Applications of the Dirac Sequences in Electrodynamics

Transcription

1 Poc of the 8th WSEAS It Cof o Mathematical Methods ad Computatioal Techiques i Electical Egieeig Buchaest Octobe Applicatios of the Diac Sequeces i Electodyamics WILHELM W KECS Depatmet of Mathematics Uivesity of Petosai Uivesitatii Steet 6 Petosai ANTONELA TOMA Depatmet of Mathematics II Uivesity Politehica of Buchaest Splaiul Idepedetei Steet 64 Buchaest MARIA DOBRIŢOIU Depatmet of Mathematics Uivesity of Petosai Uivesitatii Steet 6 Petosai Abstact: We established a method of costuctig the Diac sequeces O the basis of this method we costuct some Diac sequeces with applicatio i electical egieeig Key-Wods: distibutios theoy geealized fuctios Diac sequeces Itoductio The theoy of distibutios (geealized fuctios epesets a geeal ad uitay backgoud egadig the mathematical epesetatio of some physical quatities ad the aalysis of some discotiuous pheomea The Diac sequeces have impotat applicatios i the epesetatio of the physical quatities with puctual suppot as well as i solvig of bouday value poblems fom mathematical-physics We metio that [4] ad [7] have applied the distibutio theoy i the mechaics of the defomable solid ad electical egieeig ad also applicatio of Diac sequeces have bee give by [] [] [6] [5] [] ad othe i the electical egieeig We shall coside some applicatio of the Diac sequeces i electical egieeig ad electodyamics We shall deote by D( R the Schwatz s space of idefiitely diffeetiable fuctios with compact suppot ad by D ( R the set of liea cotiuous fuctioals defied o D( R amed as distibutios Geeal esults Defiitio Let f : R R > be a family of locally itegable fuctios ( f L ( R loc We say that the fuctios f fom a epesetative Diac family o Diac sequece if i the sese of the covegece of D ( R lim f ( x δ ( x ( This meas that ϕ D( R lim f ( x ϕ( x δ( x ϕ( x ϕ( ( If ( ( C R f ( the fom ( we obtai lim D f ( x D δ ( x ( f ( x x whee D f ( x epesets the x x x patial deivative of ode + + of the fuctio f If the eal fuctio ψ x is cotiuously i the viciity of the poit a the ψ ( x δ( x a ψ( a δ( x a (4 which epesets the filtat popety of the Diac distibutio δ ( x a D ( cocetated at the poit a Paticulaly xδ ( x Cotiuous fuctios with cetai popeties allow the costuctio of the Diac sequeces Thus accodig to [4] p 6 we ca state Popositio Let f C ( R f : R R be with the popety R f ( x The the family of fuctios f > havig the expessio x f ( x f x > R (5 fom a Diac sequeces hece lim f ( x δ ( x Fo f > we shall give the followig geealizatio of the popositio :

2 Poc of the 8th WSEAS It Cof o Mathematical Methods ad Computatioal Techiques i Electical Egieeig Buchaest Octobe Popositio Let g : R R > g ( x x h( f ( x h( f be a family of fuctios whee h( is a cotiuously fuctio with the fiite limit lim h( a ad f > The x h( lim g ( x lim f δ ( x a (6 Poof Fo ay ϕ D( x h( ( f ( x h( ϕ( x f ϕ( x Makig the chage of vaiable x u+ h( the Jacobia of the tasfomatio is x x x u u u ( x x Ju ( ( u u x x x u u u ad thus ( ( ( ( ( ( + ( f x h ϕ x f u ϕ u h du f( u [ ϕ( u+ h( ϕ( a ] du+ ϕ( a (7 whee a lim h( O the othe had sice f( u du is fiite [ ϕ( + ϕ ] f ( u u h( ( a du ( sup ϕ u+ h( ϕ( a wheefom o the basis of cotiuity of the fuctio ϕ ( u+ h( D( it yields hece [ ϕ( ϕ ] lim f( u u+ h( ( a du [ ϕ( ϕ ] lim f( u u+ h( ( a du Cosequetly fom (7 lim f x h( ϕ( x ϕ( a δ( x a ϕ( x ( ( ( hece lim f ( x h( δ( x a QED Example Let be the fuctio f : f( x We ote that ( x + f C ( f ( x f( x ad f ( x f ( x ( x + Makig the substitutio x sih u u we obtai du f( x tahu cosh u Sice the coditios fom the popositio ae fulfilled we obtai the Diac sequeces x f ( x f x > ( x + ad thus we ca wite lim ( lim f x δ ( x (8 ( x + We coside the fuctio h( vt ct( whee c > t ae costats We have lim h( ct ad h C [] ( O the basis of the popositio it yields lim f ( xvt lim δ ( xct ( xvt + We emak that the family of Diac sequeces f ( x vt > x (9 ( x vt + is used i electodyamics [] to the study of the electic ad magetic fields geeated by a poit chage q of mass m movig with costat velocity v alog the x axis if the paticle is at the oigi ( at t Example Let g : be a fuctio whee gx ( a> a ( la x + x + a We ote that g C ( g( x g( x ad thus g( x gx ( l a x + x + a

3 Poc of the 8th WSEAS It Cof o Mathematical Methods ad Computatioal Techiques i Electical Egieeig Buchaest Octobe Sice obtai x + x + a l + C we x + a a x+ x + gx ( la l a x+ x + a Cosequetly the fuctio g satisfies the coditios of the popositio hece the family of fuctios x g ( x g la x + x + a epesets a Diac sequeces Thus we ca wite lim g ( x lim δ ( x la x + x + a Applig the popositio we obtai lim g ( x vt lim la ( x vt + δ ( x ct ( x vt a + We emak that this esult was used i [] egadig the study of the gavitatioal field of a massless paticle i geeal elativity whe v c Example We coside the fuctio f : whee f( x cosh x Obviously f C ( ad f ( x f( x We have f( x f( x tahx cosh x Sice the coditios of the popositio ae satisfied it esults that the family of fuctios f ( x x > cosh x epesets a Diac sequece Cosequetly we ca wite the elatios lim f ( x lim δ ( x ( cosh x lim f ( x ct lim δ ( x ct ( cosh x ct The study of electomagetic wave pulses [] uses the fomula ( ad ( substitutig Thus lim δ ( x cosh x lim δ ( x ct cosh x ( ct We emak that the Diac sequece h ( x is obtaied fom the fuctios cosh x sequece H ( x ( + tah x by diffeetiatio Ideed H ( x h ( x cosh x Applicatios i electodyamics Let q be a poit electical chage i vacuum placed at the oigi of the othogoal efeece Oxyz If at the poit M O is placed a electical chage + the accodig to Coulomb law the foce which acts o the chage + is q q E E OM x + y +z ( whee epesets the dielectic costat of vacuum By defiitio the vectoial fuctio E : \{} give by the fomula ( is amed the itesity of the electostatic field coespodig to poit electical chage q placed at the oigi of the efeece Oxyz We ote that the vectoial fuctio E is a locally itegable fuctio sice Ω the itegal dv dv dydz exists ad it is fiite Ω Cosequetly the itesity E of the electical field geeates a fuctio type distibutio fom D ( By defiitio the fuctio V : \{} give by the fomula q V( x y z x + y + z (4 is amed the electostatic field potetial coespodig to the chage q placed at the oigi The potetial fuctio V is also a locally itegable fuctio which detemies a fuctio type distibutio fom D (

4 Poc of the 8th WSEAS It Cof o Mathematical Methods ad Computatioal Techiques i Electical Egieeig Buchaest Octobe Betwee the itesity E of the electostatic field ad the potetial V the elatio E gadv (5 Geeally if we deote by the volume desity of the electical chage epeseted by a distibutio fom D ( ad with V the field potetial the the quatities EV cosideed distibutios fom D ( satisfy the Poisso equatio Δ V (6 hece div E div( gadv Δ V The Eqs (5 ad (6 cosideed i the distibutios space costitute the fudametal equatios of the electostatics We emak that the Eqs (5 ad (6 ca be paticulaly applied o o I these cases epesets the liea ad the suface desity of the electical chage espectively We will study the case whe the electical chage q is movig liea with costat velocity v Let q be a poit electical chage of mass m movig with costat velocity v alog the x axis We admit that the electical chage q at t is at the oigi O The [] the electic field E ad the magetic field B at ay poit M ( xyz ad time t > have the expessios q E ( x vt i + yj zk + (7 μ qv B zj + yk (8 whee v ( x vt + y + z (9 c ad μ epesets the magetic pemeability of vacuum Fom (8 ad (9 we obtai B ( μ vi E ( Fom above it esults that the magetic field B is pepedicula o the i diectio of the axis x ad also o the electic field E A vey impotat poblem is the behaviou of the electic ad magetic fields whe v c c beig the speed of light i vacuum The limit case v c is equivalet with Fom the expessios of the fields E ad B it esults that lim E ad lim B do t exist i the classical sese We shall show that these limits exist i the geealized sese that meas i the distibutios space Sice the electical chage q is movig alog the Ox-axis with costat velocity v the coodiates y ad z beig costats the quatities E ad B ae vectoial distibutios with compoets fom D ( with espect to the vaiable x Fom (7 ad (8 it esults that q E lim E ( x ct i + yj + zk lim ( qμ c B lim B zj + yk lim ( Popositio Deotig y + z > we have lim lim δ ( x ct ( Poof Takig ito accout (9 ( x vt + ( x vt + (4 Deotig ad sice + it esults Thus the expessio (4 becomes > ( x vt + Cosequetly o the basis of the fomula ( lim lim lim ( x vt + δ ( x ct QED Applyig this esult ad takig ito accout ( ad ( we obtai q E lim E ( x ct i + yj + zk δ ( x ct qμ c B lim B zj + yk δ ( x ct whee y + z

5 Poc of the 8th WSEAS It Cof o Mathematical Methods ad Computatioal Techiques i Electical Egieeig Buchaest Octobe Sice uδ ( u it esults that ( x ct δ ( x ct ad thus we ca wite q E lim E δ ( x ct( yj + zk π q c B lim B μ δ ( x ct zj yk π + Betwee the fields E ad B the elatio B μci E which shows that the magetic field B is pepedicula o Ox-axis diectio as well as o the electic field E Takig ito accout the esults show i [] the Diac sequece h ( x ct is used cosh ( x ct fo studyig the electomagetic wave pulses Thus the electic fields of thee pulses ae give by the expessios E ( xt Ah( x ct E ( xt Bh ( x ct E ( xt Ch ( x ct The eegy desity of a plae electomagetic wave i vacuum is give by W E which obviously is witte usig the geealized fuctios h ( x ct h ( x ct h ( x ct [] Aguiegabiia JM Headez A Rivas M δ-fuctio covegig sequeces Am J Phys Vol 7 No pp8-85 [] Coste Robet C Poducts of some geealized fuctios NASA Techical Note D [4] Kecs Wilhelm W Theoy of distibutios with applicatios (i omaia Editua Academiei Romae [5] Kecs Wilhelm W Applicatios of the Diac sequeces i mechaics Acta Uivesitatis Apulesis Mathematics-Ifomatics Poceedigs of the Iteatioal Cofeece of Theoy ad Applicatios of Mathematics ad Ifomatics ICTAMI 4 No pp [6] Skie R Weil JA A itoductio to geealized fuctios ad thei applicatio to static electomagetic poit dipoles icludig hypefie iteactios Ameica Joual of Physics pp777-9 [7] Toma Atoela Mathematical methods i elasticity ad viscoelasticity (i omaia Editua Tehică Bucuesti 4 4 Coclusios The distibutios theoy costitutes the adequate mathematical tool i the epesetatio of the physical quatities ad i the study of some pheomea with discotiuities The Diac sequeces allow the study of some physical quatities such as electic ad magetic fields whe the chages have the velocity which teds to the light speed These limit cases whe v c do t exist i the classical mae but i the geealized mae of theoy of distibutios Refeeces: [] Aichelbug PC Sexl RU O the gavitatioal field of a massless paticle Ge Relativ Gavit 4 97 pp-