Section 5: Magnetostatics

Transcription

1 ection 5: Magnetostatics In electostatics, electic fiels constant in time ae pouce by stationay chages. In magnetostatics magnetic fiels constant in time ae pouces by steay cuents. Electic cuents The electic cuent in a wie is the chage pe unit time passing a given point. If chage Q passes point P in Fig.5. pe time t the magnitue of the cuent is Q I. (5.) t By efinition, negative chages moving to the left count the same as positive chages moving to the ight. In pactice, thee is a convention to assume that the electic cuent flows in the iection of motion of positive chages. Cuent is measue in coulombs-pe-secon, o ampees (A): A=C/s. A line chage taveling own a wie at spee v (Fig. 5.) constitutes a cuent I v. (5.) This is because a segment of length vt, caying chage vt, passes point P in a time inteval t. Fig. 5. Line cuent We note that cuent is actually a vecto. A neutal wie, of couse, contains as many stationay positive chages as mobile negative ones. The fome o not contibute to the cuent. When chage flows ove a suface, we escibe it by the suface cuent ensity, K, efine as follows: Consie a ibbon of infinitesimal with l, unning paallel to the flow (Fig. 5.). If the cuent in this ibbon is I, the suface cuent ensity is K I. (5.) l In wos, K is the cuent pe unit with-pepenicula-to-flow. In paticula, if the mobile suface chage ensity is an its velocity is v, then K v. (5.4) In geneal, K will vay fom point to point ove the suface, eflecting vaiations in an/o v. Fig. 5. uface cuent

2 When the flow of chage is istibute thoughout a thee-imensional egion, we escibe it by the volume cuent ensity J efine as follows: Consie a tube of infinitesimal coss section a, unning paallel to the flow (Fig. 5.). If the cuent in this tube is I, the volume cuent ensity is J I. (5.5) a In wos, J is the cuent pe unit aea-pepenicula-to-flow. If the mobile volume chage ensity is an the velocity is v, then J v. (5.6) Fig. 5. Volume cuent Accoing to Eq.(5.5), the cuent cossing a suface can be witten as J n. (5.7) I Ja a In paticula, the total chage pe unit time leaving a volume V is Jna J. (5.8) V Because chage in conseve, whateve flows out though the suface must come at the expense of that emaining insie: J t. (5.9) t V V V The minus sign eflects the fact that an outwa flow eceases the chage left in V. ince this applies to any volume, we conclue that J. (5.) t This is the pecise mathematical statement of local chage consevation. It is calle the continuity equation. The continuity equation playe an impotant ole in eiving Maxwell s equations as will be iscusse in electoynamics. Magnetostatics eals with steay cuents which ae chaacteize by no change in the net chage ensity anywhee in space. Consequently in magnetostatics / t an theefoe J. (5.) We note hee that a moving point chage oes not constitute a steay cuent an theefoe cannot be escibe by laws of magnetostatics.

3 Biot an avat Law It is convenient to escibe magnetic phenomena in tems of a magnetic fiel B. Biot an avat (in 8), fist, an Ampee (in 8-85), in much moe elaboate an thoough expeiments, establishe the basic expeimental laws elating the magnetic fiel B to the electic cuents an the law of foce between cuents. The magnetic fiel is elate to the cuent as follows. Assume that l is an element of length (pointing in the iection of cuent flow) of a filamentay wie that caies a cuent I an is the cooinate vecto fom the element of length to an obsevation point P, as shown in Fig Then the magnetic fiel B at the point P is given by I l B () I. (5.) 4 4 We note that eq.(5.) epesents an invese squae law, just as is Coulomb s law of electostatics. Howeve, the vecto chaacte is vey iffeent. The constant /4 is given in I units, whee is the 7 pemeability of fee space: 4 N / A. In these units B itself comes out in newtons pe ampee- T N/ A m. mete o teslas (T): Fig. 5.4 Elemental magnetic inuction B ue to cuent element l. In oe to fin the magnetic fiel pouce by a wie caying cuent I, we nee to integate ove the wie so that B () I 4 l. (5.) This expession is known as the Biot an avat law. s Example: a staight infinite wie. By symmety the magnetic fiel pouce by a staight infinite wie epens only on the istance fom the O wie s an is oiente pepenicula to the wie. In Fig.5.5 the vecto l I points into the page an has the magnitue l cos. ince - s l l stan we have l an also s cos, so cos cos. Theefoe, Fig. 5.5 s cos s I I B I 4 s cos 4s s / / cos cos / /. (5.4)

4 The Biot an avat law (5.) can be genealize to the case of suface an volume cuents. Fig. 5.6 Fo a volume cuent one has to eplace Il in eq. (5.) by J()al, whee J() is the volume cuent ensity, a is the coss-sectional aea of the filament, an l is the magnitue of l (see Fig.5.6). Then taking into account the fact that al = is a volume element, we obtain imila, fo suface cuents we have B () 4 ( ) B () 4 J ( ) K The Biot an avat law is an analog of the Coulomb s law in electostatics.. (5.5) a. (5.6) Ampee s expeiments i not eal iectly with the etemination of the elation between cuents an the magnetic fiel, but wee concene athe with the foce between cuent-caie wies. It was expeimentally foun that the foce expeience by a wie caie cuent I in the pesence of a magnetic fiel B is given by F I B I l B. (5.7) By substituting the expession fo the magnetic fiel (5.) pouce by anothe wie we fin the foce between two wies caying cuents I an I F II 4 l l. (5.8) This is the oiginal fom of the Ampee s law. This expession can be witten on a moe simple fom if we apply it to two cuent loops. Using the vecto ientity a ( bc) b( ac) c( ab ) we obtain F l l ll II 4 C C. (5.9) Accoing to Newton, the foce of the loop caying cuent I shoul be equal an opposite to the foce on the othe loop, implying cetain symmeties in the integan above. The secon tem in the numeato changes sign une the intechange of an (which is consistent with Newton s low), but the fist tem oes not. It appeas that the fist tem is ientically equal to zeo: ll l l l. (5.) C C C C C C Hee the last step follows fom the fact that we integate ove a close path. Thus we fin that the foce between the two cuent loops may be witten simply as F ll II 4 C C. (5.) 4

5 Example: foce between paallel wies uppose we have two paallel wies a istance apat caying cuents I an I an wish to fin the foce pe unit length acting on one of them. At a point on the one caying cuent I, thee is a fiel B pouce by the othe cuent which is iecte pepenicula to the plane containing the wies. This fiel is given by eq.(5.4) the integal ove the souces in the othe wie, Accoing to eq. (5.7), the foce is B I. (5.) II. (5.) F I Bl l The total foce is, not supisingly, infinite, but the foce pe unit length is f II. (5.4) In case of a volume cuent istibution the magnetostatic foce pouce by a magnetic fiel object caying a volume cuent ensity J ( ) is as follows: B on a F I B J () B. (5.5) This equation is actually a consequence of the Loentz foce affecting a moving chage paticle. The Loentz foce is f v B, (5.6) q whee q an v ae a chage an velocity of the paticle. Accoing to this equation the foce on the chage ensity () moving with velocity v () is the integal F v() B (). (5.7) Taking into account that by efinition J ( ) v ( ) ( ), we aive at eq. (5.5). Diffeential Equations of Magnetostatics an Ampee s Law To obtain iffeential equations fo the magnetic fiel we ewite Eq.(5.5) as follows: B () ( ) 4 J whee we have use the ientity. Now using the ientity, (5.8) a a a, (5.9) whee an a ae abitay scala an vecto functions espectively, we can wite J ( ) J ( ) J ( ) J ( ). (5.) Hee we took into account that the cul is taken ove the unpime cooinates an theefoe J ( ). This allows us to wite Eq.(5.8) in the fom 5

6 ( ) B () 4 J. (5.) It follows immeiately fom hee that This is the fist iffeential equation of magnetostatics. Now we calculate the cul of B: Using the ientity B. (5.) ( ) B () 4 J. (5.) aa a, (5.4) which is vali fo any abitay vecto fiel a, expession (5.) can be tansfome into B () ( ) ( ) 4 J 4 J. (5.5) If we use an (5.6) 4, (5.7) the integals in (5.5) can be witten: Integation by pats yiels B 4 J J. (5.8) ( ) ( ) 4 J ( ) J ( ) B J. (5.9) The fist integal in backets vanishes because the cuent istibution is localize in space. This is because accoing to the ivegence theoem it is euce to the integal ove suface bouning the volume of integation. ince we integate ove all space an since thee ae no cuents at infinity the fist integal is zeo. The secon integal is also equal to zeo because we consie steay-state magnetic phenomena fo which J. Theefoe we obtain B J. (5.4) This is the secon equation of magnetostatics. This is a iffeential fom of Ampee s law. The integal fom of the Ampee s law can be obtaine by applying tokes s theoem. Integating the nomal component of the vectos in the left- an ight-han sie of Eq.(5.4) ove open suface shown in Fig.5.7 we obtain: Bna Jn a. (5.4) 6

7 Fig. 5.7 Using the tokes s theoem it can be tansfome into Bl Jna. (5.4) C ince the suface integal of the cuent ensity is the total cuent I passing though the close cuve C, Ampee's law can be witten in the fom: C Bl I. (5.4) Just as Gauss s law can be use fo calculation of the electic fiel in highly symmetic situations, so Ampee's law can be employe in analogous cicumstances. Example: a staight infinite wie. Ampee s law allows calculating the magnetic fiel pouce by an infinite staight wie in a much moe simple way as compae to the iect calculation using eq. (5.). By symmety the magnetic fiel pouce epens only on the istance fom the wie s an is oiente pepenicula to the wie. Using the Ampee s law we can wite C Bl I, (5.44) whee C is a cicle of aius s centee on the wie, an I is the cuent cossing the suface subtene by the cicle which is exactly the cuent is the wie. ince B is constant on the cicle we immeiately obtain o which is ientical to the esult (5.4). Vecto Potential The basic iffeential laws of magnetostatics ae sb I, (5.45) B I s, (5.46) B, (5.47) B J. (5.48) Accoing to eq. (5.47) the ivegence of B is zeo which implies that thee ae no souces which pouce a magnetic fiel. Thee exist no magnetic analog to electic chage. Accoing to eq. (5.48) a magnetic fiel culs aoun cuent. Magnetic fiel lines o not begin o en anywhee to o so woul equie a nonzeo ivegence. They eithe fom close loops o exten out of infinity. Now the poblem is how to solve iffeential equations (5.47) an (5.48). If the cuent ensity is zeo in the I s 7

8 egion of inteest, B pemits the expession of the magnetic fiel B as the gaient of a magnetic scala potential, B M. Then (5.47) euces to the Laplace equation fo M, an all ou techniques fo hanling electostatic poblems can be bought to bea. A lage numbe of poblems fall into this class, but we will efe iscussion of them until late. A geneal metho of attack is to exploit equation (5.47). If B eveywhee, B must be the cul of some vecto fiel A(), calle the vecto potential: B () A (). (5.49) We have, in fact, aleay witten B in this fom (5.). Eviently, fom (5.), the geneal fom of A is A ( ) 4 J. (5.5) () () The ae gaient of an abitay scala function shows that fo a given magnetic inuction B, the vecto potential can be feely tansfome accoing to A () A () (). (5.5) This tansfomation is calle a gauge tansfomation. uch tansfomations on A ae possible because (5.49) specifies only the cul of A. The feeom of gauge tansfomations allows us to make A have any convenient functional fom we wish. If (5.49) is substitute into the equation (5.48), we fin o A J. (5.5) A A J. (5.5) If we now exploit the feeom implie by (5.5), we can make the convenient choice of gauge, A. (5.54) To pove that this is always possible, suppose that ou oiginal potential, A, is not ivegenceless. If we a to it the gaient of some scala function, so that A A, the new ivegence is If we assume now that A, then we have AA. (5.55) A, (5.56) which is epesents a Poisson equation with espect to. The equation can be solve using stana methos that we have iscusse in electostatics. Thus we can always fin such that gives as A. Using this gauge in eq.(5.5), we fin that each ectangula component of the vecto potential satisfies the Poisson equation, A J. (5.57) Fom ou iscussions of electostatics it is clea that the solution fo A in unboune space is ( ) A () 4 J, (5.58) i.e. = constant. It is easy to see by taking iectly the ivegence of eq. (5.58) that inee A. 8

9 Magnetic ipole moment Now consie asymptotic behavio of the vecto potential. Let the istibution of cuent be confine to a volume V by a suface. We take to outsie of V, while is necessay insie. We examine the vecto potential at lage istance, stating fom expession (5.58). Expaning we obtain Theefoe, Now we simplify this equation. Fist we show that Fo the localize cuent istibution. (5.59) A () ( ) ( ) 4 J J. (5.6) V J ( ). (5.6) x ij ( ) x ij ( ) na, (5.6) since thee ae no cuents cossing suface. On the othe han, xj ( ) x J ( ) x J ( ) J ( ), (5.6) i i i i whee x i is the i component of vecto, an we took into account that iection an J ( ). Theefoe, i ( ) i( ) x is a unit vecto along x i xj J. (5.64) Compaing eq.(5.6) an eq. (5.6) we see that fo any component i of the cuent ensity J i ( ), (5.65) which poves eq. (5.6). The fist tem in eq. (5.6) coesponing to the monopole tem in the electostatic expansion is theefoe absent. econ we show that J ( ) ( ) J. (5.66) Using the elation J ( ) J ( ) J ( ), (5.67) We can wite A ( ) ( ) ( ) 4 J J. (5.68) Consie the j component of the fist integal in eq. (5.68) J x x xj x x xj, (5.69) j i j i i j i i i whee we use eq.(5.6). Taking the last integal by pats we obtain i 9

10 i i i Hee we took into account that xx j ij xx j ij na J x x xx J x xj x x xj J, (5.7) j i j i i i j i i j j. Genealizing to all thee components we have V ue to no cuents cossing suface J J. (5.7) It easy to see fom. Eqs.(5.67) an (5.7) that eq. (5.66) is satisfie. Thus we finally fin fo the vecto potential A () ( ) 4 J. (5.7) It is customay to efine the magnetic moment ensity o magnetization as an its integal as the magnetic moment m: M () J (). (5.7) m () J. (5.74) The vecto potential epesents theefoe the magnetic ipole vecto potential: m 4 A (). (5.75) This is the lowest nonvanishing tem in the expansion of A fo a localize steay-state cuent istibution. The magnetic fiel B outsie the localize souce can be calculate iectly by evaluating the cul of (5.75). Using the ientity an taking into account that m is constant, we obtain ( ab) a( b) b( a) ( b) a( a) b. (5.76) m m B() 5. (5.77) 4 Fa away fom any localize cuent istibution the magnetic fiel is that of a magnetic ipole of ipole moment given by (5.74). The equation (5.77), howeve, is not complete fo a point magnetic ipole. In this case thee is an aitional tem that takes into account the fiel at the oigin epesenting a eltafunction fiel simila to the point electic ipole. We may fin the magnitue an the iection of this singula fiel by a moe caeful analysis of what happens as. To this en it is useful to wite the vecto potential of the ipole as follows m A (). (5.78) 4 We can the wite m m m B () A () 4 4. (5.79)

11 The last tem on the ight-han sie is just 4 m ( ). The fist one we have seen befoe, as it is the same as the electic fiel of an electic ipole an leas to eq.(5.77) when. It has also a singulaity at. tat by integating this tem ove a small sphee of aius centee at the oigin an then take the limit : whee we have use the ientity, vali fo any scala function f(x): m m a n, (5.8) V f () nf() a, (5.8) is the suface enclosing the omain V an n is the usual outwa unit nomal. Continuing, we have m m 4 n a ˆ a m. (5.8) Hence we fin that m contains the singula piece 4 m. Putting it into Eq.(5.79), we conclue that the elta-function piece of the magnetic fiel is 8 m, an hence the total fiel of the magnetic ipole is m m 8 B() m () 5. (5.8) 4 Consequences of the pesence of the elta-function piece ae obseve in a atomic hyogen whee the magnetic moment of the electon inteacts with that of the nucleus, o poton. Without this inteaction, all total-spin states of the atom woul be egeneate. As a consequence of the inteaction, the tiplet (spinone) states ae aise slightly in enegy elative to the singlet (spin-zeo) state. The splitting is small even on the scale of atomic enegies, being about -5 ev. The elta-function pat of the fiel also plays an impotant ole in the scatteing of neutons fom magnetic mateials. If the cuent is confine to a plane, but othewise abitay, loop, the magnetic moment can be expesse in a simple fom. If the cuent I flows in a close cicuit whose line element is l, (5.7) becomes I m l. (5.84) Fo a plane loop such as that in Fig..6, the magnetic moment is pepenicula to the plane of the loop. ince ½ l a, whee a is the tiangula element of the aea efine by the two ens of l an the oigin, the loop integal gives the total aea of the loop. Hence the magnetic moment has magnitue, egaless of the shape of the cicuit. m I Aea, (5.85) Fig.5.8

12 Foces on a Localize Cuent Distibution If a localize cuent istibution is place in an extenal magnetic fiel B(), it expeiences a foce accoing to Ampee s law. The geneal expession fo the foce is given by F J() B. (5.86) If the extenal magnetic fiel vaies slowly ove the egion of cuent, a Taylo seies expansion can be use to fin the ominant tem s in the foce. We expan the applie fiel aoun some suitably chosen oigin (within the cuent istibution) so that B B B ( ).... (5.87) The foce that the fiel exets on a localize cuent istibution locate aoun the oigin is then expane as follows: FB J ( ) J ( ) B ( ).... (5.88) Now, the fist integal in the last line vanishes fo a localize steay state cuent istibution (thee can't be any net flow of chage in any iection), an we can manipulate the integan in the final integal as follows: ( ) ( ) ( ) B B B. (5.89) an we may suppose that B is ue entiely to extenal souces so that B ( ) aoun the oigin. Thus we fin that the foce is F J ( ) B ( ) ( ) ( ) ( ) ( ) B J B J. (5.9) Now, we can wite the last integal as Futhe, ( ) () ( ) () ( ) () B J B J B J. (5.9) B( ) J( ) B( ) J( ). (5.9) This fact can be emonstate by consieing the i component of this integal: B( ) ( ) B( ) J( ) Ji xi. (5.9) x x x B( ) J( ) B( ) J( ) B( ) J( ) Hence fom eqs.(5.9), (5.9), an (5.9) we fin i i i ( ) ( ) ( ) ( ) F B J B J. (5.94) B ( ) m m B ( ) m B ( ) m B( ) m B( ) m B( ) Along the way in this eivation we have mae use of the facts that the ivegence an cul of B ae zeo in the egion nea the oigin. The final esult has the fom of the gaient of a scala function, F mb. (5.95)

13 whee the gaient is to be evaluate at the cente of the cuent istibution. Notice in paticula that thee is no foce if the applie magnetic inuction is unifom. Moe geneally, the foce is in the iection of the gaient of the component of B in the iection of m. The potential enegy of a pemanent magnetic moment (ipole) in an extenal magnetic fiel can be obtaine fom the expession fo the foce (5.95). If we intepet the foce as the negative gaient of the potential enegy U, we fin U mb. (5.96) This is well-known esult which shows that the ipole tens to oient itself paallel to the fiel in the position of lowest potential enegy.