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the Eur J Phys In11 11990) 9 4 9 8 . Prlnted 1 I 94 UK Electrostatic charges in v x B fields:the Faraday diskandthe rotatingsphere Paul Lorrain Department of Geological Sciences, McGill University, Montreal, Quebec, Canada, H3A 2A7 and Departamento de Fisica, Facultad de Ciencias, Universidad de Murcia, Espinardo, Murcia, Spain Received 16 June 1989 Abstract. When a conductor moves in a magnetic field, this vector is not zero, then the divergence of E is not zero either and the conductor carries electrostatic charges whose fields are as important as v x B. The Faraday disk and a conducting sphere rotating in a uniform magnetic field serve as examples. This little known effect plays a fundamental role in magnetohydrodynamic phenomena. RCumB. A l’interieur d’un conducteur en mouvement dans un champ magnetique, v x B agit comme une source distribuee. Si la divergence de ce champ n’est pas nulle, la divergence de E est non-nulle egalement et le conducteur porte des charges electrostatiques dont le champ est aussi important que v x B. Le disque de Faraday, et une sphere conductrice tournant dans un champ magnetique uniforme, servent a illustrer notre propos. Cet effet peu connu joue un r d e essentiel dans les phenomenes magnetohydrodynamiques. 1. Introduction 2. Electrostatic charges in v x 6 fields Imagine a body that moves ata velocity Y in a region where there exist an electric field E and a magnetic field B. Then anelectric charge Q inside the body feels a force Q(E Y x B). Thus, inside the moving body, the Y x B field acts like the electric field ofadistributed source. We are concerned here with conducting media that move in magnetic fields. We shall see that they carry electrostatic charges whose field is just as important as v x B. Indeed, there are many cases where the two fields cancel each other exactly at every point. The Faraday disk and a conducting sphere rotating in a magnetic field will serve as examples, but this little known effect plays a fundamental role in magnetohydrodynamics. Another article will discuss the function ofthese electrostaticcharges in thedynamo mechanism that generates the Earth’s magneticfield. We first consider a general velocity v , but most of our discussion will concerntherotation of rigid bodies, either connected toa stationary circuit or not. We perform our calculations both with respect to a fixed reference frame and with respect to a rotating frame. It is well known that conductors do not support an electric spacecharge;anyextrachargedeposited insidemoves outtotheperipheryalmostinstantaneously (Lorrain et al 1988, p 75). However, few physicists realise that conductors do carry anelectric space charge when subjected to a v x B field whose divergence is not equal to zero. If the conductor is isolated, then it also carries a compensating surface charge. v x B acts like a distributed source. If the divergence of + 2.1. Fixed reference frame Choose a fixed reference frame S. With respect t o S, the electric field strength is E and the magnetic flux density B. If now a point inside a body of conductivity Q moves a t a velocity Y with respect to S, then the electric current density at that point, again with respect to S, is J = u(E + Y x B). (1) The body need not be rigid and all the terms can be both space and time dependent. The Y x B field acts like the electric field of a distributed source (Lorrain et a1 1988, p 494). 0143-08071901010094 + 05 $03.50@ 1990 IOP Publlshlng Ltd 8 The European Physlcal Soclety Electrostatic charges in Y x B fields 95 Nowtakethe divergenceof both sides. Under steady-state conditions, the divergenceof J is zero and, if the medium is homogeneous, V * E =-V.(U X B). (2) Thus, if the divergenceof v x B is not zero, there exists an electrostatic space charge of density Q = &,%V E = -&,%V = &,%[v (V X (v X B) - B * (V B) X V)]. (3) (4) Observe that the chargedensity is independent of the conductivity. The relative permittivity of good conductors is not measurable, butit is presumably of the order of three, like that of ordinary dielectrics. We use p for the distance to theaxis of rotation, e (varrho) for the space charge density, a for the conductivity,and s (varsigma) forthe surface charge density. With an idealdielectric, J and a would be both zero, equation (1) would be meaningless, and these expressions for e would not apply. One author argues that, in conducting media such as plasmas in space, the charge density is in fact zero because oftheabundance of chargecarriers.He concludes that, under steady-state conditions, E = 0. That is of course a misconception; the above equations apply in any medium and E # 0 if the divergence of v x BZO. We shall see that, if the conductor is isolated, surface charges compensate for the space charge. If the fields are time dependent, andif the medium is not uniform, then there still exists an electric space charge, but there are further terms on the right-hand side of equation (4). It follows from equation (1) that E = -(v x B) + J/a. (5) Thevectors J and B arenotindependent;they satisfy the Maxwell equation V X B=hJ (6) assumingthatthemedium is non-magnetic,and under steady-state conditions. Now consider a rotating body. If a rigid axisymmetric conducting body rotates at an angular velocity W about the z axis in any magnetic field, then v = up$ 4 vx v = 2wi (7) where and 2 are the unit vectors pointing, respectively, in the positive 4 and z directions.The azimuthalcurrentcomesfromtherotation of the spacecharge,and possibly alsofromaninternal source. If there is no internally generated azimuthal current, then the conductor need not be axisymmetric. Let us calculate the curl of J for a passive rotating conductor that is not necessarily axisymmetric. From equation (l), in a homogeneous medium, VxJ=a[VxE+Vx(vxB)] + = a[-aB/at v X (W X B)]. Understeady-stateconditionsthetimederivative vanishes and V x J = aV x (v x B). (9) (10) (1 1) Suppose now that our rotating conductor is not connected to a stationary circuit through sliding contacts.Supposealsothatthemagnetic field is both uniform and parallel to the axis of rotation. Both W and B are constant. Then VxJ=aVx(wp$xBi)=O. (12) If the magneticfield is not uniform, butaxisymmetric, then it turns out that the curl is again zero: V xJ=v(V*B)=O. (13) Now, in our isolated rotating conductor, currents can only flow around closed circuits. Then the line integral of J around anyclosed circuit is equal tozero and there are no induced currents: J=O E= -(W X B). (14) In this instance, the field of the electrostatic charges cancels the v x B field exactly at every point. This is not a new result (van Blade1 1984), but it is important for thefollowing reason. Since there are no induced currents in the rotating conductor, the magnetic field is not disturbed and the lines of B are not dragged alongby the moving conductor, as one would expect from Alfven’s ‘theorem’ (Alfven and Falthammar 1963). Another paper will discuss this ‘theorem’ at some length. If the applied magnetic field is symmetrical about an axis that forms an angle with the axis of rotation, then we can consider it to be composed of an axisymmetric field B,,, plus a transverse field B , f . Then + V x J = aV x [v x (B,,, B , i ) ] = aV x ( w p ~ , $x a) = aV x (wpB,cos~i) (15 ) (16) (17) and there are induced currents in the conductor. 2.2. Rotating reference frame Rotating reference frames are troublesomein relativity because they are non-Euclidian and because clock rate depends on the distance to the axis of rotation (Msller 1974). Fortunately, the situationhere is simple because we need onlycomparethe valuesof the various variables at, andin the immediate neighbourhood of, a given point in the two frames. We use unprimed variables in the inertial reference frame S, as above. Primed variables at a given point P’in the rotating reference frame S’ apply in the local inertial reference frame S” occupied momentarily by P’ (Msller 1974). Thentherelations between the P Lorrain 96 primed and unprimed variables are those of special relativity (Lorrain et a1 1988, pp 287, 303, 321). We disregard terms of the order of v2/c2,where c is the speed of light. Then E' = E + v J' = J X B =J'/d a' = a B' = B. 't (18) (19) Also, in a uniform medium, v X J' = av X E T= - a a ~ p t t . (20) Recalling now that, in an isolated moving conductor, there areno induced currents if the curl of the current density is zero, this equation shows that there are induced currents in an isolated moving conductor only if the magnetic induction in the reference frame of the conductor is time dependent. This is in agreement with equations (13) and (14). The space charge density is the same in the two reference frames if the azimuthal current results only from the motion of the space charge. 3. The Faraday disk Figures 1 and 2 show schematic diagrams of a homopolar motor and of a homopolar generator, as viewed from a fixed reference frame (Lorrain et a1 1988, p 399t). Both devices are also called Faraday disks. We have shown single brushes on the axis and on the rim, but there are in fact continuous brushes all around. Also, we have not shown the coils that generate theapplied magnetic fields B. At any point in the disks, v 2 << c'. Assume that the axial magnetic field B is that of a long solenoid, and is thus uniform. This assumptionis not realistic, but it makes the calculation simple and instructive. We may set J+ = 0 in equation (8) because, as we shall see below, the convection current of the rotating space charge generates an opposing Flgure 2. Homopolar generator. axial magnetic field that is negligible. We disregard the magnetic field of the radial current through the disks because it is azimuthal, which makes its v x B equal to zero. Set B, = B in equation (8). Say B and CO are the same in figures 1 and 2. Then v x B and Q are the same and, from equation (8), the space charge density is uniform and negative: e= ( 2~1E) , Q w B . This space charge density is independent of the conductivity, except if a were zero, and it is independent of whether the switch SW in figure 2 is open or closed. Therotation of this space charge generates an azimuthal convection current of density ea whose magnetic field opposes the above B. Since that B is uniform, it is easy to calculate the magnetic induction B, of this azimuthal convection current at the centre of the disk. Setting R and S equal to the outer radius and to the thickness of the disk, respectively, we find that, for S << R, Figure 1. Homopolar motor. since w2sR is of the order of v 2 , and vz/c2<< 1 by hypothesis. Now disregard end effects. This is equivalent to stretching the disk into a long cylinder. This assumption does not invalidate the above approximation. Then Observe that, here, E, = - ( U X B). ?Note that in this reference B is negative. The v x B field 'pumps' conduction electrons in the - j direction, and their space charge cancels the v x B field exactly at every point in both figures 1 and 2, whether the switch SW of figure 2 is open or closed. The simplest case is that of the generator, with SW Electrostatic charges in Y x B fields 97 open. Then J = 0, E+v X B=O (25) E = Ep. A surface charge of density E , E ~ ~ compensates BR for the negative space charge withoutaffecting the electric field inside the cylinder. Now close the switch SW in the circuit of the generator.Conduction electrons escape at the axis into the external circuit, where the v x B is zero, and return at the rim. The field J/a in the disk points in the direction of the rim. The case of the motor is similar, except that Jpoints in the - j direction: current flows in the direction of v x B in the generator, and in the opposite direction in the motor. The current in the homoplanar motor depends not only on the characteristics of the motor, but also on the source that feeds it and on its mechanical load, while the current in the generator depends similarly on the applied torque and on the load resistance. Let us now calculate the potentials V and A for the motor. From Lorrain et al (1988, p 349), under steady-state conditions and for a long solenoid, aA/at = o A = (BPI216 (26) Since J = Z/(2nps), where the current Z is independent of the radius, The quantity p. is a constant of integration; we have arbitrarily set the second term equal to zero at p = po. In the rotating reference frame, from Lorrain et a1 (1988, pp 287, 303, 321),and from equations (26) and (2819 Figure 3. 4. Conducting sphere rotating in a uniform magnetic field Figure 3 shows a passive conducting sphere of radius R rotating at anangular velocity w in a uniform magnetic field B parallel to the axis of rotation. The sphere is uncharged. As in section 2.1, there are no induced currents and equations (14) apply. It will be instructive to discuss this simple case (see also Thomson 1893, Mason and Weaver 1929, van Blade1 1984). We assume that the sphere is rigid: a further paper will discuss the effectof differential rotation with reference to the Earth's core. We can again disregard theazimuthalcurrent associated with the rotation of the space charge. Then equation (21) applies: the space charge is uniform throughout the sphere, and negative. As we shall see, positive surface charges maintain the net charge equal to zero. This negative space charge establishes a radial electric field The electrostatic field of the space charge now canfirst cels the v x B field only to acertainextent, because the electric field is along - P , while the v x B field is along fi, and second because the magnitudes of the two fields are not equal. Equations(14) still apply, however: the combined electrostatic fields ofthe space and surface charges cancel the v x B field. We can find the surface density c of electrostatic charge as follows. From the Gauss law, w2p2 =A --A C2 op2J + --In-$. c2 0 p Po - In this last equation the second term on the right is proportional to v 2 / c 2and is therefore negligible. The third term in negligible for the same reason because J is proportional to the sum of two terms, E v x B, that are of the same order of magnitude. Thus A' = A aA'/at' = o E' = - V V ' . (33) + c = Er.outside - Er'% Er,ins>de. (35) Inside the sphere, in cylindrical coordinates, E= - ( V X B) = - ~ p B j (36) or, in spherical coordinates, E = - wr sin BB(sin BP + cos 06) (37) Er,insrde = - wrB sin2B. Also, inside the sphere, the electric fieldis radial P Lorrain 98 and the potential along the axis is uniform. Call that potential V,. Then, on the surface, from equation (361, Vsurface = V, + wBp2/2= V, + wBR2sin28/2. (39) Outside the sphere, the space charge density is zero, Laplace’s equation applies, and we can expand V as a series of Legendre polynomials (Lorrain et al 1988, p 225): show that the net surface charge is equal to minus the space charge. The value of the potential inside the sphere follows from equations (36) and (43): (47) The potential is equal to zero on an axial cylindrical surface whose radius is equal to (2/3)’’2R. 5. Conclusion The first term, which is the monopole term, implies the existence of a net charge on the sphere. That potential is of no interest; it would apply only if the sphere carried a net charge. The second, dipole, term implies a charge asymmetry between the upper and lower halves of the sphere. This term is also zero because both E and E, are symmetrical about the equatorial plane. Let us therefore set thethird, or quadrupole, term of equation (40) equal to the V of equation (39) at r = R: V ’ B, 3cos28 - 1 + +oBR2sin20= R3 2 We have therefore shown that, if aconducting medium moves in a magnetic field, and if the divergence of the ZJ x B field is not zero, then there exists an electrostatic space charge and its associated electric field. If the conductor is isolated, then surface charges correct for any charge imbalance. If the conductor is isolated and if the magnetic field isaxisymmetric, then the fieldof the electrostatic charges cancels ZJ x B exactly at every point, there are no induced currents, the rotating conductor does not affect the magnetic field, and the Alfven ‘theorem’ does not apply. (41) Acknowledgment Upon matching corresponding terms, we find that B2 = -wBRS/3 V, = -wBR2/3. I thank David Crossley and Oliver Jensen for several fruitful discussions. (43) References Thus Finally, fromequations (35), (38), and (45), the surface charge density is given by = E,,wBR[(Q + E,)sin20 - I]. (46) The surface charge density is positive in the region of the equator, and negative near the poles. It is easy to Alfven H and Falthammar C-G 1963 Cosmical Electrodynamics (Oxford: Clarendon) p 189 Lorrain P, Corson D R, and Lorrain F 1988 Electromagnetic Fields and Waves 3rd edn (NewYork: Freeman)Mason M and Weaver W 1929 The Electromagnetic Field (New York: Dover) Moller C 1914 The Theory of Relativity 2nd edn (London: Oxford University Press) ch 8 and 9 Thomson J J 1893 Notes on Recent Researches on Electricity and Magnetism (Oxford: Clarendon) P 534 van Blade1 J 1984 Relativity and Engineering (Berlin: Springer) p 287 INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 22 (2001) L1–L2 www.iop.org/Journals/ej PII: S0143-0807(01)13153-3 LETTERS AND COMMENTS Comment on electrostatic charges in v × B fields Dragan V Redžić Faculty of Physics, University of Belgrade, POB 368 11001 Beograd, Yugoslavia Received 22 May 2000 Abstract A basic error is rectified in Lorrain’s paper (1990 Eur. J. Phys. 11 94) on conducting media that move in magnetic fields. Lorrain (1990) gave an interesting discussion concerning conducting media that move in magnetic fields, focusing on steady-state conditions and disregarding terms of the order of v 2 /c2 . It seems, however, that some of his results are either wrong or derived in a wrong way for the following reason. In his argument the author used the well-known constitutive equation for a linear isotropic medium at rest, D = ε0 εr E , as if it were valid for the medium in motion too. That, of course, is not so; in the first-order theory for a moving non-magnetic medium discussed by Lorrain, the correct constitutive equation is D = ε0 E +ε0 (εr −1)(E + v × B ) (see, for example, Rosser (1964)). Therefore, one must rectify Lorrain’s results accordingly. For example, one gets D = ε0 E inside a homogeneous uniformly rotating isolated conductor in a magnetic field that is symmetrical about the axis of rotation. (Under steady-state conditions E + v × B = 0 inside the conductor since the conduction current density vanishes, as Lorrain pointed out.) As one can see, Lorrain’s equations concerning rotating bodies will become correct after substituting εr = 1 in them and thus the troublesome relative permittivity of good conductors disappears from final results (compare Jefimenko (1989)). By the way, if Lorrain’s equation (46) were valid one could, in principle, measure the relative permittivity of good conductors. Also, it should be mentioned that Lorrain inadvertently used the same symbol J for both the conduction and the total current densities. For example, in the Maxwell equation ∇ × B = µ0 J that is valid for a moving homogenous non-magnetic medium under steadystate conditions and when the curl of (E + v × B ) × v is zero (Rosser 1964), J is the total current density that includes the convection as well as the conduction current densities. The reader of Lorrain’s paper must therefore be cautious with J . Especially, it appears that under steady-state conditions the divergence of the conduction current density in the general case is not zero inside a moving medium. (The divergence of the total current density is always zero under steady-state conditions.) Consequently, in the general case the space charge density inside a moving homogeneous not necessarily rigid body satisfies the equation = ε0 [v · (∇ × B ) − B · (∇ × v )] − (ε0 εr /σ )∇ · (v ) (1) at variance with Lorrain’s equation (4). For a rigid axisymmetric conductor that rotates uniformly in a magnetic field that is symmetrical about the axis of rotation (the z axis), the divergence of v is zero and the above equation together with ∇ × B = µ0 J imply Lorrain’s equation (8), without εr , of course. As one can see, this is true for an isolated conductor as well as for the Faraday disk connected to a stationary circuit through sliding contacts. The present author believes that the above remarks could add to the pedagogical value of Lorrain’s fine article. 0143-0807/01/010001+02$30.00 © 2001 IOP Publishing Ltd Printed in the UK L1 L2 Letters and Comments References Jefimenko O D 1989 Electricity and Magnetism 2nd edn (Star City: Electret Scientific Company) ch 12 Lorrain P 1990 Eur. J. Phys. 11 94–8 Rosser W G V 1964 An Introduction to the Theory of Relativity (London: Butterworths) p 337 INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 22 (2001) L3–L4 www.iop.org/Journals/ej PII: S0143-0807(01)16453-6 LETTERS AND COMMENTS Electrostatic charges in v × B fields Paul Lorrain Earth and Planetary Sciences, McGill University, Montréal, Québec, Canada Received 17 August 2000 Abstract Professor D V Redžić corrects an error that appeared in my 1990 paper on electrostatic space charges in v × B fields. These charges are almost invariably disregarded, despite the fact that they play a crucial role throughout magnetohydrodynamics. Many thanks to Professor D V Redžić for his corrections to my 1990 paper. The constitutive equation for D in a moving medium is indeed the one that he gives, and not simply D = r 0 E , as I assumed. The Rosser book that he refers to is a solid reference, except for the fact that it uses vectors only sparingly. The book by Penfield and Haus (1967) is also a good reference on moving media, except that it does not state the constitutive equations, and that it makes no reference to space charges. Thus, as Professor Redžić correctly points out, if E + (v × B ) = 0, then, surprisingly, D = 0 E . So the space-charge density inside a conductor that rotates in an axial, steady, magnetic field is given by = −0 ∇·(v × B ). (1) As to the convection current density v , it is negligible if v 2 c2 , at least in the case of the Faraday disc as I have shown, and presumably more generally. So my equation (4) is in general correct, after substituting 1 for r . Two points are worth noting. First, these space charges play an essential role whenever conductors move in magnetic fields. The reason is that J = σ [E + (v × B )] (2) and the E usually comes from . If you disregard the space charge, then you disregard E and you get into absurd situations. For example, it has been known since Thomson (1893) that there are no induced currents in a conducting sphere that rotates in an axial magnetic field. This is extremely mystifying because J = 0, despite the fact that v × B = 0! Moffatt (1978), who does not know about the existence of , resorts to the following ‘explanation’: as the sphere rotates, the lines of B , inside, are stationary with respect to the sphere, and ‘disconnect and reconnect’ at the surface! Second, very few authors know about the existence of this space charge and of its field. Moffatt is just one example. Among the many thousands of papers and books on the magnetohydrodynamics of the Earth, of the Sun, and of space, no author takes into account this electric field. Rarely, an author will deduce the expression for (setting r = 1), but then disregard its electric field. Parker (1979) finds on page 46 of his book the correct expression for (setting r = 1), but then goes on to say that, since ‘the universe is saturated with electric charge’, ‘a slight rearrangement takes care of the induced electric field’. In other words, he is saying that the Maxwell equation for the divergence of E does not apply! That statement is not too surprising on his part because, throughout his book, he also disregards the Maxwell equation for the curl of B . 0143-0807/01/010003+02$30.00 © 2001 IOP Publishing Ltd Printed in the UK L3 L4 Letters and Comments References Lorrain P 1990 Eur. J. Phys. 11 94–8 Moffatt H K 1978 Magnetic Field Generation in Electrically Conducting Fluids (Cambridge: Cambridge University Press) Parker E N 1979 Comical Magnetic Fields (Oxford: Clarendon) Penfield P Jr and Haus H 1967 Electrodynamics of Moving Media [Research Monograph No. 40] (Cambridge, MA: MIT) Redžić D V 2001 Eur. J. Phys. 22 L1 Rosser W G V 1964 An Introduction to the Theory of Relativity (London: Butterworths) p 337 Thomson J 1893 Notes on Recent Researches on Electricity and Magnetism (Oxford: Clarendon)