ABOUT THE LORENTZ RECIPROCITY RELATION IN ELECTROMAGNETIC FIELDS

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1 ABOUT T LORNTZ RIPROITY RLATION IN LTROMAGNTI FILD UGO ROMAN Deicate to prof. ANDRI ŢUGULA member of the Acaemy on his birthay Key wors: ectromagnetic fie armonic steay-state Maxwe s equations Vectoria ientities Lorentz reciprocity reation. Noe reciprocity reations simiar to that obtaine by Lorentz in eectromagnetic fies eoing in harmonic steay-state in inear motioness isotropic an with osses meium are estabishe.. INTRODUTION Let be an eectromagnetic fie which eoes in harmonic steay-state in a inear motioness isotropic an with osses meium. If two istinct harmonic steay-states of this fie characterize by the compex ectors i i are consiere Maxwe s equations referring to each of these states are [] ( ) jω jω σ i () ( ) jω. jω σ i () Aso D D B 0 B 0 (3) are the state equations an Gh. Asachi Technica Uniersity Jassy -mai: ai_aru005@yahoo.com Re. Roum. ci. Techn. Éectechn. et Énerg p Bucarest 008

2 54 ugo Rosman D D J J σ σ ( i) B ( ) B i (4) represent the materia equations of the eectromagnetic fie in the two consiere states. ere σ are the compex materia constants of the meium [] consiere with osses the rest of notations being the usua ones. These equations may be coupe successiey with the foowing ectoria ientities [3] ( M N ) N M M N ( MN ) ( M ) N ( N ) M M N N ( M N ) M N N M ( N ) M ( M ) N (5) M (6) (7) where M N are rea or compex ectors. If eqs. () (4) are coupe with ectoria ientity (5) the Lorentz reciprocity reation is obtaine making successiey the substitutions M N M N (8) namey [4] ( ) σ( ) i i (9) ( ) A σ ( ) i i (0) representing the oca an the goba form of this reation. The aim of this paper is to connect eqs. () (4) to ectoria ientities (6) (7) obtaining by this way noe Lorentz type reciprocity reations usefu in the stuy of an eectromagnetic fie.. FIRT RIPROITY RLATION In orer to obtain the oca an goba form of this reation the substitutions (8) into ectoria ientity (6) are successiey performe obtaining ( ) ( ) ( ) σ ( ) jω i jω ()

3 3 About the Lorentz reciprocity reation in eectromagnetic fies 55 ( ) ( ) ( ) σ ( ) jω i jω () where eqs. () an () were taken into account. Aing reations () an () it resuts ( ) ( ) ( ) ( ) ( ) σ( ). i i xpression (3) may be consiere as representing the oca form of the first Lorentz type reciprocity reation in eectromagnetic fie. As regars the goba form of the same reation is aantageous to appy beforehan the operator in both members of reation (3) ( ) [ ( ) ( ) ( ) ( ) ] σ( ). i i aing in iew the ectoria ientity (5) an eqs. ( ) ( ) it resuts finay ( ) [ ( ) ( ) ( ) ( ) ] σ jω ( ) i i After that it is necessary to integrate expression (5) in the oume boune by the cose surface namey ( ) A [( ) [( ) ( ) ( ) ] A jωσ( ) σ ( ). i i ]. [ i i i i. xpression (6) represents the goba form of the first Lorentz type reciprocity reation in the stuie eectromagnetic fie. (3) (4) (5) (6) 3. OND RIPROITY RLATION As in the preious section the substitutions (6) are performe this time in ectoria ientity (7) obtaining

4 56 ugo Rosman 4 ( ) ( ) ( ) (7) ( ) ( ) ( ). (8) Using the ectoria ientity [3] ( M N ) M N N M (9) where M an N eqs. (3) an (4) ea to ( ) ( ). (0) onsequenty reations (7) an (8) become ( ) () ( ) ( ) ( ) ( ) ( ). Aing reations () an () it resuts () ( ) ( ) ( ) ( ) ( ) ( ) ( ) (3) which can be consiere as representing the oca form of the secon Lorentz type reciprocity reation. It is usefu to obsere that compex ectors

5 5 About the Lorentz reciprocity reation in eectromagnetic fies 57 ( ) ( ) ( ) ( ) (4) are each of them of soenoia type ( 0). The goba form of this secon Lorentz type reciprocity reation may be obtaine using the same proceeing as that utiize in the preious section of this paper concerning the goba form of the first Lorentz type reciprocity reation. Beforehan reation (3) is written in the form ( ) ( ) ( ) ( ) ( ) ( ) [ ]. (5) Finay reation ( ) ( ) ( ) ( ) ( ) ( ) [ ] A A (6) is obtaine. Another goba form of this reation may be written as ( ) ( ) ( ) [( ) ( ) ( ) ( ) ]. A (7) where is an open surface boune by the cose cure. 4. PARTIULAR A When either the meium is a perfect insuator ( ) 0 σ or this one is homogeneous ( ) 0 i i reations (3) an (6) become

6 58 ugo Rosman 6 ( ) ( ) ( ) ( ) ( ) (8) ( ) A [( ) ( ) ( ) ( ) ] A In the same time when the meium is homogeneous ( 0) reations (3) (6) (7) become ( ) ( ) ( ) ( ) ( ) ( ) (9) (30) ( ) A [( ) ( ) ( ) ( ) ] A ( ) ( ) [( ) ( ) ( ) ( ) ] A. A In the same particuar case ectors (4) are of the form ( ) ( ) ( ) ( ) (3) (3) (33) Reations (30) (3) (3) (33) become more simper when moreoer ONLUION Utiizing Maxwe s equations of an eectromagnetic fie which eoes in harmonic steay-state in a inear motioness isotropic non-homogeneous an

7 7 About the Lorentz reciprocity reation in eectromagnetic fies 59 with osses meium as we as certain ectoria ientities some new reciprocity reations are estabishe simiar to that obtaine by Lorentz. An eient connection between those reciprocity reations an those obtaine by m. Nicoau [5] [6] or by us in a preious paper [7] may be obsere. Receie on 8 December 007 RFRN.. Rosman. Petrescu Bazee teoriei câmpuui eectromagnetic III ectromagnetismu Gh. Asachi Tehn. Uni. Iaşi m. Nicoau Propagarea uneor eectromagnetice I it. Acaemiei Bucharest I. Biju. óos P.P. TeoorescuTehnici e cacu ectoria cu apicaţii it. Tehnică Bucharest A. Lorentz Theory of ectrons an its Appication to the Phenomena of Light an Raiant eat Doer Pub. Inc. NY m. Nicoau Reaţii e reciprocitate şi e conserare în eectricitate Bu. şt. Aca. Rom. Şt. matem. şi fiz. IV 4 (95). 6. m. Nicoau Generaizarea unor reaţii e reciprocitate şi conserare în eectricitate t. şi cerc. e fiz. V - pp (954). 7.. Rosman ur queques nouees reations e réciprocité en champs éectromagnétiques Bu. Inst. Poitehnic Iaşi L(LIV) 3-4 s. ect. nerg. ectron. pp. 7 3 (004).