Several Rules about the Magnetic Moment of Rotational Charged Bodies

Transcription

1 IES ONLINE, VOL. 3, NO. 6, Severa ues about the Magnetic Moment of otationa Charged Bodies Guo-Quan Zhou Department of hsics, Wuhan Universit, Wuhan 43007, China Abstract A strict and deicate anaog reation between the magnetic moment of a rotationa charged bod and the rotation inertia of a rigid bod about a fied ais has been found in this paper. Based on this anaog reation, man rues and s about cacuating the rotation inertia of a rigid bod can be transpanted to fied of eectromagnetism and used to cacuate the magnetic moment of a rotationa charged bod. Some principes or s are etended, generaied, iustrated, and enumerated. Some reated eampes are isted in the paper. DOI: 10.59/IES EFACE eference [1] gives the anaog reation between the magnetic moment [3 5] of a rotationa charged bod and the rotation inertia of a rigid bod about a fied ais, which is ver stimuating and pedagogica. According to this paper, a rotationa charged bod with an eement eectric charge of dq, a constant anguar speed of ω (the sie of ω) about a fied ais, and a rotation radius of r, must have a definite magnetic moment of foowing magnitude: d m = diπr or = (ω/π) dq πr = (ω/) r dq d( m /ω) = r dq Its direction is parae or anti-parae to ω according to the positive or negative sign of dq respective. The above epression has a strict and deicate anaog reation to the rotation inertia epression dj = r dm for an eement mass of dm about a definite ais: m /ω J, dq dm, q m Let us pa more attention to the anaog reation of m /ω J, which is more appropriate than the corresponding one given in the reference []. Because on m /ω is the phsica quantit that independent of ω and on impressed b the distribution of eectric charge and position of ais. Thus there is a strict anaog reation between m /ω and J. Nevertheess, on the same time we shoud notice that, charged conductor which is in a state of a eectrostatic equiibrium on has its charge distributed on its surface. So we must treat the rotationa charged soid conductor as a charged she, anaogica to the hoow rigid bod of the same shape. Other anaog reationship between the two circumstances is: Q(the tota charge vaue) m(the tota mass), C e (the center of eecric charge) C m (the mass center), σ e (eectrica charge area densit) σ m (mass area densit), ρ e (eectrica charge bod densit) ρ m (mass bod densit), λ e (eectrica ine densit) λ m (mass ine densit). It is obvious that man computation rues and propert of rotation inertia for a rigid bod can be transpanted to that of the magnetic moment for a rotationa charged bod. Just as in reference [6], the rotation inertia for an triange rigid pane with homogeneous mass distribution about its center-of-mass ais vertica to this rigid pane can be formuated as: J ABC(Cm) =

2 IES ONLINE, VOL. 3, NO. 6, m(a +b +c )/36. According to the anaog reation given as above, we can immediate deduce the magnetic moment formua for a rotationa charged triange pate with homogeneous charge distribution about its center-of-charge ais vertica to the pate to be: mabc(ce) = Q(a +b +c )ω/7. This paper etends and appies the concusions drawn from references [6 9] to the case for cacuation of magnetic moment of a rotationa charged bod, tabuates and enumerates reated eampes of those s and makes some further discussion.. THE AALLEL-AXIS THEOEM FO THE MAGNETIC MOMENT OF A OTATIONAL CHAGED BODY The parae-ais [6 9] of rotation inertia for a rigid bod is universa to a cases of an shape and an mass distribution. Then there is some reviews are in order: once we know the rotation inertia of rigid bod about its center-of-mass ais, we can cacuate its rotation inertia about an other parae ais. Based on above anaog reation, first of a, we can define the eectrica charge center r c = 1 rdq. And suppose the vaue of magnetic moment of a rotationa Q charged bod about one of its ais passing through its charge center is mc (the corresponding anguar speed is ω), the distance between the two parae aes is d, the tota charge vaue is Q, then foowing concusion hods: ω m = mc + Qd (1) ω c In need of simpicit and conciseness, we can take ω = ω c (but notice that ω and ω c are independent of each other), and attain a formua: m = mc + ωq d (1 ) This is the parae-ais for the magnetic moment of a rotationa charged bod with an shape and an eectric charge distribution (a voume, surface, curve or discrete points distribution). No matter what a kind of charged bod it is, the above formua awas hods for a rigid bod or a kind of iquid matter, a conductor or a kind of eectrosis. Its appication eampes can be found in Tabe 1. There is a ver impotent deduction for (1 ): if the tota charge vaue of a rotationa charged bod is ero, then m = mc and it has nothing to do with distance d. That is to sa: for a bod with a tota ero charge and a nonero charge densit, its magnetic moment about an ais with an direction is equa to that of a charged bod about the center-ofcharge ais. 3. THE OTHOGONAL-AXES THEOEM FO THE MAGNETIC MOMENT OF A OTATIONAL CHAGED BODY For a thin sice of rigid pate of ero thickness, its rotation inertia about the,, aes respective satisfies the so-caed orthogona-aes [7, 9]. (provided the o coordinate pane is ocated upon the rigid pate): J = J + J Based on the anaog reation, for a thin pate with an arbitrar surface charge distribution, its magnetic moments about the,, aes with an anguar speed of ω, ω, ω respective wi satisf the foowing orthogona-aes : m ω + m ω = m ω () For simpicit and conciseness, without oss of generait, we can aso assume ω = ω = ω, and attain foowing equation: m + m = m ( ) Its appication eampes can be found to be eampe (3) in Tabe 1. Now et us pa more attention to the so-caed generaied orthogona-aes pubished in one of m papers [7]. For

3 IES ONLINE, VOL. 3, NO. 6, a coumn rigid bod with an arbitrar shape of transversa section and of uniform mass distribution aong its generatri, with a height of L, a mass of m, with its Z ais parae to its generatri and the coordinate pane of o pumb to its generatri ine, amputating it to be two parts, each with a ength of L 1 and L respective (L 1 + L = L), as Fig. 1 shows. Then the foowing generaied orthogona-aes hods J + J = J + 3 m ( L 1 L 1 L + L ) Figure 1: A coumn with height of L. According to the anaog reation, for a uniform charged rotationa coumn which has a tota charge vaue of Q and anguar speeds of ω, ω, ω about the,, aes respective, its corresponding magnetic moments must satisf the foowing orthogona-aes when foowing substitution is taken: ω m J, Q m. m /ω + m /ω = m /ω Q(L 1 L 1 L + L ) (3) For simpicit and conciseness, assuming that ω = ω = ω = ω (notice that ω is not the resutant anguar veocit of ω, ω, ω ), then m + m = m + ω 3 Q(L 1 L 1 L + L ) (3 ) When L 1 = 0, L = L, we have m + m = m + ω 3 QL (3 ) (The end face orthogona aes ) When L 1 = L = L, we have m + m = m + ω 1 QL (3 ) (The orthogona aes on the midde transversa section) We can get its appication eampes such as No. 4, No. 5, No. 6 in Tabe. 1 b means of anaog transition from reference [7]. What shoud be paid more attention to is that when the charged coumn has a tota ero charge vaue and a nonero charge densit, the generaied orthogona-aes (3), (3 ) reduce to the origina orthogona-aes (), ( ). That is to sa, its magnetic moment is independent of the coumn ength. It is ver important and usefu to cacuate the magnetic moment of a rotationa charged bod, especia for a charged bod with a four-degree smmetr ais (in terms of group theor). The eampes Nos. 5 and 6 in Tabe 1 are its appication.

4 IES ONLINE, VOL. 3, NO. 6, Tabe.1. The tabe of cacuation rues and eampes of magnetic moment for rotationa charged bodies. (rovided a matter is even charged with tota charge vaue of Q and with a constant voume (surface, inear) charge densit) Eampes Form of charged bodies ues and s Formua of magnetic moment No. 1.The thin and straight rod ( inear) The anaog reation and definition m 1 Q ω 4 No..The rectanguar pate ( surface distri. ) The paraeais m 1 Q ω 6 No. 3. The circuar and thin pate ( surface distri. ) No. 4. The cuboid ( voume distri. ) No. 5.The soid Cinder (voume distri. and aia sm.) No. 6.The thin and hoow cinder (surface distri. and aia sm.) o c a b The orthogonaaes The etended orthogona - aes The etended orthogona aes The etended orthogonaaes { 1 m Qω ( mthe simiar) 8 1 m Q ω 4 1 m Qac ( ) ω 4 (, the simiar) m m thesimiar m ( Q QL ) m m ω 1 1 ( Q QL ) ω 4 4 No. 7.The soid gobe ( voume distri. and spherica sm.) The originmoment m 1 Q ω 5 No. 8.The thin Spherica crust ( surface distri. and spherica sm.) The originmoment m 1 Q ω 3 4. THE OIGIN-MOMENT THEOEM For those rigid bodies of arbitrar shape and mass distribution, random seected coordinate sstem O-XYZ, the rotation inertia J, J, J about the aes X, Y, Z satisf the foowing equation J + J + J = r dm(= J o ) Here J o is defined as the origin moment [8, 9] (aso caed the center moment). r is the distance between the mass eement and the origin point. The above epression is caed origin moment or center moment, which has been mentioned in man references [8, 9]. And its atest manifestation can be found in reference [8].

5 IES ONLINE, VOL. 3, NO. 6, According to the simpe anaog reation given before, we have J ω m, J ω m, J ω m and dm dq Then we can get the origin-moment reated to moments of m, m, m m + m + m = r dq (4) ω ω ω For simpicit and conciseness, assuming that ω = ω = ω (= ω). Then m + m + m = ω r dq (4 ) Especia for those rotationa charged bodies of spherica smmetr, their voume charge densit and eement charge can be epressed as ρ e = ρ e (r), dq = 4πr ρ e (r)dr. Then m + m + m = 4πω r 4 ρ e (r)dr But due to the spherica smmetr m = m = m, we have m = m = m = 4 3 πω r 4 ρ e (r)dr (4 ) Its appication eampes are enumerated in Tabe 1, or No. 7, No CONCLUSIONS Such a perfect and deicate anaog reation between the magnetic moment of a rotationa charged bod and the rotation inertia of a rigid bod supp us with a powerfu too to cacuate precise the magnetic moments of rotationa charged bodies with a kinds of shape. Formuas such as (1 ) (4 ) can a be viewed as s to dea with a kinds of probem invoved in magnetic moment cacuation of rotationa charged bodies, especia for those rotationa bodies of smmetrica charge distribution. Tabe 1 gives some concrete iustrations of these formuas. Such cacuation is ver necessar and meaningfu for research and teaching of the eectromagnetic fied and even for investigating of space such as the aunching of sateites and spaceships. ACKNOWLEDGMENT Supported b Nationa Science Foundation Grant. No EFEENCES 1. Zhou, G.-Q., Charge moment tensor and the magnetic moment of rotationa charged bodies, rogress in Eectromagnetics esearch, Vo. 68, , New York, Xu, D.-F and Z.-S. Yu, The cacuation of magnetic moment for a rotationa charged bod rotating around a fied ais, Universit hsics, V16, No. 4, 3 4, Higher Education ress, Beijing, 1997, (Ch). 3. Cheng, D. K., Fied and Wave Eectromagnetism, 10 13, Addison-Wese ubishing Compan, New York, Lim, Y. K., Introduction to Cassica Eectrodnamics, Word Scientific, Badassare, D. B., Cassica Theor of Eectromagnetism, rentic Ha, Enge-wood Ciffs, New Jerse, Zhou, G.-Q., Computing rotationa inertia of rigid bodies b smmetric operation and dimensiona theor, Journa of Wuhan Universit, Vo. 9, No. 5, 90 94, Eng. Ed., 1996, (Ch). 7. Zhou, G.-Q., The etension of orthogona aes about moment inertia of rigid bod, hsics Buetin, Vo. 8, No., 4 5, 1998, (Ch). 8. Ma, T.-J., A New Theorem to Cacuate the Moment Inertia of igid Bodies hsics and Engineering, Vo., No. 1, 8 9, Qinghua Universit ress, Beijing, 1995, (Ch). 9. Chen, J.-F., Cassica Mechanics, 165, ress of China Universit of Sci. & Tech., He Fei, 1990, (Ch).