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eISSN: 2576-4543

Physics & Astronomy International Journal

Mini Review Volume 3 Issue 1

A note on the Maxwell Equations of electromagnetic irradiation in a holed spatial region

Luiz CL Botelho

Department of Applied Mathematics, Mathematics Institute, Fluminense Federal University, Brazil

Correspondence: Luiz CL Botelho, Department of Applied Mathematics, Mathematics Institute, Fluminense Federal University, Rua Mario Santos Braga, CEP 24220-140 Niter

Received: October 23, 2018 | Published: January 11, 2019

Citation: Botelho LCL. A note on the Maxwell Equations of electromagnetic irradiation in a holed spatial region. Phys Astron Int J. 2019;3(1):26-27. DOI: 10.15406/paij.2019.03.00151

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Abstract

We make relevant and important comments on the electromagnetic irradiation in a holed spatial region by writing the Maxwell equations on light of the Helmoholtz Theorem for vectorial fields on 3.

Keywords: electromagnetic irradiation, Helmholtz theorem, landau Gauge, propagator

The non space-time approach for electromagnetic irradiation in a holed spatial region-Maxwell equations

mechanical level is to consider as first and fundamental dynamical equations describing the phenomena of electromagnetic irradiation the first order system of Partial Differential Equations for the physical (measurable) strenght electromagnetic field; the full Maxwell equations on the vacuum written below:

.E=4πρ      (1a)

× H = 4π c J E + 1 c t E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuGHhis0gaWcai abgEna0kqadIeagaWcaiaai2dadaWcaaqaaiaaisdacqaHapaCaeaa caWGJbaaaiqadQeagaWcamaaBaaaleaacaWGfbaabeaakiabgUcaRm aalaaabaGaaGymaaqaaiaadogaaaWaaSaaaeaacqGHciITaeaacqGH ciITcaWG0baaaiqadweagaWcaaaa@49A6@         (1b)

div H =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGKbGaaeyAai aabAhaceWGibGbaSaacqGH9aqpcaaIWaaaaa@3D22@  (1c)

H × E + 1 c H t =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaSaacq GHxdaTceWGfbGbaSaacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGJbaa amaalaaabaGaeyOaIyRabmisayaalaaabaGaeyOaIyRaamiDaaaacq GH9aqpcqGHWaamaaa@44C4@  (1d)

In classical electrodynamics and in the situation of the existence of compact holes on the 3 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1uy0H MmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab=1risnaaCaaaleqabaGa aG4maaaakiaacYcaaaa@4413@ one must apply the Helmoholtz theorem first for the vector field ( E , H ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGabmyray aalaGaaGilaiqadIeagaWcaiaaiMcaaaa@3B8D@ , before writing the dynamical wave equations for the potentials, as we have earlier.1

Namely by a direct application of the Helmoholtz theorem to the magnetic vector field:

H =( × H H )+(grad ϕ H )+ H top MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaSaaca aI9aGaaGikaiqbgEGirBaalaGaey41aqRabmisayaalaWaaSbaaSqa aiaadIeaaeqaaOGaaGykaiabgUcaRiaaiIcaqaaaaaaaaaWdbiaabE gacaqGYbGaaeyyaiaabsgapaGaeqy1dy2aaSbaaSqaaiaadIeaaeqa aOGaaGykaiabgUcaRiqadIeagaWcamaaCaaaleqabaGaaeiDaiaab+ gacaqGWbaaaaaa@4E08@    (2)

where H top MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaSaada ahaaWcbeqaaiaabshacaqGVbGaaeiCaaaaaaa@3B9F@  is the unrotational harmonic magnetic topological field piece and ϕ H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzdaWgaa WcbaGaamisaaqabaaaaa@3A78@  the mag- netic scalar potential suppose from here on to be given explicitly as boundary conditions. On basis of the Helmoholtz theorem for the magnetic vector field eq(2), one has the identity for eq(1d)

×( E + 1 c A H t )=( 1 c t grad ϕ H ) 1 c H top t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuGHhis0gaWcai abgEna0kaaiIcaceWGfbGbaSaacqGHRaWkdaWcaaqaaiaaigdaaeaa caWGJbaaamaalaaabaGaeyOaIyRabmyqayaalaWaaSbaaSqaaiaadI eaaeqaaaGcbaGaeyOaIyRaamiDaaaacaaIPaGaaGypaiaaiIcacqGH sisldaWcaaqaaiaaigdaaeaacaWGJbaaamaalaaabaGaeyOaIylaba GaeyOaIyRaamiDaaaacaqGNbaeaaaaaaaaa8qacaqGYbGaaeyyaiaa bsgapaGaeqy1dy2aaSbaaSqaaiaadIeaaeqaaOGaaGykaiabgkHiTm aalaaabaGaaGymaaqaaiaadogaaaWaaSaaaeaacqGHciITceWGibGb aSaadaahaaWcbeqaaiaabshacaqGVbGaaeiCaaaaaOqaaiabgkGi2k aadshaaaaaaa@5F98@           (3)

Let us point out that is exactly at eq(3) that the presence of compact holes on 3 alters significantly the usual analysis to write the equations for the electromagnetic potentials. On basis of eq(3) one can only conclude that there is an electric field scalar potential ϕE such that

E+1cAHtgradϕE+ε^ϕH,Htop    (4)

Here ε^ϕH,Htop is an additional (not yet explicited) non-zero functional of the magnetic potential ϕH and of the harmonic magnetic fields Htop as solution of the first order partial differential eq (3) supposed to have non trivial solutions (a further study of this mathematical issue will appears elsewhere ).2

By substituting eq(4) and eq (2) into eq(l-b) and eq(l-a), one gets our somewhat elec- tromagnetic wave equations for the electromagnetic potentials with “topological sources” in a the “Holed” space 3.

( A H +graddiv A H )+( 1 c 2 2 t 2 A H + 1 c t grad ϕ E ) = 4π c J + 1 c t ε ^ ( ϕ H , H top ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaGikai abgkHiTiabgEGirlqadgeagaWcamaaBaaaleaacaWGibaabeaakiab gUcaRabaaaaaaaaapeGaam4zaiaadkhacaWGHbGaamizaiaaykW7ca WGKbWdaiaabMgacaqG2bGabmyqayaalaWaaSbaaSqaaiaadIeaaeqa aOGaaGykaiabgUcaRiaaiIcadaWcaaqaaiaaigdaaeaacaWGJbWaaW baaSqabeaacaaIYaaaaaaakmaalaaabaGaeyOaIy7aaWbaaSqabeaa caaIYaaaaaGcbaGaeyOaIyRaamiDamaaCaaaleqabaGaaGOmaaaaaa GcceWGbbGbaSaadaWgaaWcbaGaamisaaqabaGccqGHRaWkdaWcaaqa aiaaigdaaeaacaWGJbaaamaalaaabaGaeyOaIylabaGaeyOaIyRaam iDaaaapeGaam4zaiaadkhacaWGHbGaamiza8aacqaHvpGzdaWgaaWc baGaamyraaqabaGccaaIPaaabaGaaGypamaalaaabaGaaGinaiabec 8aWbqaaiaadogaaaGabmOsayaalaGaey4kaSYaaSaaaeaacaaIXaaa baGaam4yaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadshaaaGafq yTduMbaKGbaSaacaaIOaGaeqy1dy2aaSbaaSqaaiaadIeaaeqaaOGa aGilaiqadIeagaWcamaaCaaaleqabaGaaeiDaiaab+gacaqGWbaaaO GaaGykaaaaaa@787B@          (5-a)

ϕ E 1 c t (div A H )=4πρdiv ε ^ ( ϕ H , H top ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabgE Girlabew9aMnaaBaaaleaacaWGfbaabeaakiabgkHiTmaalaaabaGa aGymaaqaaiaadogaaaWaaSaaaeaacqGHciITaeaacqGHciITcaWG0b aaaiabgIcaOiaabsgacaqGPbGaaeODaiaadgeadaWgaaWcbaGaamis aaqabaGccqGHPaqkcqGH9aqpcqGH0aancqaHapaCcqaHbpGCcqGHsi slcaqGKbGaaeyAaiaabAhacuaH1oqzgaqcgaWcaiabgIcaOiabew9a MnaaBaaaleaacaWGibaabeaakiaaiYcaceWGibGbaSaadaahaaWcbe qaaiaabshacaqGVbGaaeiCaaaakiabgMcaPaaa@5E43@  (5-b)

By following the usual protocols, one must fix a gauge on the above set of equations for the Electromagnetic potentials ( A H , ϕ E ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcaceWGbb GbaSaadaWgaaWcbaGaamisaaqabaGccaaISaGaaGjbVlabew9aMnaa BaaaleaacaWGfbaabeaakiaaiMcaaaa@406A@ in order to make them mathematically soluble.2

We choose thus the well-known and decoupling (and free of mathematical problem) Landau Gauge

div A H + 1 c ϕ E t =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabsgacaqGPb GaaeODaiqadgeagaWcamaaBaaaleaacaWGibaabeaakiabgUcaRmaa laaabaGaaGymaaqaaiaadogaaaWaaSaaaeaacqGHciITcqaHvpGzda WgaaWcbaGaamyraaqabaaakeaacqGHciITcaWG0baaaiabg2da9iaa icdaaaa@47B8@  (6)

After implementing this gauge fixing, one gets the dynamical equations satisfied by the Electromagnetic potentials in the gauge eq(6).

Δ A H 1 c 2 A H t 2 = 4π c J E 1 c t ε ^ ( ϕ H , H top ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgs5aejqadg eagaWcamaaBaaaleaacaWGibaabeaakiabgkHiTmaalaaabaGaaGym aaqaaiaadogaaaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaa GcceWGbbGbaSaadaWgaaWcbaGaamisaaqabaaakeaacqGHciITcaWG 0bWaaWbaaSqabeaacaaIYaaaaaaakiaai2dacqGHsisldaWcaaqaai aaisdacqaHapaCaeaacaWGJbaaaiqadQeagaWcamaaBaaaleaacaWG fbaabeaakiabgkHiTmaalaaabaGaaGymaaqaaiaadogaaaWaaSaaae aacqGHciITaeaacqGHciITcaWG0baaaiqbew7aLzaajyaalaGaaGik aiabew9aMnaaBaaaleaacaWGibaabeaakiaaiYcaceWGibGbaSaada ahaaWcbeqaaiaabshacaqGVbGaaeiCaaaakiaaiMcaaaa@5DB4@                (7-a)

Δ ϕ E 1 c 2 2 ϕ E t 2 =4πρ+div ε ^ ( ϕ H , H top ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgs5aejabew 9aMnaaBaaaleaacaWGfbaabeaakiabgkHiTmaalaaabaGaaGymaaqa aiaadogadaahaaWcbeqaaiaaikdaaaaaaOWaaSaaaeaacqGHciITda ahaaWcbeqaaiaaikdaaaGccqaHvpGzdaWgaaWcbaGaamyraaqabaaa keaacqGHciITcaWG0bWaaWbaaSqabeaacaaIYaaaaaaakiaai2dacq GHsislcaaI0aGaeqiWdaNaeqyWdiNaey4kaSIaaeizaiaabMgacaqG 2bGafqyTduMbaKGbaSaacaaIOaGaeqy1dy2aaSbaaSqaaiaadIeaae qaaOGaaGilaiqadIeagaWcamaaCaaaleqabaGaaeiDaiaab+gacaqG WbaaaOGaaGykaaaa@5CA1@               (7-b)

We note the presence of news effectives electrical current and electrical charge densities on the eq(7-a) and eq(7-b) due to the ( (harmonic” pieces of the magnetic field.

Let us stress again that the topological harmonic magnetic configuration Htop  on the Helmoholtz decomposition eq(2) must be given as boundary conditions and must be suppose fully explicited.

As a result of eq(l-c), now with a non trivial magnetic source Jm for sake of generality, one obtains that:

div(× A H +grad ϕ H + H top )=div J n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabsgacaqGPb GaaeODaiaaiIcacqGHhis0cqGHxdaTceWGbbGbaSaadaWgaaWcbaGa amisaaqabaGccqGHRaWkqaaaaaaaaaWdbiaadEgacaWGYbGaamyyai aadsgapaGaeqy1dy2aaSbaaSqaaiaadIeaaeqaaOGaey4kaSIabmis ayaalaWaaWbaaSqabeaacaqG0bGaae4BaiaabchaaaGccaaIPaGaaG ypaiaabsgacaqGPbGaaeODaiqadQeagaWcamaaBaaaleaacaWGUbaa beaaaaa@53BD@     (8)

As a result one has

ϕ H = Δ 1 (div( H top + J m )) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMnaaBa aaleaacaWGibaabeaakiaai2dacqGHsislcqGHuoardaahaaWcbeqa aiabgkHiTiaaigdaaaGccaaIOaGaaeizaiaabMgacaqG2bGaaGikai qadIeagaWcamaaCaaaleqabaGaaeiDaiaab+gacaqGWbaaaOGaey4k aSIabmOsayaalaWaaSbaaSqaaiaad2gaaeqaaOGaaGykaiaaiMcaaa a@4C57@                 (9)

Where Δ 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgs5aenaaCa aaleqabaGaeyOeI0IaaGymaaaaaaa@3B5B@  is the Green function of the Laplacean operator on the 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xhHi1aaWbaaSqabeaa caaIZaaaaaaa@43C1@  with compac holes inside, certainly a highly non trivial problem even on the “Riemann surface” case of 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xhHi1aaWbaaSqabeaa caaIYaaaaaaa@43C0@ .2

Another useful relation is obtained from eq(3) and the unrotationality of H top MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaSaada ahaaWcbeqaaiaabshacaqGVbGaaeiCaaaaaaa@3B9F@

×( E + 1 c A t )=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgEGirlabgE na0kabgEGirlabgIcaOiqadweagaWcaiabgUcaRmaalaaabaGaaGym aaqaaiaadogaaaWaaSaaaeaacqGHciITceWGbbGbaSaaaeaacqGHci ITcaWG0baaaiabgMcaPiabg2da9iaaicdaaaa@48DA@  (10)

which means that again under the imposed condition rot ( H top )0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcaceWGib GbaSaadaahaaWcbeqaaiaabshacaqGVbGaaeiCaaaakiaaiMcacqGH HjIUcaaIWaaaaa@3FF9@ , the following equation

Δ( E + 1 c A t )+γραδ(div( E + 1 c A t ))=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabgs 5aejabgIcaOiqadweagaWcaiabgUcaRmaalaaabaGaaGymaaqaaiaa dogaaaWaaSaaaeaacqGHciITceWGbbGbaSaaaeaacqGHciITcaWG0b aaaiabgMcaPiabgUcaRabaaaaaaaaapeGaeq4SdCMaeqyWdiNaeqyS deMaeqiTdqMaaGPaVlabgIcaO8aacaqGKbGaaeyAaiaabAhacqGHOa akceWGfbGbaSaacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGJbaaamaa laaabaGaeyOaIyRabmyqayaalaaabaGaeyOaIyRaamiDaaaacqGHPa qkcqGHPaqkcqGH9aqpcaaIWaaaaa@5DB0@  (11)

It is now worth call attention for the standard gradient solution of eq(11)

E + 1 c A t =(grad ϕ E ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadweagaWcai abgUcaRmaalaaabaGaaGymaaqaaiaadogaaaWaaSaaaeaacqGHciIT ceWGbbGbaSaaaeaacqGHciITcaWG0baaaiaai2dacqGHsislcaaIOa aeaaaaaaaaa8qacaWGNbGaamOCaiaadggacaWGKbWdaiabew9aMnaa BaaaleaacaWGfbaabeaakiaaiMcaaaa@49FF@               (12)

Note that under the solution eq(12) the usual dynamical equations for the electromagnetic potentials eq(7-a) and eq(7-b) with ε ^ ( ϕ H , H top )0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbew7aLzaajy aalaGaaGikaiabew9aMnaaBaaaleaacaWGibaabeaakiaaiYcaceWG ibGbaSaadaahaaWcbeqaaiaabshacaqGVbGaaeiCaaaakiaaiMcacq GHHjIUcaaIWaaaaa@4542@ , are obtained.

Let us thus point at that on the quantum mechanical world the first basic equations are the potential wave equations eq(5-a) and eq(5-b) with ε ^ ( ϕ H , H top )0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbew7aLzaajy aalaGaaGikaiabew9aMnaaBaaaleaacaWGibaabeaakiaaiYcaceWG ibGbaSaadaahaaWcbeqaaiaabshacaqGVbGaaeiCaaaakiaaiMcacq GHHjIUcaaIWaaaaa@4542@ , with the meaning that on the quantum world, the space is fully 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xhHi1aaWbaaSqabeaa caaIZaaaaaaa@43C1@ , without any non trivial differ- ential topological structure immersed on it for mathematical consistency of its dynamical equations and the physical assumptions in Quantum Mechanics-non relativistic case. As an pedagogical comment we write the path integral expression for the mild point rule Feynman propagator of a charged particle in the presence of a magnetic constant, but Gaussian random potential A ˜ H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadgeagaacam aaBaaaleaacaWGibaabeaaaaa@39ED@  and zero magnetic field H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaSaaaa a@3896@ :

B A = A 3 y 2 i + A 3 x 2 j Helmoholtz vector MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGcbGbaSaada WgaaWcbaGaamyqaaqabaGccaaI9aGaeyOeI0YaaGraaeaadaqadaqa amaalaaabaGaamyqamaaBaaaleaacaaIZaaabeaakiaadMhaaeaaca aIYaaaaaGaayjkaiaawMcaaiqadMgagaWcaiabgUcaRmaabmaabaWa aSaaaeaacaWGbbWaaSbaaSqaaiaaiodaaeqaaOGaamiEaaqaaiaaik daaaaacaGLOaGaayzkaaGabmOAayaalaaaleaaqaaaaaaaaaWdbiaa dIeacaWGLbGaamiBaiaad2gacaWGVbGaamiAaiaad+gacaWGSbGaam iDaiaadQhacaqGGaGaamODaiaadwgacaWGJbGaamiDaiaad+gacaWG Ybaak8aacaGL34paaaa@5A15@               (13)

A H = A 3 K H 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGabmyqay aalaWaaSbaaSqaaiaadIeaaeqaaOGaaGypaiaadgeadaWgaaWcbaGa aG4maaqabaGcceWGlbGbaSaaaeaaceWGibGbaSaacqGHHjIUcaaIWa aaaaa@40C5@

Namelly:

G(x1,x2,T,[AH])=xx1xTx2DF[xσ)]exp(i0T12m(σ)dσ)×exp{ie2c0Tdσ[dxidσEijs(xjBAsxl))](σ)} 

We can see that the averaged Feynman propagator with x ==( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWG4bGbaSaacq GH9aqpcaaI9aGaaGikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaaI SaGaaGjbVlaadIhadaWgaaWcbaGaaGOmaaqabaGccaaISaGaaGjbVl aadIhadaWgaaWcbaGaaG4maaqabaGccaaIPaaaaa@464B@ and y =( y 1 , y 2 , y 3 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaarqqr1ngBPrgifH hDYfgaiuaacqWFaCFEceWG5bGbaSaacqGH9aqpcaaIOaGaamyEamaa BaaaleaacaaIXaaabeaakiaaiYcacaaMe8UaamyEamaaBaaaleaaca aIYaaabeaakiaaiYcacaaMe8UaamyEamaaBaaaleaacaaIZaaabeaa kiaaiMcaaaa@4C19@  is explicitly given by

E A {G( x , y t,[ A H ])}= G 0 (( x 1 , x 2 );( y 1 , y 2 ),T) × E A 3 e ie c A 3 0 T dσ dz dσ = G 0 (( x 1 , x 2 );( y 1 , y 2 ),T)× ( π μ 2 )exp {[ 1 2 μ 2 e 2 c 2 2 ( x 3 y 3 ) 2 ]} dampingfactor MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadweada WgaaWcbaGaamyqaaqabaGccqGH7bWEcaWGhbGaaGikamaaFiaabaGa aeiEaaGaay51GaGaaGilamaaFiaabaGaaeyEaaGaay51GaGaaGjbVl aadshacaaISaGaaGjbVlaaiUfadaWhcaqaaiaadgeaaiaawEniamaa BaaaleaacaWGibaabeaakiaai2facaaIPaGaaGyFaiaai2dacaWGhb WaaSbaaSqaaiaaicdaaeqaaOGaaGikaiaaiIcacaWG4bWaaSbaaSqa aiaaigdaaeqaaOGaaGilaiaaysW7caWG4bWaaSbaaSqaaiaaikdaae qaaOGaaGykaiaaiUdacaaIOaGaamyEamaaBaaaleaacaaIXaaabeaa kiaaiYcacaaMe8UaamyEamaaBaaaleaacaaIYaaabeaakiabgMcaPi abgYcaSiaaysW7caWGubGaaGykaaqaaiaaxMaacqGHxdaTcaWGfbWa aSbaaSqaaiaadgeadaWgaaqaaiaaiodaaeqaaaqabaGcdaGadaqaai aadwgadaahaaWcbeqaaiabgkHiTmaalaaabaGaamyAaiaadwgaaeaa caWGJbWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacq WFpeY=aaaaaOGaamyqamaaBaaaleaacaaIZaaabeaakmaapedabeWc baGaaGimaaqaaiaadsfaa0Gaey4kIipakiaadsgacqaHdpWCdaWcaa qaaiaadsgacaWG6baabaGaamizaiabeo8aZbaaaiaawUhacaGL9baa aeaacaWLjaGaaGypaiaadEeadaWgaaWcbaGaaGimaaqabaGccaaIOa GaaGikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaaISaGaaGjbVlaa dIhadaWgaaWcbaGaaGOmaaqabaGccaaIPaGaaG4oaiaaiIcacaWG5b WaaSbaaSqaaiaaigdaaeqaaOGaaGilaiaaysW7caWG5bWaaSbaaSqa aiaaikdaaeqaaOGaeyykaKIaeyilaWIaaGjbVlaadsfacaaIPaGaey 41aqlabaGaaCzcaiaaiIcadaGcaaqaaiabec8aWjabeY7aTnaaCaaa leqabaGaaGOmaaaaaeqaaOGaaGykaiGacwgacaGG4bGaaiiCamaaye aabaGaaG4EaiabgkHiTiaaiUfadaWcaaqaaiaaigdaaeaacaaIYaaa aiabeY7aTnaaCaaaleqabaGaaGOmaaaakmaalaaabaGaamyzamaaCa aaleqabaGaaGOmaaaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaGc cqWFpeY=daahaaWcbeqaaiaaikdaaaaaaOGaaGikaiaadIhadaWgaa WcbaGaaG4maaqabaGccqGHsislcaWG5bWaaSbaaSqaaiaaiodaaeqa aOGaeyykaKYaaWbaaSqabeaacaaIYaaaaOGaaGyxaiaai2haaSqaai aadsgacaWGHbGaamyBaiaadchacaWGPbGaamOBaiaadEgacaaMc8Ua amOzaiaadggacaWGJbGaamiDaiaad+gacaWGYbaakiaawEJ=aaaaaa@D3AE@ (15)

Here G0((x1,x2),(y1,y2),T)  denotes the free propagation on the plane and 2 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1uy0H MmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab=1risnaaCaaaleqabaGa aGOmaaaakiaac6caaaa@4414@

It is curious that the “damped” Ahronov-Rhom effect predicts by eq(12) means dissipation on quantum mechanics by a zero classical magnetic field but with a random constant vector potential and non-zero Helmoholtz potential vector, a new physical quantum phenomena predicts by us.

Finally we point out for sake of comparasion with eqs(7a−7b) that the Maxwell Equations eq(l-a)-eq(l-d) take the following form, if one uses directly the Helmoholtz theorem for the electric and magnetic fields

E =(× A E )+(grad ϕ E )+ E top MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGfbGbaSaaca aI9aGaaGikaiabgEGirlabgEna0kqadgeagaWcamaaBaaaleaacaWG fbaabeaakiaaiMcacqGHRaWkcaaIOaGaam4zaiaadkhacaWGHbGaam izaiabew9aMnaaBaaaleaacaWGfbaabeaakiaaiMcacqGHRaWkceWG fbGbaSaadaahaaWcbeqaaiaabshacaqGVbGaaeiCaaaaaaa@4DBC@         (16-a)

H =(× A H )+(grad ϕ H )+ H top MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaSaaca aI9aGaaGikaiabgEGirlabgEna0kqadgeagaWcamaaBaaaleaacaWG ibaabeaakiaaiMcacqGHRaWkcaaIOaGaam4zaiaadkhacaWGHbGaam izaiabew9aMnaaBaaaleaacaWGibaabeaakiaaiMcacqGHRaWkceWG ibGbaSaadaahaaWcbeqaaiaabshacaqGVbGaaeiCaaaaaaa@4DC8@       (16-b)

The set of our proposed news equations for the Electromagnetic Field for a Holed space 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xhHi1aaWbaaSqabeaa caaIZaaaaaaa@418A@ takes the form (plus suitable boundary conditions)(*)

Δ ϕ E =4πρdiv E top MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHuoarcqaHvp GzdaWgaaWcbaGaamyraaqabaGccqGH9aqpcaaI0aGaeqiWdaNaeqyW diNaeyOeI0IaaeizaiaabMgacaqG2bGabmyrayaalaWaaWbaaSqabe aacaqG0bGaae4Baiaabchaaaaaaa@48C5@  (17-a)

Δ ϕ H =div( H top ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHuoarcqaHvp GzdaWgaaWcbaGaamisaaqabaGccaaI9aGaeyOeI0IaaeizaiaabMga caqG2bGaaGikaiqadIeagaWcamaaCaaaleqabaGaaeiDaiaab+gaca qGWbaaaOGaaGykaaaa@45C0@                           (17-b)

Δ A H +grad(div A H )= 4π c J + 1 c t (rot A E +grad ϕ E + E top ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislcqGHuo arceWGbbGbaSaadaWgaaWcbaGaamisaaqabaGccqGHRaWkcaWGNbGa amOCaiaadggacaWGKbGaaiikaiaabsgacaqGPbGaaeODaiqadgeaga WcamaaBaaaleaacaWGibaabeaakiaaiMcacaaI9aWaaSaaaeaacaaI 0aGaeqiWdahabaGaam4yaaaaceWGkbGbaSaacqGHRaWkdaWcaaqaai aaigdaaeaacaWGJbaaamaalaaabaGaeyOaIylabaGaeyOaIyRaamiD aaaacaaIOaGaamOCaiaad+gacaWG0bGaaGPaVlqadgeagaWcamaaBa aaleaacaWGfbaabeaakiabgUcaRiaadEgacaWGYbGaamyyaiaadsga cqaHvpGzdaWgaaWcbaGaamyraaqabaGccqGHRaWkceWGfbGbaSaada ahaaWcbeqaaiaabshacaqGVbGaaeiCaaaakiaaiMcaaaa@65DE@ (17-c)

Δ A H +grad(div A E )= 1 c t (rot A H +grad ϕ H + H top ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislcqGHuo arceWGbbGbaSaadaWgaaWcbaGaamisaaqabaGccqGHRaWkcaWGNbGa amOCaiaadggacaWGKbGaaiikaiaabsgacaqGPbGaaeODaiqadgeaga WcamaaBaaaleaacaWGfbaabeaakiaaiMcacaaI9aWaaSaaaeaacaaI XaaabaGaam4yaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadshaaa GaaGikaiaadkhacaWGVbGaamiDaiaaykW7ceWGbbGbaSaadaWgaaWc baGaamisaaqabaGccqGHRaWkcaWGNbGaamOCaiaadggacaWGKbGaeq y1dy2aaSbaaSqaaiaadIeaaeqaaOGaey4kaSIabmisayaalaWaaWba aSqabeaacaqG0bGaae4BaiaabchaaaGccaaIPaaaaa@60AE@          (17-d)

Usually one considers the mathematically well define A. Sommerfeld irradiation conditions at the spatial infinity.3

In this case E top MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGfbGbaSaada ahaaWcbeqaaiaabshacaqGVbGaaeiCaaaaaaa@3B9C@ and H top MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaSaada ahaaWcbeqaaiaabshacaqGVbGaaeiCaaaaaaa@3B9F@ should be prescribed as boundary conditions for possible well-posedness of the irradiation problem associated to the above written set of PDE’s. A complete mathematical and physical study will appears elsewhere in a more detailed paper with applications.

Acknowledgments

We are thankfull to CNPq for a fellowship and to Professor José Helayel-CBPF and W. Rodrigues-IMEC-UNICAMP.

Conflict of interest

Authors declare there is no conflict of interest.

References

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