FI 2201 Electromagnetism

Transcription

1 F Eectomagnetism exane. skana, Ph.D. Physics of Magnetism an Photonics Reseach Goup Magnetostatics MGNET VETOR POTENTL, MULTPOLE EXPNSON

2 Vecto Potentia Just as E pemitte us to intouce a scaa potentia V in eectostatics E V So, B in magnetostatics, aows us to intouce a vecto potentia B since iv cu is zeo. How o we cacuate the vecto potentia fo a given cuent t? The govening equation is obtaine fom mpee s Law B J Howeve, this iffeentia equation is not easy to sove. exane. skana Eectomagnetism Vecto Potentia Reca that in eectostatic potentia, we can a any constant to the potentia function, since gaient of a constant is zeo, without ateing the physica E. We can o a simia thing, i.e. a a function to the given vecto potentia, whose cu vanishes. Fom vecto cacuus, we know that cu of a ga vanishes, hence we can a a gaient of a scaa function to the vecto potentia that wi not ate the physica B λ We can use this feeom of efining the vecto potentia to simpify the govening equation of J exane. skana Eectomagnetism

3 Vecto Potentia Fom the oigina vecto potentia, we choose a scaa function λ such that i.e. λ o λ We can fin such scaa function λ? Note that the govening equation fo λ is simia to the Poisson s equation of the scaa eectostatics potentia V ρ V ε Which fo ρ goes to zeo at infinity, the soution is given as ρ( V( τ ε V exane. skana Eectomagnetism 5 Vecto Potentia Hence, by anaogy, we can wite that the soution fo λ povie that goes to zeo at infinity as ( λ τ V f it oes not goes to zeo at infinity, thee ae othe means to fin λ (just as the case of ρ oes not goes to zeo at infinity. n othe wos, it is aways possibe to fin a scaa function λ such that the vecto potentia is ivegenceess. This is because the efinition B ony specifies the cu of, we sti have the feeom of efining the ivegence of, an zeo is the simpest choice. exane. skana Eectomagnetism 6

4 Vecto Potentia hoosing this specia choices of offsets to make cetain pobems easie to set up, is cae fixing the gauge. Setting is cae choosing a ouomb gauge. Othe gauge that is usefu in eectoynamics is the Loentz gauge, i.e. setting V ε t whee V is the scaa potentia. exane. skana Eectomagnetism 7 Vecto Potentia Reca the govening equation fo the vecto potentia ( J With the ouomb gauge, i.e. fixing, then the govening equation fo the vecto potentia becomes J which again simia to the Poisson s equation fo the scaa potentia. Hence, assuming that J goes to zeo at infinity, then by anaogy, the soution fo the vecto potentia is ( J τ V exane. skana Eectomagnetism 8

5 Vecto Potentia Fo ine an suface cuents ( a Exampe 5. an 5. S exane. skana Eectomagnetism 9 What is Vecto Potentia Unike the eectostatic potentia V, the magnetic vecto potentia is a vecto. Since is soenoia it can t have a scaa potentia. Thus B is much ess usefu in magnetostatic cacuations than V is in eectostatics. Unike V, which we think of as wok pe unit chage, thee s no obvious mechanica intepetation of. The est mechanica intepetation is pobaby : momentum pe unit chage (times c. Fo instance, the canonica momentum of a chage patice in an eectomagnetic fie is q p canonica p c This canonica momentum appeas fequenty in the equations of quantum mechanics. exane. skana Eectomagnetism 5

6 What is Vecto Potentia Futhe, in eativistic quantum mechanics whee a quantities ae epesente as a fou-vecto, the fouvecto potentia of eectomagnetism is given as ( V,x,y, z c exane. skana Eectomagnetism Magnetostatic Bounay onitions Just ike eectic fie, that suffes a iscontinuity at a suface chage, the magnetic fie is iscontinuous at a suface cuent. Use Gauss theoem with a pi-box Gaussian suface : ε B beow B beow S B a Bbeow The noma components of the magnetic fie is continuous. exane. skana Eectomagnetism 6

7 Magnetostatic Bounay onitions Fo the tangentia components, appy mpee s Law using a vey thin, ε, e oop integation, so that the uppe an owe ine segments wi be immeiatey above an beow the suface cuent. ε B beow B beow B B B enc above beow The tangentia components of the magnetic fie that is pepenicua to the suface cuent is iscontinuous. exane. skana Eectomagnetism Magnetostatic Bounay onitions The tangentia components of the magnetic fie that is paae to the suface cuent is continuous. B enc Babove Bbeow ε B beow B beow We can summaize these bounay conitions as foows B B nˆ above beow nˆ is the upwas unit vecto noma to the suface cuent. exane. skana Eectomagnetism 7

8 Magnetostatic Bounay onitions Like the scaa potentia in eectostatics, the vecto potentia of magnetostatic is continuous as any bounay. above beow Noma components of the vecto potentia ae continuous as guaantee by, whie B in the fom a B a Φ S B means that the tangentia components ae continuous (fux though an mpeian oop of vanishing thickness is zeo. Howeve, the eivative of inheits the iscontinuity of B above beow nˆ n n n exane. skana Eectomagnetism 5 S Mutipoe Expansion of Vecto Potentia Just ike in the case of appoximate eectostatic scaa potentia, an appoximate vecto potentia fo a ocaize cuent istibution can be obtaine fom mutipoe expansion. onsie an abitay oop that caies a cuent, its vecto potentia at point is ( ( O exane. skana Eectomagnetism 6 8

9 9 Mutipoe Expansion of Vecto Potentia We can expan / as foows. n n P nseting the Legene expansion on the vecto potentia we get P n n n n n Eectomagnetism exane. skana 7 P P P Mutipoe Expansion of Vecto Potentia Thus, the vecto potentia can be witten as the foowing expansion The fist tem is cae the monopoe tem, the secon is the ipoe tem, the thi is the quaupoe tem etc. Howeve note since ispacement aoun a oop vanishes quaupoe ipoe monopoe Eectomagnetism Howeve, note since ispacement aoun a oop vanishes, the monopoe tem aso vanishes, in accoance with exane. skana 8 B

10 Magnetic Dipoe Fo points fa away fom the oop compae to its size, we obtain a goo appoximation fo by using just the fist (o fist two nonvanishing tems. (Fo points e by, one wou nee moe tems fo the same accuacy. This is, of couse, the same usefu behavio we saw in the mutipoe expansion of V. O The ipoe tem is ipoe ( ( ˆ exane. skana Eectomagnetism 9 Magnetic Dipoe Note that [ ( ˆ ] ( ˆ ( ˆ [ ( ˆ ] ( ˆ ( ˆ Howeve, [( ˆ ] Thus, ( ˆ ( ˆ so note that, ( B B( ( B, hence ˆ ( ˆ ( ˆ ( ˆ O exane. skana Eectomagnetism

11 Magnetic Dipoe Thus, ( ˆ ( ˆ ˆ Hence, the ipoe tem can be witten as m ˆ ipoe ( ˆ ˆ 8 Whee m is cae the magnetic ipoe moment. ompae this magnetic ipoe potentia with eectic ipoe potentia p ˆ V ip ε exane. skana Eectomagnetism Magnetic Dipoe Futhe note that α Thus sinα sin m a a ( α aea of tiange S m a a Exampe 5. exane. skana Eectomagnetism

12 Magnetic Dipoe Magnetic fie of a pue magnetic ipoe can be obtaine as foows Z m ˆ m sin m ipoe ˆ ϕ X ϕ Then, m B ˆ ˆ ipoe ( ipoe ( sin gain, see the simiaity with eectic fie of a pue eectic ipoe p E ( ˆ sin ˆ ip Vip ε exane. skana Eectomagnetism Y Magnetic Dipoe an Eectic Dipoe The simiaity of Magnetic ipoe an Eectic ipoe goes futhe. We can cacuate the toque pouce by the extena magnetic fie on the magnetic ipoe. Magnetic fie tens to aign the magnetic ipoes, in the same way as extena eectic fie aigns the eectic ipoes. Thus, we can wok out foces, toques, enegies, an even a ot about magnetostatics in magneticay-poaizabe matte in stict anaogy with eectostatics. exane. skana Eectomagnetism

13 Summay of Magnetostatics The eations between cuent ensity J, magnetic fie B an magnetic vecto potentia in magnetostatics can be summaize in the foowing iagam : ( J τ V J J B ( J ˆ B τ B J B B exane. skana Eectomagnetism 5 Summay of Eectostatics Just ike the eations between chage ensity ρ, eectic fie E an eectic potentia V in eectostatics : ρ( V( τ ε V V ρ ρ ρ E ˆ τ ε ρ E E V ε ε E V V E exane. skana Eectomagnetism 6 O V E