Materials Physics and Mechanics 23 (2015) Received: March 27, 2015 ELECTROMAGNETIC FIELD IN MOVING SPACE WITH SPHERICAL ENCLOSURE

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1 Matials Physis ad Mhais 3 ( Rivd: Mah 7 5 EECTROMAGNETIC FIED IN MOVING SPACE WITH SPHERICA ENCOSURE V.A. Vstyak.A. Igumov D.V. Talakovskiy 3* Mosow Aviatio Istitut (Natioal Rsah Uivsity Volokolamskoy shoss 4 А-8 GSP -3 Mosow 5993 Russia Rsah Istitut fo Mhais of obahvsky Stat Uivsity of Nizhi Novgood 3 Pospkt Gagaia (Gagai Avu BDG 6 Nizhy Novgood 6395 Russia 3 Istitut of Mhais (omoosov Mosow Stat Uivsity Mihuiski pospkt Mosow 99 Russia *-mail: tdvhom@mail.u Abstat. Th ustady axisymmti poblm big osidd is latd to th idtifiatio of th ompots of ltomagti fild i movig spa with sphial losu filld with homogous isotopi oduto. Th modl usd iluds Maxwll s quatios ad galizd Ohm s law. I od to fid a solutio a sis xpasio i gd polyomials ad apla tim tasfom a applid. Itgal pstatios of th ompots of th ltomagti fild with G s futio kls hav b gatd.. Itodutio As of today th poblm latd to wav popagatio i ltomagtolasti bodis has b maily osidd i gad to statioay poblms (s xampl []. Wh sahig o-statioay posss i suh bodis (xampl [] th small paamt mthod is ovit to b usd pstd by mhaial ad ltomagti fild itatio fato. At th sam tim a solutio is quid fo th auxiliay poblm of idtifiatio of th ltomagti fild paamts i aoda with th pdtmid motio law. This poblm is osidd blow i spt of a pla with sphial losu. It also has its ow idpdt valu i tms of fo xampl sah o th motio of vaious aiafts afftd by ltomagti fild.. Sttig up th poblm It is assumd that th spa with th sphial losu of adius R is filld with homogous isotopi oduto ad movs i aoda with th pdtmid law. Axisymmti hag of th ltomagti fild i it is dsibd usig th Maxwll's quatios ad galizd Ohm's law i th sphial oodiat systm ϑ π π < ϑ π : η ( γ ( η ( γ H = j + E H si si = j + E ( E E tg ; = H E+ E + E + E = ρ ( j = E + ρ u γ j = E + ρ v γ. ( 5 Istitut of Poblms of Mhaial Egiig

2 3 V.A. Vstyak.A. Igumov D.V. Talakovskiy Fom ow o dots stad fo tim divativs whil th vaiabl aft th oma i th low idx idiats its divativ. Th followig dimsiolss quatitis a also usd (if th taig is th sam th pim ospods to dimsioal aalogus: t R u v H µ 4πρ τ = = = = = = = u v H ρ E ε E E E j j µε 4πσ E = E = j = j = = γ = E E σe σe ε η wh t - tim; u ad v E ad E j ad j - adial ad tagtial displamts ompots of lti fild ad ut dsity vtos; H - magti fild vto ozo ompot; ρ - sufa hag dsity; ad - spd of light ad tsio wav popagatio; ε ad µ - dilti ad magti odutivity fatos; σ - lti odutivity fato; ad E - lti fild lia dimsio ad itsity. Th Ohm s law liaizatio was pfomd i gad to iitial ltomagti fild (whih has a ospodt additioal idx with th followig ompots: E = E E = E H = H = H. ϑ ϑ At th losu bouday th followig lti fild itsity is spifid: E ( τ =. (3 = All th quid futios a boudd whil th iitial oditios a homogous: E = E = E = E = H = H =. (4 τ= τ= τ= τ= τ= τ= Equatios ( ( lad to th followig quatio futh usd as th basi: η H + γh = H H si + η ρv ρ u ( si ( si H= H + H ad th followig hag dsity atio: si ( si ρ + γρ = ρ u ρ v. (6 3. Itgal solutios I od to solv iitial bouday valu poblm (3-(5 with osidatio of ( futios 3 E ρ j u ad E H j v a xpadd i gd P ( x ad Ggbau C ( x polyomials aodigly (two sis a giv fo xampl:. (7 ( τ = ( τ ( os ( τ = si ( τ 3 ( os E E P H H C = = This lads to th followig quatios i gad to boudd fatos H ( ( ( ( ( η H + γh = H + ηlh uv H = H mh m= ( + lh ( uv = ( ρv + ρ u. τ : (5 (8

3 Eltomagti fild i movig spa with sphial losu 33 Ratios fo oth ltomagti fild ompot xpasio fatos follow fom fomulas ( ad (6: ( ( η E + γe = H η v η E + γe = m H η u ; (9 ρ ρ + γρ = l u v l u v = ρ u + m ρ v. ( ρ I th last atio ρ a th fatos of sis (7 fo futio Bouday ad iitial oditios (3 ad (4 with osidatio of th fist quality i (9 is tasfomd as follows: H =η h v τ τ h v = v + + γ ( = ρ. E = E = E = E = H = H =. ( τ= τ= τ= τ= τ= τ= Nxt w apply apla tim tasfom τ ( s mas its paamt whil idx poits to its viw [3] to atios (8-( with osidatio of oditios (: ( ( s η H = H + η sl u v s = s s + γ ; (3 H η s + γ E = H η sv η s + γ E = mh η su ; (4 ( s γ ρ ( slρ u v + = ; (5 ( =η ( ( = + ( + γ H h v s s h v sv s =. (6 It is ovit to pst th solutio fo bouday-valu poblm (3 (6 as itgals: ( =η ( ξ ( ξ ( ξ ξη ( ( H s s G sl u s v s d G sh v s s. (7 H H H H GH ( ξ s ad GH ( th poblms ( δ ( x - Dia dlta futio [3]: H H H s a th G s futios i.. boudd solutios of G s η G = δ ξ G = ; (8 H H H = = G s η G = G =. (9 Simila (7 pstatios fo futios E ( s ad E ( s may b obtaid fom qualitis (4. Oigial pstatio (7 has th followig viw (astisk stads fo tim ovolutio: H( τ =η GH( ξτ lh u ( ξτ v ( ξτ dξη GH( τ h v ( τ ( τ. ( Similaly futios E ( τ ad E ( sufa hag xpasio fatos follows fom (5: ( ( ( τ may b pstd. Th fomula fo th ρ τ =l u v γτ ρ τ τ. (

4 34 V.A. Vstyak.A. Igumov D.V. Talakovskiy 4. G s futios Th solutios fo bouday-valu poblms (8 ad (9 hav th followig viw ( h( x - Havisid futio [3]: GH ( ξ s = ξ H ( ξ s h( ξ + H ( ξ s h( ξ ξ s = ηss ηs ηs Z ηξs Y ηs ; H 3 ( η ( η ( η G s = s Z s Y s (3 H 3 wh Z ( z = z K+ ( z Z ( z = z I+ ( z S ( x y = Z ( y Y4 ( x Y3 ( x Z ( y 3 3 Y3( z = z ( + K+ ( z zk+ 3( z Y4( z = z ( + I+ ( z + zi+ 3( z H K+ ( z ad I+ ( z a modifid Bssl futios.. Wh buildig fomulas ( ad (3 th Bssl futio xpssios though Z z Y z : lmtay futios [4] w usd fom whih follow th blow listd qualitis fo Z ( z ad Y ( z 3 4 = π = ( ( π ( = π = ( ( π ( z z z Z z z R z Z z z R z R z z z z Y3 z z R3 z Y4 z z R3 z R3 z wh! R z A z A R z R z R z R z R z R z. ( + k ( k = k k= 3 = = k + k = k! k! I this as G s futios i ( ad (3 obtai th followig viw: ( k ( ( ( k s + H ξ = H ξ H = H ( ( s k = s G s G s G s τ wh + H ( ξ s = ( R ( ηξ s R ( η s D ( ξη s ξ s = R ηξs R ηs R η s D ξη s 3( ( = ( η 3 ( η + + = τ ( ξ = η ( ξ τ ( ξ = η ( + ξ H H G s R s R s D xy x y. Thi oigials may b aalytially foud usig opato alulus thoms. Howv si fo al mdiums η << sious hallgs ais duig alulatios assoiatd with th small paamt. That is why quasistati aalogus of G s futios with η = will b futh usd. 5. Quasistati solutio I this vsio quatios i bouday poblms (8 ad (9 a simplifid ad hav th ( followig fudamtal systm of solutios: +. Th simila to p. 3 w om to th followig fomulas fo G s futios: (

5 Eltomagti fild i movig spa with sphial losu ( ξ = ( ξ = ξ ( ξ ( ξ + ( ξ ( ξ GH s GH GH h GH h ( H ( ξ = G H( s = GH = ξ Th fomula (7 is also simplifid by substitutig kls GH ( ξ s ad GH ( s with futios G ( ξ ad G (. Its oigial ( is substitutd with th followig quality: H H ( τ =η ( ξ ( ξτ ( ξτ ξη ( τ ( τ H G l u v d G h v. (5 H H H Th algoithm is followd by th summatio of sis pstd as (7 ad th idtifiatio of th ut dsity ompot vto usig fomulas (. 6. Exampl 4 t's assum that th spa matial is alumiium ( η = ; γ= 5 6 whil th oth 3 iitial valus a pstd as: = ρ = a u =τ+ os v=τ+ si. Th th ltomagti fild ompots alulatd usig fomulas (7 ( (4 ad (5 a dtmid as follows: 3 a 3 3 H ( τ = η w + ( + τh( τ si 3 ρ τ 3γ γτ τ os. 5 ( = aw ( + γτ h 7. Colusios A aalytial solutio of th axisymmti o-statioay poblm is pstd latd to th idtifiatio of th ompots of th ltomagti fild i th spa filld with homogous isotopi oduto with sphial losu. Its movmt law ad fild distibutio at th avity bouday a pdtmid. It is show that th kls of th itgal pstatios built may b substitutd with thi quasistatial aalogus whih sigifiatly simplifis th valuatio of th ospodt itgals. Th solutio obtaid may sv as th basis fo th sah of poblms with mo omplx gomty. Akowldgmts Fiaial suppot fo this wok has b povidd by Russia Foudatio fo Basi Rsah (pojt ad th Russia Fdatio Psidt gat NSh Rfs [] V.T. Gihko A.F. Ulitko N.A. Shulga Mhais of latd filds i th ostutio lmts. Т.5. Elti lastiity (Naukova dumka Kiv 989. [] D.V. Talakovski V.A. Vstyak A.V. Zmskov I: Eylopdia of Thmal Stsss (Spig Dodht Hidlbg Nw Yok odo 4 Vol. p. 64. [3] A.G. Goshkov A.. Mdvdsky.N. Rabiskky D.V. Talakovsky Wavs i Cotiuum Mdia (Fismatlit Мosow 4. [4] A.G. Goshkov D.V. Talakovskiy Tasit Aohydolastiity of Sphial Bodis (Spig-Vlag Bli-Hidlbg-Nw Yok. 35 (4