Section 1: Main results of Electrostatics and Magnetostatics. Electrostatics

Transcription

1 Chage density ection 1: ain esults of Electostatics and agnetostatics Electostatics The most fundamental quantity of electostatics is electic chage. Chage comes in two vaieties, which ae called positive and negative, because thei effects tend to cancel. Chage is conseved: it can not be ceated o destoyed. Chage is quantized. The fact is that electic chages come only in discete lumps integes multiples of the basic unit chage e. Howeve, the fundamental unit of chage is so tiny that in macoscopic applications this chage quantization can be ignoed. Continuous distibution of chage is descibed in tems of the chage density. The chage density ρ() is the local chage pe unit volume, so that the total chage in volume V is Q= dq= ρ() dv = ρ() d. (1.1) V V V Fig.1.1 O V d ρ() If chage q is localized in a point of space, say, this chage is called a point chage. In this case the chage density is given in tems of the delta function: ρ() = qδ ( ). (1.) Fo N point chages q i located at positions i (i = 1 N) the chage density is given by N ρ() = q δ ( i). (1.) i= 1 In addition to volume chage density we will also be dealing with suface and line chage densities. The suface chage density σ ( ) is given the local suface chage pe unit suface, such that the chage in a small suface aea da at point is then dq = σ ( ) da. imilaly, the line chage density λ( ) is given by the local line chage pe unit length dl, such that the chage in a small element of length dl at point is then dq = λ( ) dl. Coulomb s law All of electostatics oiginates fom the quantitative statement of Coulomb s law concening the foce acting between chaged bodies at est with espect to each othe. Coulomb, in a seies of expeiments, showed expeimentally that the foce between two small chaged bodies sepaated in ai a distance lage compaed to thei dimensions vaies diectly as the magnitude of each chage, vaies invesely as the squae of the distance between them, is diected along the line joining the chages, and is attactive fo the oppositely chaged bodies and epulsive if the bodies have the same type of chage. i 1

2 Theefoe, accoding to Coulomb s law the foce acting on chage q located at point due to chage q 1 located at point 1 is qq1 1 F () =, (1.4) whee is the pemittivity of fee space. 1 Although the thing that eventually gets measued is a foce, it is useful to intoduce a concept of an electic field. The electic field is defined as the foce exeted on a unit chage located at position vecto. It is a vecto function of position, denoted by E. F = qe, (1.5) whee F is the foce, E the electic field, and q the chage. Accoding to Coulomb the electic field at the point due to point chages q i located at positions i (i=1 N) is given by N 1 E () = q i = i 1 i i, (1.6) which is the supeposition of field poduced by the individual point chage. The electic field can be visualized in tems of lines stating on positive chages and teminating on negative chages. ρ(') d ' - ' E() Fig.1. O ' If thee is continuous distibution of the chage it can be descibed by a chage density ρ ( ) as is shown schematically in Fig.1.. In this case the sum in Eq.(1.6) is eplaced by an integal: 1 1 E () = dq ρ( ) d =. (1.7) V V imilaly we can wite expessions fo the field poduced by suface and line chage densities. Gauss s law Thee is an impotant integal expession, called Gauss s law, which elates the flux of electic field acoss a closed suface and the electic chage inside this suface electic E(). The flux of electic field though a suface is the integal of the nomal component of the electic field ove the suface, En da, whee n is the nomal to this suface. The flux ove any closed suface is a measue of the total chage inside. Indeed fo chages inside the closed suface field lines stating on positive chages should teminate on negative chages inside o pass though the suface. On the othe hand, chages outside the suface will contibute noting to the total flux, since its field lines pass in one side and out the othe. This is the essence of Gauss s law. Fo a continuous chage density ρ(), Gauss s law takes the fom: 1 En = () =, (1.8) da ρ d Q V

3 whee V is the volume enclosed by and Q is the total chage inside the volume V. This equation is one of the basic equations of electostatics. Note that it depends upon the invese squae law fo the foce between chages, the cental natue of the foce, and the linea supeposition of the effects of diffeent chages. Theefoe, Gauss s law is the consequence of Coulomb s law. Diffeential fom of Gauss s Law Gauss s law (1.8) can be thought of as being an integal fomulation of the law of electostatics. We can obtain a diffeential fom by using the divegence theoem. The divegence theoem states that fo any well-behaved vecto field A() defined within a volume V suounded by the closed suface the elation A nda = d A (1.9) V holds between the volume integal of the divegence of A and the suface integal of the outwadly diected nomal component of A. The equation in fact can be used as the definition of the divegence. The divegence theoem allows us to wite Eq.(1.8) as follows ρ E d=. (1.1) V ince this equation is valid fo an abitay volume V, we can put the integand equal to zeo to obtain ρ E =, (1.11) which is the diffeential fom of Gauss s law of electostatics. cala Potential The single equation (1.11) is not enough to specify completely the thee components of the electic field E(). A vecto field can be specified completely if its divegence and cul ae given eveywhee in space. Thus we look fo an equation specifying cul E as a function of position. uch an equation follows diectly fom Coulomb s law (1.7) because it can be witten as follows: 1 1 E () = ρ( ) d. (1.1) ince the cul of the gadient of any well-behaved scala function of position vanishes ( ψ =, fo all ψ), it follows immediately fom this equation that Ε =. (1.1) Eq. (1.1) is the consequence of the cental natue of the foce between chages and of the fact that the foce is a function of elative distances only. Accoding to Eq.(1.1) the electic field (a vecto) is deived fom a scala by the gadient opeation. ince one function of position is easie to deal with than thee, it is wothwhile to define the scala potential Φ() by the equation: E = Φ. (1.14) Then the scala potential is given in tems of the chage density by 1 ρ( ) Φ () = d. (1.15)

4 whee the integation is ove all the space, and Φ is abitay only to the extent that a constant can be added to the ight-hand side of this equation. The scala potential has a physical intepetation when we conside the wok done on a test chage q in tanspoting it fom one point A to point B in the pesence of an electic field E(), as is shown in Fig. 1.. Fig.1. The foce acting on the chage at any point is F = qe, so that the wok done in moving the chage fom A to B is B B B B W = F dl= q E dl= q Φ dl = q dφ= q Φ Φ. (1.16) A A A A ( B A ) Fo a localized chage distibution it is convenient to place a efeence point at infinity so that the potential at point is defined as follows: Φ () = E dl. (1.17) This shows that qφ can be intepeted as the potential enegy of the test chage in the electostatic field. It also follows fom eq. (1.16) that E d l =. (1.18) This is consistent with Ε = because of the tokes's theoem saying that if A() is a well-behaved vecto field, is an abitay open suface, and C is the closed cuve bounding, then C ( ) A dl= A nda. (1.19) whee dl is a line element of C, n is the nomal to, and the path C is tavesed in a ight-hand scew sense elative to n. Fo a chage density ρ( ) placed in the extenal potential Φ( ) the potential enegy is U = ρ( ) Φ() d. (1.) This is diffeent fom the potential enegy of a chage distibution ρ( ) which is defined as the enegy which is equied to build this distibution by binging the chages fom infinity. In this case Φ( ) is the potential poduced by the chage distibution ρ( ) and the enegy is 1 W = Φ ρ() () d, (1.1) Whee ½ appeas due to double counting. This enegy can also be ewitten in tems of electic field alone W = E d all. (1.) space 4

5 Poisson and Laplace Equations We see that the behavio of an electostatic field can be descibed by the two diffeential equations: ρ E =, (1.) Ε =. (1.4) the latte equation being equivalent to the statement that E is the gadient of a scala function, the scala potential Φ: E = Φ. (1.5) Equations (1.) and (1.5) can be combined into one patial diffeential equation fo the single function Φ(x): ρ Φ=. (1.6) This equation is called the Poisson equation. In egions of space that lack a chage density, the scala potential satisfies the Laplace equation: Φ=. (1.7) Dipole potential At a lage distance fom a localized change distibution the electostatic potential exhibits a multipole natue. This follows fom eq.(1.15) if we assume that.the fist non-vanishing tem gives a monopole contibution to the potential: whee Q Φ () =, (1.8) is the total chage. If Q =, the potential is given by a dipola tem whee p is the dipole moment Enegy in a slowly vaying electic field Q= ρ( ) d, (1.9) 1 p Φ () =, (1.) p ( ) d. (1.1) = ρ If the extenal electic field vaies slowly ove the egion whee a chage distibution ρ( ) is localized one expend the electostatic enegy given by (1.) d U = ρ( ) Φ() in multipoles. By expending the potential Φ( ) aound the oigin of coodinates in a Taylo seies we obtain = (1.) Φ ( ) =Φ () + Φ( ) +... =Φ() E() +..., (1.) 5

6 so that U ρ() d Φ() ρ() d E() = QΦ p E. (1.4) The fist tem is a monopole contibution and the second tem is the dipole contibution to the electostatic enegy. Highe ode tems ae not included. Dielectics If electic field is applied to a medium made up of lage numbe of atoms o molecules, the chages bound in each molecule will espond to applied field which will esults in the edistibution of chages leading to a polaization of the medium. The polaization P(') is defined as the dipole moment pe unit volume. Polaization is a macoscopic quantity. The potential poduced by polaization P(') is Φ () = As was shown in Phys.91, this eq. can be ewitten ( ) 1 ( ) P V d. (1.5) 1 P ( ) 1 P ( ) n Φ () = d + da. (1.6) V As follows fom this expession, the polaization of the medium poduces an effective chage which can be intepeted as bound chage o polaization chage. Thee ae two contibutions to the bound chage bulk and suface. The bulk polaization chage density is given by The suface polaization chage density is ρ () = P(). (1.7) P σ P () = P() n. (1.8) If the integation in eq.(1.6) is pefomed ove all space the second tem can fomally be omitted (it is due to the abupt change of the polaization on the suface of the dielectic). We can, theefoe, make a geneal statement that the pesence of the polaization poduces an additional polaization chage so that the total chage density becomes ρtotal = ρ fee + ρp = ρ P. (1.9) The espective electostatic equations involve a macoscopic electic field which is the aveage ove volume which includes a lage numbe of atoms. The accuate pocedue fo the macoscopic aveaging will be discussed late. Taking into account eq.(1.9) we can wite the divegence of E as follows: It is convenient to define the electic displacement D, 1 E= [ ρ P ]. (1.4) D= E+ P, (1.41) Because this field is geneated is geneated by fee chages only. Using the electic displacement the Gauss s law takes the fom D =ρ. (1.4) In the integal fom it eads as follows: Dn da = ρ() d. (1.4) V 6

7 This is paticulaly useful way to epesent Gauss s law because it makes efeence only on fee chages. Connecting D and E is necessay befoe a solution fo the electostatic potential o fields can be obtained. Fo a linea esponse of the system the displacement D is popotional to E, whee is the electic pemittivity. D= E+ P = E, (1.44) If the dielectic is not only isotopic, but also unifom, then is independent of position. The Gauss s law (1.4) can then be witten ρ E =. (1.45) In this case all poblems in that medium ae educed to those with no electic polaization, except that the electic fields poduced by given chages ae educed by a facto /. The eduction can be undestood in tems of a polaization of the atoms that poduce fields in opposition to that of the given chage. One immediate consequence is that the capacitance of a capacito is inceased by a facto of / if the empty space between the electodes is filled with a dielectic with dielectic constant /. Bounday conditions Fo solving electostatics poblems one needs to know bounday conditions fo the electic field. Conside a bounday between diffeent media, as is shown in Fig.1.4. The bounday egion is assumed to cay idealized suface chage σ. Conside a small pillbox, half in one medium and half in the othe, with the nomal it to its top pointing fom medium 1 into medium. Accoding to the Gauss s law D n da =σ A, (1.46) whee the integal is taken ove the suface of the pillbox and A is the aea of the pillbox lid. In the limit of zeo thickness the sides of the pillbox contibute nothing to the flux. The contibution fom the top and A D D n, esulting in bottom sufaces to the integal gives ( ) D 1 D D = σ, (1.47) 1 whee is the component of the electical displacement pependicula to the suface. Eq. (1.47) tells us that thee is a discontinuity of the D at the inteface which is detemined by the suface chage. A E D σ E 1 D 1 l Fig. 1.4 chematic diagam of the bounday suface between diffeent media. 7

8 Now we conside a ectangula contou C such that it is patly in one medium and patly in the othe and is oiented with its plane pependicula to the suface. ince the cul of electic field is zeo we have E d l =. (1.48) Fo the ectangula contou C of infinitesimal height this integal is equal to ( 1) component of electic field paallel to the suface. This implies that 1 i.e. the tangential component of electic field is always continuous. The electostatic potential is continuous acoss the bounday. E E l, whee E is the E = E, (1.49) Electic cuents agnetostatics In electostatics electic fields which ae constant in time ae poduced by stationay chages. In magnetostatics magnetic fields that ae constant in time ae poduces by steady cuents. If the mobile volume chage density is ρ and the velocity is v, then J = ρv. (1.5) Fig. 1.5 The cuent cossing a suface can be witten as In paticula, the total chage pe unit time leaving a volume V is I = Jda = J n da. (1.51) V ( J) J nda = d. (1.5) Because chage in conseved, whateve flows out though the suface must come at the expense of that emaining inside: d ρ J d= d= d. (1.5) dt t ( ) ρ V V V The minus sign eflects the fact that an outwad flow deceases the chage left in V. ince this applies to any volume, we conclude that ρ J =. (1.54) t This is the pecise mathematical statement of local chage consevation. It is called the continuity equation. 8

9 agnetostatics deals with steady cuents which ae chaacteized by no change in the net chage density anywhee in space. Consequently in magnetostatics ρ / t = and theefoe J =. (1.55) We note hee that a moving point chage does not constitute a steady cuent and theefoe cannot be descibed by laws of magnetostatics. Biot and avat Law agnetic phenomena is convenient to descibe in tems of a magnetic field B. Biot and avat (in 18), fist, and Ampee (in ), in much moe elaboate and thoough expeiments, established the basic expeimental laws elating the magnetic field B to the electic cuents and established the law of foce between one cuent and anothe. These expeiments can be summaized in two elatively simple expessions. The magnetic field poduced by a cuent distibution of density J ( ) at point is given by ( ) μ ( ) B () = d J. (1.56) whee μ is the pemeability of fee space. This is Biot and avat law which is an analog of the Coulomb s law in electostatics. The magnetostatic foce poduced by a magnetic field B( ) on a object caying a volume cuent density J () is ( ) F= J() B d. (1.57) Fo a moving chaged paticle this equation is actually epesents the Loentz foce f = q( v B ), (1.58) whee q and v ae a chage and velocity of the paticle. Diffeential Equations of agnetostatics and Ampee's Law The magnetic field (1.56) can be witten in the fom It follows immediately fom hee that μ ( ) B () = J This is the fist diffeential equation of magnetostatics. d. (1.59) B =. (1.6) Calculating the cul of B fom eq. (1.59) fo steady-state cuents, J =, leads to B=μ J. (1.61) This is the second equation of magnetostatics. This is a diffeential fom of Ampee s law. The integal fom of the Ampee s law can be obtained by applying tokes s theoem. Integating the nomal component of the vectos in the left- and ight-hand side of Eq.(1.61) ove open suface shown in Fig.1.6 we obtain: B nda = μ J nda. (1.6) 9

10 Fig. 1.6 Using the tokes s theoem it can be tansfomed into B dl= μ J nda. (1.6) C ince the suface integal of the cuent density is the total cuent I passing though the closed cuve C, Ampee s law can be witten in the fom: C B dl=μ I. (1.64) Just as Gauss s law can be used fo calculation of the electic field in highly symmetic situations, so Ampee's law can be employed in analogous cicumstances. Vecto Potential The basic diffeential laws of magnetostatics ae B =, (1.65) B=μ J. (1.66) Accoding to eq. (1.65) the divegence of B is zeo which implies that thee ae no souces which poduce a magnetic field. Thee exist no magnetic analog to electic chage. Accoding to eq. (1.66) a magnetic field culs aound cuent. agnetic field lines do not begin o end anywhee to do so would equie a nonzeo divegence. They eithe fom closed loops o extend out of infinity. Now the poblem is how to solve diffeential equations (1.65) and (1.66). If the cuent density is zeo in the egion of inteest, B = pemits the expession of the magnetic field B as the gadient of a magnetic scala potential, B = Φ. Then (1.65) educes to the Laplace equation fo Φ, and all ou techniques fo handling electostatic poblems can be bought to bea. A lage numbe of poblems fall into this class, as was discussed in Phys. 91. A geneal method of attack is to exploit equation (1.65). If B = eveywhee, B must be the cul of some vecto field A(), called the vecto potential: B () = A (). (1.67) We have, in fact, aleady witten B in this fom (1.59). Evidently, fom (1.59), the geneal fom of A is A μ ( ) J () = d+ λ(). (1.68) The added gadient of an abitay scala function Ψ shows that fo a given magnetic induction B, the vecto potential can be feely tansfomed accoding to A () A () + λ(). (1.69) This tansfomation is called a gauge tansfomation. We will discuss gauge tansfomations in detail late. 1

11 uch tansfomations on A ae possible because (1.67) specifies only the cul of A. The feedom of gauge tansfomations allows us to make A have any convenient functional fom we wish. If (1.67) is substituted into the equation (1.66), we find o A =μ J. (1.7) ( A) A=μ J. (1.71) If we now exploit the feedom implied by (5.9), we can make the convenient choice of gauge, A =. (1.7) In this case each ectangula component of the vecto potential satisfies the Poisson equation, A= μ J. (1.7) Fom ou discussions of electostatics it is clea that the solution fo A in unbounded space is μ ( ) A () = J d It is easy to see by taking diectly the divegence of eq. (1.74) that indeed A =. agnetic dipole moment. (1.74) At a lage distance fom a localized cuent distibution we can conside the asymptotic behavio of the vecto potential. We have shown in Phys. 91 that the fist non-vanishing contibution is given by the magnetic dipole tem μ m A () =. (1.75) whee m is the magnetic moment: 1 m= [ ()] d J. (1.76) Foces on a Localized Cuent Distibution If a localized cuent distibution is placed in an extenal magnetic field B(), it expeiences a foce accoding to Ampee s law. The geneal expession fo the foce is given by eq. (1.57) ( ) F= J() B d. (1.77) If the extenal magnetic field vaies slowly ove the egion of cuent, a Taylo seies expansion can be used to find the dominant tem in the foce. We expand the applied field aound some suitably chosen oigin within the cuent distibution ( ) ( ) ( ) = B = B + B( ) (1.78) The foce that the field exets on a localized cuent distibution is then expanded as follows: ( ) d ( ) F= B J ( ) + J ( ) B ( ) d+.... (1.79) Now, the fist integal in the last line vanishes fo a localized steady state cuent distibution (thee can't be any net flow of chage in any diection). The second integal afte some tansfomations (see Phys. 91) gives = 11

12 F= ( m B ), (1.8) whee the gadient is to be evaluated at the cente of the cuent distibution and m is the magnetic moment (1.76). Notice in paticula that thee is no foce if the applied magnetic induction is unifom. oe geneally, the foce is in the diection of the gadient of the component of B in the diection of m. The potential enegy of a pemanent magnetic moment in an extenal magnetic field can be obtained fom the expession fo the foce (1.8). If we intepet the foce as the negative gadient of the potential enegy U, we find U = m B. (1.81) This is well-known esult which shows that the dipole tends to oient itself paallel to the field in the position of lowest potential enegy. agnetic Fields in atte In the pesence of matte, atomic electons give ise to effective atomic cuents (o bound cuents) which poduce the obital magnetic moment. In addition, electons have thei intinsic magnetic moments due to electon s spin. Thus matte has the magnetic dipole moment. The magnetic dipole moment pe unit volume is known as the magnetization. The magnetization poduces a magnetic field which can be descibed though the vecto potential. In the magnetized object, each volume element total vecto potential is given by - ' A() μ A () = d caies a dipole moment ( ) ( ) d ( )d (Fig. 1.7), so the. (1.8) d ' ' Fig.1.7 O It was shown in Phys.91 couse that eq. (1.8) can be tansfomed to μ ( ) ( ) n A () = d + da. (1.8) The fist tem looks like the potential of a volume cuent density, while the second tem looks like the potential of a suface cuent density whee n is the nomal unit vecto. With these definitions J =, (1.84) K = n, (1.85) 1

13 μ J( ) K( ) A () = d + da. (1.86) V What this means is that the potential (and hence also the field) of a magnetized object is the same as would be poduced by a volume cuent J = thoughout the mateial, plus a suface cuent K = n, on the bounday. Instead of integating the contibutions of all the infinitesimal dipoles, as in Eq. (1.8), we fist detemine these bound o magnetization cuents, and then find the field they poduce, in the same way we would calculate the field of any othe volume and suface cuents. Notice the stiking paallel with the electostatics: thee the field of a polaized object was the same as that of a polaization volume chage ρ P = P plus a polaization suface chage σ P = P n. Incidentally, like any othe steady cuent, J obeys the consevation law: J =, (1.87) because the divegence of a cul is always zeo. If the integation in eq. (1.86) is pefomed ove all space the second tem can be omitted (it appeas in this integation due to the abupt change of the magnetization on the suface). That implies that in geneal the effective cuent density in a magnetic medium epesents a sum of fee cuents and magnetization cuents, J+. Consequently a new macoscopic equation fo the magnetic field aveaged ove lage numbe of atoms in medium is μ ( ) B= J+. (1.88) The tem can be combined with B to define a new macoscopic field H, 1 H= B. (1.89) μ Theefoe, the macoscopic equations become H = J, (1.9) B =. (1.91) Eq.(1.9) is the Ampee s law fo magnetostatics with magnetized mateials. It epesents a convenient way to find magnetic field H using only fee cuents. In the integal fom the Ampee s law eads H d l = I, (1.9) whee I is the electic fee cuent passing though the loop. To complete the desciption of macoscopic magnetostatics, thee must be a constitutive elation between H and B. Fo diamagnetic and paamagnetic mateials and not too stong fields the elation is linea and is given by whee μ is the pemeability of the mateial. Bounday Conditions ( ) B= μ H+ = μh. (1.9) Just as the electic field suffes a discontinuity at a suface chage, so the magnetic field is discontinuous at a suface cuent. Only this time it is the tangential component that changes. Indeed, if we apply B = in the integal fom Β n da =. (1.94) 1

14 to a thin gaussian pillbox staddling the suface (Fig. 1.8), we obtain whee B B B1 =. (1.95) is the component of the magnetic field B pependicula to the suface. Eq. (1.95) tells us that B is continuous at the inteface. The pependicula component of H is howeve discontinuous if the magnetization of the two media a diffeent: ( ) H H =. (1.96) 1 1 As fo the tangential components, fom Ampee s law H = J an ampeian loop unning pependicula to the cuent (Fig. 1.8) yields H d l= ( H H1) l = Kl. (1.97) o H H = K. (1.98) 1 Thus the component of H that is paallel to the suface but pependicula to the cuent is discontinuous in the amount K. A simila ampeian loop unning paallel to the cuent eveals that the paallel component is continuous. These esults can be summaized in a single fomula: 1 ( ) H H = K n whee n is a unit vecto pependicula to the suface, pointing upwad. A. (1.99) B H K B 1 H 1 l Fig. 1.8 agnetic scala potential If the cuent density vanishes in some finite egion in space, i.e. J =, eq.(1.9) becomes This implies that we can intoduce a magnetic scala potential H =. (1.1) Φ such that H = Φ, (1.11) just as E = Φin the electostatics. Assuming that the medium is linea and unifom (i.e. the magnetic pemeability is constant in space) eq.(1.11) togethe with eqs. (1.9) and (1.91) lead to the Laplace equation fo the magnetic scala potential: Φ =. (1.1) Theefoe, one can use methods of solving diffeential equations to find the magnetic scala potential and theefoe the magnetic fields H and B. 14

15 In had feomagnets the situation becomes simple. In this case the magnetization is lagely independent on the magnetic field and theefoe we can assume that is a given function of coodinates. We can exploit eqs. B= μ ( H+ ) and = B to obtain and hence ( ) B= μ H+ =, (1.1) H =. (1.14) Now using the magnetic scala potential (1.11) we obtain a magnetostatic Poisson equation: whee the effective magnetic chage density is given by The solution fo the potential Φ = ρ, (1.15) ρ =. (1.16) Φ it thee ae no bounday sufaces is 1 ρ ( ) 1 ( ) Φ () = d = d. (1.17) In solving magnetostatics poblems with a given magnetization distibution which changes abuptly at the boundaies of the specimen it is convenient to intoduce magnetic suface chage density. If as had feomagnet has volume V and suface we specify () inside V and assume that it falls suddenly to zeo at the suface. Application of the divegence theoem to ρ (1.16) in a Gaussian pillbox staddling the suface shows that the effective magnetic suface change density is given by σ = n, (1.18) whee n is the outwadly diected nomal. Then instead of (1.17) the potential is epesented as follows 1 ( ) 1 n ( ) Φ () = d + da. (1.19) V An impotant special case it that of unifom magnetization thoughout the volume V. Then the fist tem vanishes; only the suface integal ove σ contibutes. As an example conside a slab of magnetic mateial which has a unifom magnetization oiented ethe paallel (Fig. 1.9a) o pependicula (Fig.1.9b) to the sufaces of the slab. The slab is infinite in the plane. We need to calculate the magnetic fields H and B eveywhee in space. (a) (b) H Fig. 1.9 ince electic cuent J =, H = and H =. This implies that ρ = plays a ole of magnetic chage density, and H can be found like electic field E in electostatics. In case (a), since =const, ρ = =, and theefoe H = eveywhee in space. Theefoe B = outside the slab and B= μ inside the slab. In case (b) magnetization ceates positive suface chage, σ =+, on the top suface and negative suface chage, σ =, on the bottom suface. These chages geneate magnetic field, H =, opposite to the magnetization within the slab and no field outside, H =. This makes field B zeo eveywhee in space. 15