Section 6: Magnetostatics

Transcription

1 agnetic fieds in matter Section 6: agnetostatics In the previous sections we assumed that the current density J is a known function of coordinates. In the presence of matter this is not aways true. The atoms in matter have eectrons that give rise to effective atomic currents, the current density of which is a rapidy fuctuating quantity. Ony its average over a macroscopic voume is known. Furthermore, the atomic eectrons contribute to intrinsic magnetic moments in addition to those resuting from their orbita motion. A these moments can give rise to dipoe fieds that vary appreciaby on the atomic scae. The process of averaging the microscopic quantities to obtain the macroscopic description on magnetic fieds in matter wi be discussed in the second part of E. The resut of this averaging is a macroscopic magnetic fied B which obeys the same equation B = (6.) as its microscopic anaog. Then we can sti use the concept of a vector potentia A whose cur gives B. In the presence of a magnetic fied, matter becomes magnetized; that is the dipoes acquire a net aignment aong some direction. There are two mechanisms that account for this magnetic poarization: () paramagnetism: the dipoes associated with the spins of unpaired eectrons experience a torque tending to ine them up parae to the fied; () diamagnetism: the orbita speed of the eectrons is atered in such a way as to change the orbita dipoe moment in a direction opposite to the fied. Whatever the cause, we describe the state of magnetic poarization by the vector quantity = magnetic dipoe moment per unit voume. is caed the magnetization; it pays a roe anaogous to the poarization P in eectrostatics. Note that ferromagnetic materias are magnetized in the absence of appied fied. Beow we wi not worry about how the magnetization got there it coud be paramagnetism, diamagnetism, or even ferromagnetism we sha take as given, and cacuate the fied this magnetization itsef produces. Suppose we have a piece of magnetized materia; the magnetic dipoe moment per unit voume,, is given. What fied does this object produce? The vector potentia of a singe dipoe m is given by μ m r Ar () =. In the magnetized object, each voume eement d carries a dipoe moment r r ( )d (Fig. 6.), so the tota vector potentia is μ Ar () = ( ) ( ) r r r r dr. (6.) r - r' A(r) d r' r r' Fig.6. O

2 Using the identity eq. (6.) can be written in the form Integrating by parts we obtain: r = r r μ Ar () = ( ) r r dr (6.), (6.4) where we used the identity μ r ( ) Ar () = ( ) d d r, (6.5) r r r ( ψ ) ψ ( ψ) a = a + a. (6.6) Now we rewrite the second integra in (6.5) using the divergence theorem. For arbitrary vector fied v(r) and a constant vector c the divergence theorem gives V dr v c = v c nda. (6.7) On the other hand using the cycic permutation in the tripe product we have and Therefore, eq.(6.7) can be written as Since c is arbitrary, it foows from eq.(6.) that Therefore, eq. (6.5) yieds S ( v c) = c ( v) v ( c) = c ( v ), (6.8) V ( v c) n= c ( v n ). (6.9) dr c v = c v n da. (6.) V ( v) dr = ( v n) S da. (6.) S μ r ( ) r ( ) n Ar () = d + da r. (6.) r The first term ooks ike the potentia of a voume current density, whie the second term ooks ike the potentia of a surface current density where n is the norma unit vector. With these definitions J =, (6.) K = n, (6.4) μ J( ) K( ) Ar () = d + da r. (6.5) r

3 This means that the potentia (and hence aso the fied) of a magnetized object is the same as woud be produced by a voume current J = throughout the materia, pus a surface current K = n, on the boundary. Instead of integrating the contributions of a the infinitesima dipoes, as in Eq.(6.), we first determine these bound or magnetization currents, and then find the fied they produce, in the same way we woud cacuate the fied of any other voume and surface currents. Notice the striking parae with the eectrostatics: there the fied of a poarized object was the same as that of a poarization voume charge ρ P = P pus a poarization surface charge σ P = P n. Now we discuss the physica picture for bound currents. Fig. 6.a shows a thin sab of uniformy magnetized materia, with dipoes represented by tiny current oops. Notice that if the magnetization is uniform a the interna currents cance. However, at the edge there is no adjacent oop to do the canceing. The whoe magnetized sab is therefore equivaent to a singe ribbon of current I fowing around the boundary (Fig. 6.b). Fig.6.a,b What is this current, in terms of? Say that each of the tiny oops has area a and thickness t (Fig. 6.). In terms of the magnetization, its dipoe moment is m = a t. In terms of the circuating current I, however, m = Ia. Therefore I = t, so the surface current is K = I/t =. Using the outward-drawn unit vector n (Fig. 6.b), the direction of K is convenienty indicated by the cross product: K = n. This expression aso records the fact that there is no current on the top or bottom surface of the sab; here is parae to n, so the cross product vanishes. Fig. 6. This bound surface current is exacty what we obtained above [eq. (6.4)]. It is a pecuiar kind of current, in the sense that no singe charge makes the whoe trip on the contrary, each charge moves ony in a tiny itte oop within a singe atom. Nevertheess, the net effect is a macroscopic current fowing over the surface of the magnetized object. Fig. 6.4

4 When the magnetization is nonuniform, the interna currents no onger cance. Fig. 6.4 shows two adjacent chunks of magnetized materia, with a arger arrow on the one to the right suggesting greater magnetization at that point. On the surface where they join there is a net current in the x-direction, The corresponding voume current density is therefore z Ix = [ z( y+ dy) z( y) ] dz = dydz, (6.6) y z ( J ) x =, (6.7) y By the same token, a nonuniform magnetization in the y-direction woud contribute an amount / z (Fig. 6.4b), so ( J ) y z = x z y. (6.8) In genera, then, J =, which is consistent, again, with the resut (6.). Incidentay, ike any other steady current, J obeys the conservation aw: because the divergence of a cur is aways zero. J =, (6.9) For magnetization ocaized in space in the presence of free currents J the expression for the vector potentia has the form μ ( ) + ( ) Ar () = Jr r r dr, (6.) where the integra is taken over a space. This is because the magnetization is zero at infinity. We note that the surface magnetization currents are incuded is eq.(6.) due to the abrupt change in at the boundary of the magnetized materia. Eq.(6.) impies that taking into account microscopic currents (magnetization) eads to the effective current density J+ and consequenty to a new macroscopic equation for the magnetic fied: μ B= J+. (6.) The term can be combined with B to define a new macroscopic fied H, Therefore, the macroscopic equations become H= B. (6.) μ H= J, (6.) B =. (6.4) Eq.(6.) is the Ampere s aw for magnetostatics with magnetized materias. It represents a convenient way to find magnetic fied H using ony free currents. In the integra form the Ampere s aw reads where I is the eectric free current passing through the oop. H d = I, (6.5) y 4

5 To compete the description of macroscopic magnetostatics, there must be a constitutive reation between H and B. For diamagnetic and paramagnetic materias and not too strong fieds it is customary to write the reation between the magnetization and magnetic fied in the form = χ m H. (6.6) Note that H rather than B enters this equation. The constant χm is caed the magnetic susceptibiity. It is dimensioness quantity which is positive for paramagnets and negative for diamagnets. Typica vaues are around -5. aterias that obey (6.6) are caed inear media. In view of eq. (6.) for inear media Thus B is aso proportiona to H: where μ = μ ( ) + χ m is the permeabiity of the materia. B= μ ( ) H+ = μ + χ m H. (6.7) B= μh, (6.8) For the ferromagnetic substances, eq.(6.8) must be repaced by a noninear functiona reationship, B= F( H ). (6.9) In fact, the function F(H) depends on the history of preparation of the materia which ead to the phenomenon of hysteresis. The incrementa permeabiity μ (H) is defined as the derivative of B with respect to H, assuming that B and H are parae. For high-permeabiity substances, μ/μ O can be as high as 6. Boundary Conditions Just as the eectric fied suffers a discontinuity at a surface charge, so the magnetic fied is discontinuous at a surface current. Ony this time it is the tangentia component that changes. Indeed, if we appy B = in the integra form to a thin gaussian pibox stradding the surface (Fig. 6.5), we obtain where B B Β n da =. (6.) S B B =. (6.) is the component of the magnetic fied B perpendicuar to the surface. Eq. (6.) tes us that is continuous at the interface. The perpendicuar component of H is however discontinuous if the magnetization of the two media a different: H H =. (6.) As for the tangentia components, from Ampere s aw H= J an amperian oop running perpendicuar to the current (Fig. 6.5) yieds or ( ) H d = H H = K. (6.) H H = K. (6.4) Thus the component of H that is parae to the surface but perpendicuar to the current is discontinuous in 5

6 the amount K. A simiar amperian oop running parae to the current reveas that the parae component is continuous. These resuts can be summarized in a singe formua: H H = K n where n is a unit vector perpendicuar to the surface, pointing "upward." A. (6.5) B H K B H Fig. 6.5 Boundary vaue probems in magnetostatics The basic equations of magnetostatics are B =, (6.6) H= J, (6.7) with some constitutive reation between B and H such as eq.(6.8) or (6.9). Since the divergence of B is aways equa to zero we can aways introduce a vector potentia A such that B= A. (6.8) In genera case when the reationship between H and B is non-inear the second equation (6.7) becomes very compicated even if the current distribution is simpe. For inear media with B= μh, the equation takes the form = μ A J. (6.9) Now if μ is constant over a finite region in space, in can be taken out of differentiation and the eq. (6.9) can written With the choice of gauge this becomes a Poisson equation ( A) A=μJ. (6.4) A = (6.4) A= μj, (6.4) simiar to what we had in vacuum with a modified current density μ J / μ. The situation is anaogous the treatment of uniform dieectric media where the effective charge density in Poisson equation is ε ρ/ ε. Soutions of eq.(6.4) in different inear media must be matched across the boundary surfaces using the boundary conditions. 6

7 agnetic scaar potentia If the current density vanishes in some finite region in space, i.e. J = This impies that we can introduce a magnetic scaar potentia, eq.(6.7) becomes H =. (6.4) such that H =, (6.44) just as E = in the eectrostatics. Assuming that the medium is inear [i.e. described by eq. (6.8)] and uniform (i.e. the magnetic permeabiity is constant in space) eq. (6.6) together with eq.(6.44) ead to the Lapace equation for the magnetic scaar potentia: =. (6.45) Therefore, one can use methods of soving differentia equations to find the magnetic scaar potentia and therefore the magnetic fieds H and B. In hard ferromagnets the situation becomes simper. In this case the magnetization is argey independent on the magnetic fied and therefore we can assume that is a given function of coordinates. We can expoit eqs. B= μ ( H+ ) and = B to obtain and hence B= μ H+ =, (6.46) H=. (6.47) Now using the magnetic scaar potentia (6.44) we obtain a magnetostatic Poisson equation: where the effective magnetic charge density is given by The soution for the potentia = ρ, (6.48) ρ =. (6.49) it there are no boundary surfaces is ρ ( ) ( ) () r = d = dr r. (6.5) r If is we behaved and ocaized in space, integration by parts may be performed to yied r ( ) () r = d ( ) d = ( ) dr r r r r r r r. (6.5) Here the first integra vanishes by the divergence theorem is reduces to the integra over the surface where the magnetization is zero. Changing the variabes of the differentiation, this equation can be rewritten as foows: ( ) () r = d r. (6.5) r Far from the region of nonvanishing magnetization the potentia may be approximated by () r =. (6.5) 4 d r mr r π r 7

8 Here m ( r )d is the tota magnetic moment. This is the scaar potentia of a dipoe, as we have found in the eectrostatics. In soving magnetostatics probems with a given magnetization distribution which changes abrupty at the boundaries of the specimen it is convenient to introduce the magnetic surface charge density. If a hard ferromagnet has voume V and surface S we specify (r) inside V and assume that it fas suddeny to zero at the surface S. Appication of the divergence theorem to ρ (6.49) in a Gaussian pibox stradding the surface shows that the effective magnetic surface change density is given by σ = n, (6.54) where n is the outwardy directed norma. Then instead of (6.5) the potentia is represented as foows r ( ) n r ( ) () r = d + da r. (6.55) r V An important specia case it that of uniform magnetization throughout the voume V. Then the first term vanishes; ony the surface integra over σ contributes. Exampes of boundary vaue probems in magnetostatics To iustrate different methods for the soution of boundary vaue probems in magnetostatics, we consider two exampes. The first exampe is a sab of magnetic materia which has a uniform magnetization oriented ether parae (Fig. 6.6a) or perpendicuar (Fig.6.6b) to the surfaces of the sab. The sab is infinite in the pane. We need to cacuate the magnetic fieds H and B everywhere in space. S (a) (b) H Fig. 6.6 Since eectric current J =, H = and H=. This impies that ρ = pays a roe of magnetic charge density, and H can be found ike eectric fied E in eectrostatics. In case (a), since =const, ρ = =, and therefore H = everywhere in space. Therefore B = outside the sab and B= μ inside the sab. In case (b) magnetization creates positive surface charge, σ = +, on the top surface and negative surface charge, σ =, on the bottom surface. These charges generate magnetic fied, H =, opposite to the magnetization within the sab and no fied outside, H =. This makes fied B zero everywhere in space. Another exampe is a sphere of radius a, with a uniform permanent magnetization parae to the z axis. The simpest way to sove this probem is to use the notion of magnetization charge and the magnetic scaar potentia. The probem reduces to finding the potentia for a specified change density σ ( θ ) = cosθ gued over the surface of a spherica she of radius R. We need to find the resuting potentia inside and outside the sphere. R Fig.6.7 8

9 Using the genera soution and taking into account that the potentia is finite at infinity and at the center of the sphere we have for the interior region and for the exterior region in θ = (, r ) = Ar P(cos θ), r R C (, r θ) = P(cos ), r out. (6.56) θ R. (6.57) + = r These two functions must be joined together by the appropriate boundary conditions at the surface itsef. Since the potentia is continuous at r = R we obtain that It foows from here that C (cos ) (cos θ ) AR P θ = P + = = R. (6.58) C AR + =. (6.59) The norma derivative of the potentia suffers a discontinuity at the surface so that Thus or using eq.(6.59): r out in = σ ( θ ). (6.6) r r= R C + = ( ) P(cos ) (cos ) cos θ AR P θ + = R = θ, (6.6) (+ ) AR P(cos θ ) = cos = θ. (6.6) It foows from here that the ony term which survives is the one with = so that A =, (6.6) and therefore According to eq. (6.56) the soution inside the sphere, r R, is C = R. (6.64) (, r θ) = rcosθ = z. (6.65) This impies that the magnetic fied H is H= ˆ = z =. (6.66) and the magnetic fied B is μ B= μ H+ =. (6.67) Note that B is parae to whie H is antiparae to inside the sphere. 9

10 Outside the sphere, r R, according to eq. (6.57) the potentia is This is the potentia which is produced by a point dipoe with the dipoe moment R (, r θ) = cos θ, r R. (6.68) r (, r mr θ ) =, (6.69) r R m=, (6.7) We see that for the sphere with uniform magnetization, the fieds are not ony dipoe in character asymptoticay, but aso cose to the sphere. For this specia geometry (and this ony) there are no higher mutipoes. The ines of B and H are shown in Fig The ines of B are continuous cosed paths, but those of H terminate on the surface because there is an effective surface-charge density σ.. Fig. 6.8 Lines of B and ines of H for a uniformy magnetized sphere. The ines of B are cosed curves, but the ines of H originate on the surface of the sphere where the effective surface magnetic "charge," σ, resides. agnetized Sphere in an Externa Fied; Permanent agnets Now we consider a probem of a uniformy magnetized sphere in an externa magnetic fied. We can use resuts of the preceding section, because of the inearity of the fied equations which aows us to superpose a uniform magnetic induction B = μh throughout a space. From eqs. (6.66) and (6.67) we find that the magnetic fieds inside the sphere are now μ B= B +. (6.7) H= B. (6.7) μ Now imagine that the sphere is not a permanenty magnetized object, but rather a paramagnetic or diamagnetic substance of permeabiity μ. Then the magnetization is a resut of the appication of the externa fied. To find the magnitude of we use eq.(6.8):

11 so that This gives a magnetization B= μh, (6.7) μ B + = μ B. (6.74) μ μ μ = B. (6.75) μ μ+ μ For a ferromagnetic substance, the arguments of the preceding paragraph fai. Equation (6.75) impies that the magnetization vanishes when the externa fied vanishes. The existence of permanent magnets contradicts this resut. The noninear reation B = F(H) and the phenomenon of hysteresis aow the creation of permanent magnets. We can sove equations (6.7), (6.7) for one reation between H and B by eiminating : B= μh+ B. (6.76) The hysteresis curve provides the other reation between B and H, so that specific vaues can be found for any externa fied. Equation (6.76) corresponds to a ine with sope - on the hysteresis diagram with intercept B on the y axis, as in Fig Suppose, for exampe, that the externa fied is increased unti the ferromagnetic sphere becomes saturated and then decreased to zero. The interna B and H wi then be given by the point marked P in Fig The magnetization can then be found from eq. (6.7) or eq. (6.7) with BB =. The reation (6.76) between B and H is specific to the sphere. For other geometries other reations pertain. Fig.6.9