at 19:47 (F.83) 92 A PPENDIX F F ORMULAE Draft version released 9th December 1999 at 19:47 [...]... Snapshots of the field Multiple half-wave antenna standing current 43 44 47 9.1 Motion of a charge in an electric and a magnetic field 74 Draft version released 9th December 1999 at 19:47 vii OF F IGURES Draft version released 9th December 1999 at 19:47 viii P REFACE This is a companion volume to the book Electromagnetic Field Theory by Bo Thidé The... calculate the vector field from the given magnetic field The equations connecting the potentials with the fields are ✍ ∂A ✑ ✕ φ✕ ∂t ✍✗✓ E B ✑ (2.3) A (2.4) In this problem we see that we have no boundary conditions for the potentials Also, let us use the gauge φ ✑ 0 This problem naturally divides into two parts: the part within the magnetic field and the part outside the magnetic field Let us start with... CONSTRUCTED FROM THE FIELD TENSOR Construct the dual tensor ♦ αβ ✑ 12 ε αβ γδ Fγδ of the field tensor Fµν What quantities constructed solely with the field tensor and it’s dual tensor, are invariant under Lorentz transformations? Having found these quantities you should be able to answer the questions: ✱ can a purely electric field in one inertial system be seen as a purely magnetic field in another?... oscillation mode for the E and H fields Let’s say that we have a mode α such that the E ✑ Eα field is oscillating at ω ✑ ωα ✑ 0 and ✺ that it has a k ✑ kα ✑ 0 which is parallel to the electric field, so kα ❙ Eα From (2.28a) ✺ this implies that ωα Bα ✑ kα Bα ✑ 0 ✓ Eα ✑ 0 (2.29) (2.30) ✓ So, we see that S ✑ E H ✑ 0 trivially The lesson here is that you can have time-varying fields that do not transmit energy!... Relation of EM fields in different inertial systems Now that we know that E B and E ✕ c B are Lorentz invariant scalars, let see what they say about EM fields in different ✎ inertial systems Let us say that X ✫ E B and Y ✫ E 2 ✕ c2 B2 All inertial systems must have the same value for X and Y A purely electric field in one inertial system means that B ✑ 0, so X ✑ 0 and Y ♣ 0 A purely magnetic field would... r 0 In other words it does not seem that a purely electric field can be a purely magnetic field in any inertial system 2 2 2 For a progressive wave E s B so X ✑ 0 and in a purely electric or a purely magnetic field X ✑ 0 also, but for a progressive wave E ✑ cB so Y ✑ 0 and if the other system has E✻❢✑ 0 or B✻❢✑ 0 then Y ✑ 0 force both the fields to be zero So this is not possible E ND OF EXAMPLE 3.2... quantity involving charge density and current density and the E and B fields The EM fields are of course contained in the field tensor F µν To get a vector quantity from F µν and J µ we contract these so our guess is Fµ ✑ F µν Jν (3.28) Inhomogeneous Maxwell equations The inhomogeneous Maxwell may be written as the 4-divergence of the field tensor ∂α F αβ ✑ J β (3.29) Draft version released 9th December... gauge ✍✏✎ ✍✏✎ A ✖ µε∂ φ ✒ ∂ t A✑ 0 ✖ µσ φ ✑ 0 φ ✑ 0 The Coulomb gauge is most useful when dealing with static fields Using (2.13) and (2.14, for static fields, reduces to ∇2 φ ✑ ✕ ∇2 A ✑ ρ ε ✕ µj ✍✘✎ A ✑ 0 then (2.17) (2.18) The Lorentz gauge is the most commonly used gauge for time-varying fields In this case (2.13) and (2.14) reduce to ✛ ∂ ∂2 ✕ µε 2 ✢ φ ✑ ∂t ∂t ✛ 2 ∂ ∂2 ✕ µε 2 ✢ A ✑ ∇ ✕ µσ ∂t ∂t ∇2... net inflow of electric current through the boundary surface S Hence, the continuity equation is the field theory formulation of the physical law of charge conservation E ND Draft version released 9th December 1999 at 19:47 OF EXAMPLE 1.3 ✦ Draft version released 9th December 1999 at 19:47 8 L ESSON 2 Electromagnetic Potentials and Waves 2.1 Coverage Here we study the vector and scalar potentials A and... The field tensor (components are 0, 1, 2, 3) F µν ✂ ❭❪ ❬❭ 0 E1 E2 E3 ✡ E1 0 cB3 ✡ cB2 ✡ E3 0 cB1 cB2 ✡ cB1 ❵ 0 ❫ ❴ ✡ E2 ✡ cB3 Draft version released 9th December 1999 at 19:47 ❴ (3.2) 17 18 L ESSON 3 R ELATIVISTIC E LECTRODYNAMICS 3.3 E XAMPLE 3.1 Solved examples ✌ C OVARIANCE OF M AXWELL’ S EQUATIONS Discuss the covariance of Maxwell’s equations by showing that the wave equation for electromagnetic fields ... Relationship between the field vectors in a plane wave F.2.2 The far fields from an extended source distribution F.2.3 The far fields from an electric dipole F.2.4 The far fields from a magnetic... corresponding fields in the rest system of the charge Solution We wish to transform the EM fields The EM fields in a covariant formulation of electrodynamics is given by the electromagnetic field tensor... fields and the Lagrangian density of the EM fields 4.2 Formulae used The Lagrangian for a charged particle in EM fields is L ✂☛✡ ✆ mc2 ✡ qφ ☞ qv A γ (4.1) A useful Lagrangian density for EM field