Electromagnetic waves in vacuum.
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1 leromagne waves n vauum. The dsovery of dsplaemen urrens enals a peular lass of soluons of Maxwell equaons: ravellng waves of eler and magne felds n vauum. In he absene of urrens and harges, he equaons governng eler and magne feld are: B B B B Tang he url of (3 and usng (4: ( Usng ( we ge D Alamber equaon: Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
2 leromagne waves n vauum. ah of he wo equaon pars lead o he D Alamber eq. (respevely for x and B y, and y and B x. For example, he frs par ombnes no: z y y x x B z B ( B! x ( z B B ( z x y y The lneary of D Alamber eq. enals ha he gener M plane wave an be seen as he superposon of snusodal waves. Le us he assume ha eah omponen has he form u u exp[ ( z + φ ] u Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
3 leromagne waves n vauum. The par of equaons for ( x,b y yelds B exp[ ( z + φ ] B y exp[ ( z + φ ] exp[ ( z + φb ] x exp[ ( z + φ ] x B y These equaons mus be verfed for any and z. Therefore x B y φ φb The same ondons are rue for ( y,b x. A eah (,z he magnude of he eler and magne feld s he same (n CGS-uns. Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
4 Polarzaon of M waves The wo lasses of soluons ( x,b y and ( y,b x are ndependen: hey represen he wo polarzaon modes of M radaon. As he and B felds le on a plane, hese modes orrespond o lnear polarzaons. A gener (unpolarzed M wave s a superposon of he wo modes, wh dfferen phases and ampludes. For example, nrodung he un veors along x and y: ( z, x os( z + φx ]ˆ x + y os( z + φ y ] yˆ Insead of he lnearly polarzed modes, one ould use wo rularly polarzed modes: x y π φ y φ x + Lef Crular Polarzaon (LCP and x y φ y π φ x Rgh Crular Polarzaon (RCP Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
5 Maxwell quaons n a Plasma The wave propagaon s governed by Maxwell equaons (CGS-Gauss: 4πρ B B B 4π J + We wll solve he ombned se of Maxwell equaons and eleron equaon of moon usng a perurbave approah (o frs order and normal modes. All quanes are he sum of an unperurbed (baground value and a small perurbaon: n n + n B B + B v v (quesen plasma (no exernal eler feld Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
6 Maxwell quaons n a Plasma. All perurbaons are perod and depend on spae and me as u u exp[ ( r The surfaes of onsan phase move n spae a he phase veloy v φ (suggeson: onsder he surfaes wh null phase and ompue r/ The se of dfferenal equaons beomes now a se of algebra equaons; he dfferenal operaors ransform as Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
7 The equaons of moon. Naver-Soes equaon for he elerons (ollsonless plasma: v mene + ( v v p en e + v B The spae harge densy ρ -en e and urren densy J -en e v fullfl he equaon of onnuy: ρ + J We assume ha he plasma he plasma s magnezed: a onsan and homogeneous magne feld s supermposed o he magne feld of he wave. The plasma s supposed o be sohermal, so we do no need o onsder he energy equaon. (No only: we wll see soon ha he plasma may be safely supposed o be old, wh T! Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
8 The lnearzed, normal mode equaons. The Maxwell equaons, harge onservaon and eleron equaon of moon beome: 4π en ( B ( B B 4π J e n J (3 (5 m e n v en + v B (6 (4 Noe ha now he frs wo Maxwell equaons (- follow from he las par (3-4 and harge onservaon (5: ( follows by ang he salar produ of (3 by ; ( follows by ang he salar produ of (4 by and usng (5. Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
9 The onduvy ensor. q. (6 enals a lnear relaon beween urren densy and eler feld. e n J + J me me e B or, wrng B B n and nrodung he plasma and yloron frequenes, p Ω J J n 4π xpandng he veor produ, one ges he generalzed Ohm s law and he onduvy ensor σ: J σ Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
10 The onduvy ensor. Ω Ω Ω Ω m e e n σ Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8 Ω m Im( Im( Re( Re( Re( Re( Re( j j j j J Q σ σ J The onduvy ensor s herefore an-hermean. Therefore no hea s dsspaed, as he real par of σ s sew and * j σ j σ Im( Re( j
11 n r v φ Plane M waves n a plasma. The refrave ndex of a medum s defned as he rao beween and he phase veloy By ombnng Maxwell eq. B a ( b ( a b ( a b 4 π B J one ges: 4π ( n + n J j r r j whh gves he urren as a funon of he eler feld. Mang use of he relaonshp J σ one obans 4π ( n + n σ j r r j j j Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8 (7 4π j ( nr δj + nr σ j j
12 Plane M waves n a plasma. The homogeneous se of equaons has non-rval soluons ff 4π j de ( nr δj + nr σ j Ths s he dsperson equaon of he plasma. Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
13 M waves n a plasma: B If B, he onduvy ensor redues o a salar: σ p 4π Maxwell eq. allow wo modes of propagaon: ( and ( ( + 4π J + 4π σ ( p / p Longudnal plasma waves our a he well nown plasma frequeny. The wavenumber s free! 4π ( ( n r J Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
14 M waves n a plasma: B ( ( n r Subsung J p n r 4π J p σ 4π one ges nr p The ndex of refraon of a plasma s smaller han uny. Ths mples ha he phase veloy of e.m. waves s larger han : v φ n r > Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
15 M waves n a plasma: B However, nformaon does no ravel a he phase veloy: a onnuous, monohroma, plane wave (sarng a - anno be used o onvey nformaon. Informaon (e.g. eleromagne pulses ravels a he group veloy (always smaller or equal o : v g d d In an unmagnezed plasma: v φ > n r p d n r p vg d n r Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
16 Group veloy Le us onsder a superposon of wo M waves wh slghly dfferen frequeny and wavenumber: u u os[( os[( + z ( + ] z ( ] The superposon of he wo waves s: u + u os[( + z ( + ] + os[( os[( z + ( z ] + os[( z ( z ] os( z os( z ] sn( z sn( z ] + os( z os( z ] + sn( z sn( z ] os( z os( z ] The low frequeny modulaon ravels a he speed z ( ] v g Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8
17 Group veloy d d d z d A d A d z A z ] ( [( exp ( ] [ exp (, ( + + Ψ + + ξ ξ ξ ξ For Le us onsder now a wave pae,.e. a more general superposon of M waves wh dfferen frequeny and wavenumber: d d v g /, Ambene Spazale e Srumenazone Spazale- Prof. L.Iess AA 7-8 d d z d d z z A d d d z z A d d ] sn[ ] ( [ exp ( ] [ exp ] ( [ exp ( + ξ ξ ξ
18 Group veloy Ths pulse ravels a he speed v g d d The superposon Ambene of Spazale he wo e Srumenazone waves s: Spazale- Prof. L.Iess AA 7-8
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