Physics 442. Electro-Magneto-Dynamics. M. Berrondo. Physics BYU

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1 Physis 44 Eleo-Magneo-Dynamis M. Beondo Physis BYU

2 Paaveos Φ= V + Α Φ= V Α = = + J = + ρ J J ρ = J S = u + em S S = u em S Physis BYU

3 Poenials Genealize E = V o he ime dependen E & B ase Podu of paaveos: AB = ( α + a)( β b) = αβ ab ia b+ βa αb sala veo (omplex) Physis BYU 3

4 Conside whee Φ = ( ) Φ Φ= V A sepaaing he sala pa: Φ = ( ) s V A = s V + Loenz ondiion Φ = s V + A = (hoie of gauge ) (onsain) A Physis BYU 4

5 veo pa: F= Φ Φ A = V + i A v v o, wih Loenz gauge, F= Φ Poof: sepaae E= V A B= A hen and E= ( A) = B B = Faaday Physis BYU 5

6 Maxwell equaion fo Φ (Loenz gauge) F = Φ = Z J Φ=Z J Sepaae: V A = = wave equaions in Loenz gauge genealizaion of Poisson s equaions fo ε ρ µ J V + A In vauum: ( V + A ) = Physis BYU 6

7 Equaions fo poenials in abiay gauge: whee sepaaing: F = Φ = Z J v Φ = Φ Φ v V ( A) = ρ ε A+ ( V ) + A =µ J oupled equaions fo V and A s Physis BYU 7

8 Wave Equaion (-d) Iniial disubane g(z) v f( z, = ) = gz ( ) f (,) z = g( z v) Popagaing wih speed v: afe : f( z, ) = g( z v) = g( u) l l = v Physis BYU 8

9 Example: Find ρ, J ha give ise o x > V = A = µ k ( x ) eˆ 3 x < 4 Soluion: ρ = µ k A = f ()ˆ s e3 4 f() s = s f '() s = s f "() s = ewie A as wih and s s s = x = and = ε ( x) x Physis BYU 9

10 wave equaion fo A f() s A= µ J = f ''( s) and f() s = [ ε( x) f '( s)] = f ''( s) δ( x) f '( s) x x f s = δ x f s = δ x s so () () '() 4 () sufae uen ie plane µ k4s µ K = eˆ K = keˆ x= Physis BYU

11 Gauge ansfomaions λ(,) A A' = A+ λ λ V V ' = V and sala field (abiay) A' = E' A A ' = V ' E' = V + ( λ) A ( λ) = E Cliffos: F= Φ fo abiay gauge v hange Φ Φ ' =Φ λ hen Given λ = F v emains he same Physis BYU

12 Loenz gauge: Φ = s V + A = V = ρ ε Coulomb gauge: A = insead implies E= V A edues o Poisson s eq.: V(,) = V ρ(,) E= V A= V = ρ() ε 3-d onvoluion NO eadaion!!! and ρ / ε Physis BYU

13 Review sais Geen s funion g() = 4π and (3) g() = δ ( ) = g( ) Poenial (Poisson): Φ s = () Z J () Φ () = Zg() J () s Field (Maxwell): F( ) = Z J () Physis BYU 3

14 Soluion: Review Sai Fields Geen s funion Coulomb field ˆ G() = g() = 4π F( ) = ZG() J () F( ) G( ) ρ( ') dτ ' iµ G( ) J ( ') dτ ' = ε Physis BYU 4

15 Dynamis Geen s funion (4-d hypeboli): fom g(,) = δ ( ) δ() (3) we find Φ (,) = Z g(,) J (,) as a 4-d onvoluion. The oesponding Geen s funion and field: G(,) = g(,) F(,) = ZG(,) J(,) 4-d Physis BYU 5

16 eadaion effe: i akes a ime = / fo he signal o aive g(,) = δ ( ) (eaded soluion) 4 π solves g(,) = δ ( ) δ() (3) δ + / ( ) (fo advaned) invese of D Alembeian: = g(,) as a 4-d onvoluion!!! Physis BYU 6

17 Poenial paaveo Φ (,) = V + A Loenz gauge: wih soues Φ= Z J ρ, J J(,) 4-d onvoluion: Expliily: Φ= Zg J δ ( ') Z Φ (, ) = J( ', ') dτ ' d ' = 4π Z J( ', e ) = dτ ' whee e = 4π Physis BYU 7

18 Sala and Veo Poenials ρ( ', ) V(,) = e dτ ' 4πε µ ( ', ) A (,) = J e dτ ' 4π evaluaed a ' ' = e = Physis BYU 8

19 Geen s funion fo he Fields Le us find G(, ), paaveo soluion o Maxwell equaion wih uni poin soue: (3) G(,) =δ ( ) δ() g(,) = δ ( ) 4π Defining G(,) = g(,) fom G = g = g = δ δ (3) ( ) ( ) () Physis BYU 9

20 Soluion o Maxwell equaion wih soues F = Z J J Expliily: F= ZG J fomal fom 4-d ˆ G(,) = δ δ ' 4π + + ˆ ( ) Physis BYU

21 We need ) Define EM field (dynami ase) and alulae: G(,) = g(,) = δ ( ) 4π g( ) = and s = 4π s = ( ˆ = + = + ) s = ( +ˆ ) paaveo (pojeo) Physis BYU

22 ) δ() s = ( +ˆ ) δ '() s 3) δ δ δ ( () s g ()) = ( ()) s g () + ()( s g ()) = ˆ '()( ˆ = δ s ) δ() s 4π + + adiaion δ '( α) h( α) dα = h'() so dδ (' ) e ρ( ', ') d' = ρ( ', e ) d Coulomb (eaded) Physis BYU

23 Jefimenko equaions ρ( ', ) e ρ( ', e ) J ( ', e ) E (,) = ˆ + ˆ µ dτ ' 4π ε ε and µ ( ', ) ( ', ) 4π J J e e (,) = + dτ ' B ime dependen genealizaion of Bio-Sava ˆ Physis BYU 3

24 P Physis BYU 4

25 Lienad-Wiehe Poenials poin paile wih hage q (3) hage densiy: ρ(, ) = qδ ( w( )) w() given paile ajeoy (fixed) () = w() soue (eaded) o field Physis BYU 5

26 e is deemined fom ligh one ondiion, w( ), e Elei poenial w( ) e e uen densiy: = V(,) = Zg ρ J (,) = ρv δ ( ' ) (3) V (, ) = e qδ ( ' w( ')) dτ ' d ' 4πε whee = ' Physis BYU 6

27 To alulae V inegae and dτ ' ( ') = w( ') ( ') ( ') Key poin: fis so R beomes: ( ') δ ' q + V (,) = d ' 4 πε ( ') δ (' i ) δ f( ') = f ( ) i i whee ( ') ( ') f( ') = ' + = ' + w Physis BYU 7

28 alulae denominao: = so = nˆ = nˆ w = nˆ β whee nˆ = / and β= w / κ = d f ( ') = f( ') ˆ d ' = + = n β Dopple fao q V(,) = and 4πε nˆ β e Φ (,) = q + β 4πε κ e L.W. poenials Physis BYU 8

29 Dopple fao - veloiy paaveo: κ = nˆ β u - soue-o-field paaveo a e χu = γ( nˆ β) = γκ s = γ( + v) = γ( + β) χχ = χ = ( +nˆ ) u = γ ( + β) χ = ( ') + so (denominao) (numeao) q u q + β Φ (,) = = 4πε χu 4πε nˆ β s e e Physis BYU 9

30 x Ideniy: so Life in he ligh one y δ = ( ) ligh one oesponding densiy popoional o eaded Coulomb poenial / = + + δ δ δ e adv g(,) = δ = δ 4π π e Physis BYU 3

31 Reall: ˆ G(,) = δ δ ' 4π + + ˆ ( ) 4-d onvoluion o alulae he fields: F= G J ε whee J (, ) = q + β δ ( w( )) v ( ) (3) and β = dw d Physis BYU 3

32 Expliily, he onvoluion inegal is: F= q 4πε ˆ( ) ( ˆ)( ) [ f( ')] + δ '[ f( ')] d' + n β δ n β v E (,) q nˆ d nˆ β = + 4 πε κ κ d ' κ e veo pas ( ) ( )( ) nˆ β = nˆ i nˆ β + nˆ β = nˆ β i nˆ β v v so B= n ˆ E Physis BYU 3

33 d [( nˆ βn ) ˆ β] nˆ = d ' d ( κ) = ( β nˆ β) nˆ β d ' Using: nˆ = / and spliing (veloiy) and (aeleaion) pas E= E + Ead B= B + Bad E (,) = q ( nˆ β) 3 4 πε γ ( κ ) e γ = β Physis BYU 33

34 Eveyhing evaluaed a e E = q nˆ β 4 πε γ ( nˆ β) B= n ˆ E in ems of 3 γ E ˆ ad = n ( E ) a E Non-elaivisi: and and e d a = v d q a E ˆ ad n ( E a) 4πε Coulomb Bad = n ˆ E ad Physis BYU 34

35 Coulomb poenial fo a poin hage wih onsan veloiy v w() = v e Q E ˆn β = Q Q α θ P P v E = E (,) = R() = v = β q ( nˆ β) 4 πε γ ( nˆ β) 3 3 ( e ) Physis BYU 35

36 a) R = β= ( nˆ β) b) nˆ R = ( nˆ β) = Rosα ) law of sines sinα sinθ = sinα = βsinθ β so nˆ R = R sin α = R β sin θ E (,) whee = q R β () 3/ 3 4 πε ( β sin θ ) R ( ) R() = v Physis BYU 36

37 Elei field of a paile moving a onsan veloiy Physis BYU 37

KINGS UNIT- I LAPLACE TRANSFORMS

KINGS UNIT- I LAPLACE TRANSFORMS MA5-MATHEMATICS-II KINGS COLLEGE OF ENGINEERING Punalkulam DEPARTMENT OF MATHEMATICS ACADEMIC YEAR - ( Even Semese ) QUESTION BANK SUBJECT CODE: MA5 SUBJECT NAME: MATHEMATICS - II YEAR / SEM: I / II UNIT-