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UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede LECTURE NOTES 19 LORENTZ TRANSFORMATION OF ELECTROMAGNETIC FIELDS (SLIGHT RETURN) Before continuing on with our onslaught of the development of relativistic electrodynamics via tensor analysis, I want to briefly discuss an equivalent, simpler method of Lorentz transforming the EM fields EB and from one IRF(S) to another IRF(S'), which also sheds some light (by contrast) on how the EM fields Lorentz transform vs. “normal” 4-vectors. In P436 Lecture Notes 18.5 {p. 18-22} we discussed the tensor algebra method for Lorentz transformation of the electromagnetic field e.g. in the lab frame IRF(S), represented by the EM field strength tensor F v to another frame IRF(S'), represented by the EM field strength tensor Fv via the relation: vv T FF n.b. in matrix form: FF F since is symmetric. Analytically carrying out this tensor calculation by hand can be tedious and time-consuming. If such calculations are to be carried out repeatedly/frequently, we encourage people to code this up and simply let the computer do the repetitive work, which it excels at. For 1-dimensional Lorentz transformations (only) there is a simpler, less complicated, perhaps somewhat more intuitive method. Starting with the algebraic rules for Lorentz- transforming { EB and } in one IRF(S) to { EB and } in another IRF(S') e.g. moving with relative velocity vvx ˆ with respect to IRF(S): 2 component(s): EEx x Bx Bx 11 components: EEcByyz ByyzBEc vc EEcBzzy Bzzy BEc We can write these relations more compactly and elegantly by resolving them into their and components relative to the boost direction: here, is along vvx ˆ and is perpendicular to v , defined as follows {n.b in general, v could be e.g. to xˆˆ,,yzˆ or r ˆ}: EE vc EEvBEcB 11 2 B B 11 B BvEB2 E cc Now since vvx ˆ {here} then: EE x , B Bx and: B Byyzˆ Bzˆ , E Eyyzˆ Ezˆ {and similarly for corresponding quantities in IRF(S')}. © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede Since vvx ˆ and EE x then: vE vExx ˆ 0 And likewise, since B Bx then: vB vBxx ˆ 0 . Thus, we can {safely} write: vE vE and vB vB, as long as v is always to one of the components of E and B - e.g. xˆˆ or yz or ˆ . Then we can write the Lorentz transformation of EM fields as: EE vc EEvBEcB vc B B 11 2 B BvEB2 E 11 cc This can be written more compactly in 2-D matrix form as: EM Fields: “Normal” 4-Vector: EE → EE and BB compare to x x cB cB EE10 EE x x ↔ cB 01 cB cB cB ct ct Unit Matrix Operator Matrix Scalar Matrix Thus, we see that for the EM fields vs. the 3-D space-part of a “normal” 4-vector, the vs. components are switched, B transforms “sort of” like time t, but 2 2 Lorentz boost matrices for ( EB and ) vs. 4-vectors are not the same (they are similar, but they are not identical). We can also write compact inverse Lorentz transformations (e.g. from IRF(S') rest frame → IRF(S) lab frame): EM Fields: “Normal” 4-Vector: EE → E EBB and compare to x x cB cB EE10 EE x x ↔ cB 01 cB cB cB ct ct Unit Matrix Operator Matrix Scalar Matrix 2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede For a general Lorentz transformation (i.e. no restriction on the orientation of v {arbitrary}): A.) Lorentz transformation from IRF(S) → IRF(S'): 2 EEcB E vc vc 1 2 1 2 B BE B 11 c 1 Or: 1 EE 1 cB cB 1 1 operator matrix B.) Inverse Lorentz transformation from IRF(S') → IRF(S): 2 EEcB E 1 Switch , E E and 2 1 B B in above relations B BE B c 1 Or: 1 EE 1 cB cB x 1 1 operator matrix Electrodynamics in Tensor Notation So now that we know how to represent the EM field in relativistic tensor notation (as FGvv or ), we can also reformulate all laws of electrodynamics (e.g. Maxwell’s equations, the Lorentz force law, the continuity equation {expressing electric charge conservation}, etc. ) in the mathematical language of tensors. In order to begin this task, we must first determine how the sources of the EM fields – the electric charge density (a scalar quantity) and the electric current density J (a 3-D vector quantity) Lorentz transform. The electric charge density QV = charge per unit volume (Coulombs/m3) © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede Imagine a cloud of electric charge drifting by. Concentrate on an infinitesimal volume V containing charge Q moving at (ordinary) velocity u : Then: QV = charge density (Coulombs/m3) And: Ju = current density (Amps/m2). A subtle, but important detail: If there is only one species (i.e. kind / type) of charge carrier contained within the infinitesimal volume V, then all charge carriers travel at the same (average / mean) speed u . However, if there are multiple species (kinds / types) of charge carriers (e.g. with different masses) and/or different signs of charge carriers contained within the the infinitesimal volume V 2 2 (e.g. electrons e with rest masses mec and protons p with rest masses mpc ) then the different constituents / species must be treated separately in the following: If N species: th Current density Juiii for the i species (i = 1, . ., N), the electric charge density ii QV NN And: JJiii u ii11 We also need to express and J in terms of the proper charge density 0 = volume charge density defined in the rest frame of the charge Q, IRF(S0). The infinitesimal rest volume / proper volume = V0 {defined in the rest/proper frame IRF(S0)} The proper charge density: QV ← Recall that electric charge Q (like c) is 00 a Lorentz invariant scalar quantity Because the longitudinal direction of motion undergoes Lorentz contraction from the rest frame IRF(S0) in the Lorentz transformation → another reference frame, e.g. lab frame IRF(S) 1 1 u Then: VV 0 where: Vwd and: Vwd , where: and: 0000 u 2 u c u 1 u If the Lorentz transformation is along (i.e. || to) the length, 0 of the infinitesimal volumes 1 Then: 0 and the components of the volumes are unchanged: ww0 , dd0 . u 1 QQ Then if: VV 0 → uu 0 Ju uu00 u u u VV0 d Recall that the 3-D vector associated with the proper velocity is: uu J 0 d 4 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede dx0 dt The zeroth (i.e. temporal/scalar) component of the proper 4-velocity is: 0 cc ddu The corresponding zeroth (i.e. temporal/scalar) component of the current density 4-vector J is: 00 Jcccc00uu 0 0 2 The current density 4-vector is: JJJcJcJJJ,,,,,x yz (SI units: Amps/m ) 0 0 Then: J 000, where: ,,uuc u u cu , u cu ,,, xyz u u constant scalar quantity J is a proper four vector, i.e. J = proper current density 4-vector. Thus: JJ JJ is a Lorentz invariant quantity. Is it ??? 2 22 1 u cu 222 JJJJ 2 222222 cuuu 2 c c c 00 uxyz 0uu22 0 0 1122 u2 cc 2 22 22 2 JJ JJ00 c 0 0 c Since: c Yes, JJ JJ is a Lorentz invariant quantity! The 3-D continuity equation mathematically expresses local conservation of electric charge (using differential vector calculus): rt, Jrt, rt, = scalar point function, JJrt , = 3-D vector point function t xˆˆ yz ˆ (in Cartesian coordinates) x yz We can also express the continuity equation in 4-vector tensor notation: n.b. 3 i 00 repeated J J y J J 11 c JJ J x z and: ( Jc0 ) indices x yz xi tc t ct x0 implies i1 summation! 33JJii00 J J 3 JJ 0 0 Then: J J 0 ii00 = t t ii11xx x x 0 xx J Thus: J or J 0 0 Continuity equation (local charge conservation) t t x © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede J Physically, note that is the 4-dimensional space-time divergence of the current density x J 4-vector JcJ , . The 4-current density JcJ , is divergenceless because 0 . x The 4-vector operator is called the 4-D gradient operator, (a.k.a the quad operator x or “quad” for short). However, because the 4-D gradient operator functions like a covariant x 4-vector, e.g. when it operates on contravariant J (or any other contravariant 4-vectors), it is often alternatively given the shorthand notation .
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