A New Derivation of Dirac's Magnetic Monopole Strength

Magnetic monopole derivation
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  This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: content was downloaded on 21/08/2014 at 02:28Please note that terms and conditions apply. A new derivation of Dirac's magnetic monopole strength View the table of contents for this issue, or go to the  journal homepage for more 2003 Eur. J. Phys. 24 111( usMy IOPscience  I NSTITUTE OF  P HYSICS  P UBLISHING  E UROPEAN  J OURNAL OF  P HYSICS Eur. J. Phys.  24  (2003) 111–114 PII: S0143-0807(03)36133-1 A new derivation of Dirac’s magneticmonopole strength P V Panat Department of Physics, University of Pune, Pune-411 007, Maharashtra, India Received 24 April 2002, in final form 18 November 2002Published 13 January 2003Online at Abstract A new derivation of the strength of Dirac’s magnetic monopole is presentedwhich does not require an explicit form of the magnetic induction in terms of  g , the magneticpolestrength. The derivationessentially uses a modificationof Faraday’s law of induction and quantization of angular momentum. 1. Introduction It was usual in introductory texts of earlier days to postulate two types of magnetic pole (orsources), namely the magnetic ‘north pole’ and the magnetic ‘south pole’. This is done inanalogy with electrostatics where two types of charge occur. In electricity, the two types of charge are designated as positive and negative charges and are quantized with the basic unitof   | e | =  4 . 8  ×  10 − 10 esu (or 1.6  ×  10 − 19 C). In magnetism, however, it is found that, nomatter how small one makes pieces of magnet, north and south poles are inseparable. If suchan isolated pole was to be discovered, then, in analogy with electrostatics, it would becomenatural to postulate a force law of 1 / r  2 type, where the isolated poles would experience acentral force. However,we see no isolated magnetic poles. We thereforesay that no magneticmonopole has been hitherto discovered. We always see that a magnet manifests at least asa dipole. This means that the magnetic flux threaded by any closed surface is zero. Thisremainstrue evenin a time-varyingsituation. Maxwellincorporatedit as oneof his equations,namely  ∇ ·   B  =  0. Under these circumstances, we are forced to use an Amperian modelin which the magnetic field in the matter is produced by a multitude of tiny rings of electriccurrent distributed throughoutthe magnetic material, which leads us to infer that the matter isconstituted of electric charges. So far, this conclusion is not challenged by any experimentalevidencebecause, so far, nomagneticmonopolehas beenfoundexperimentally. Dirac, truetohisreputationofextractingaphysicalmeaningfromapparentlyanomaloussituations(e.g.e + ),was fascinated by the possibility of the existence of the two types of magnetic monopole. Herelated the magnetic monopole strength to the electric charge strength (electronic charge)  e via quantum mechanics.Dirac’s idea of a magnetic monopole has lured many top-ranking physicists since itsenunciationin 1931 [1]. This is chiefly fortwo reasons, namely,(1) it is intrinsically beautifuland (2) even if one magnetic monopole exists in the universe, that would entail the discrete 0143-0807/03/020111+04$30.00 © 2003 IOP Publishing Ltd Printed in the UK 111  112 P V Panat (quantized) nature of electric charge. Despite the failure of experimental endeavors to track downafreemagneticmonopole,manytheoreticalaswellasexperimentalpapersstill continueto appear in the literature. Dirac’s srcinal argument hinges on two basic assumptions [2]:(1) there should be least deviation from the accepted form of electromagnetic theory and (2)the existenceof a vectorpotential    A (  r  , t  )  is assumedwhichnaturallymanifestsin the minimalcoupling of the electron monopole interaction Hamiltonian  H  int   as  H  int   = e  −  emc   p ·   A ,  (1)where the monopole strength  g  enters through the vector potential    A (  r  , t  ) ,    being a scalarpotential. Here, we use the Gaussian (cgs) system of units. Subsequently, Dirac invented astring of magnetic dipoles attached to the monopole extending up to infinity where anothermonopoleofstrength − g  isattached. Theshapeofthestringisarbitrary. Eachshapegivesriseto a vector potential. All of these shapes of the string give rise to different vector potentials,which are gauge equivalent. The monopole strength  g  appears in the calculation of the vectorpotential    A (  r  , t  ) .Diracthencalculatedthesolidanglesubtendedbythesurfaceoftwo configurationsofthestrings at the point of observation. This solid angle times  g  is Dirac’s gauge function whichappears in quantum mechanics in the phase of the electronic wavefunction. The single-valuednature of the wavefunction then gives the well known value of lowest magnetic monopolestrength in terms of   e ,  c  and ¯ h , as2 egc = ¯ h .  (2)It appears that in this derivation, the idea of string must be invoked because    B  = ∇ ×   A must hold. Moreover, to get    A  in terms of   g , one needs some length measure such that theassociated magneticmoment is expressedin terms of   g . This length is a lengthof the string of dipolesofDirac. Themonopolefieldis thendefinedas thedifferencebetween    B  and    B  where   B  is the magnetic induction on the string [2]. Another argument is given by Goldhaber [3];it discusses scattering of an electron in the field of a magnetic monopole. He tacitly assumesthe impulse approximation, in conjunction with the Coulomb-type force law    B  =  gr  2 ˆ r   andcalculates the angular momentum imparted to the electron in the process of scattering withthe assumption that the smallest change in the angular momentum (orbital) of the electron isquantized in units of ¯ h . Goldhaber equates the change of angular momentum of the electronto ¯ h  andgets Dirac’s conditionofequation(2). Goldhaber’sargumentsgivethe correctfactorsof equation (2), which were missing in the earlier derivations of Saha [4] and of Wilson [5]. 2. A new derivation of Dirac’s result We now present an alternative derivation which does not invoke the specific form of theCoulomb-type relationship between    B  and  g . In Goldhaber’s derivation, the scattering centreis the monopole and it is the electron that gets scattered. We present an alternative approachin which the electronis stationaryand the monopoleis movingin the field of the electron witha velocity   v  (along the  y -axis) so large in magnitude that its trajectory is hardly altered. Letthe coordinates of the electron be  ( 0 , 0 , b ). At any instant  t  , the magnetic charge density  ρ m and magnetic current density    J  m  are ρ m  =  g δ(  x  )δ(  z )δ(  y − v t  )  (3 a )   J  m  =  g  vδ(  x  )δ(  z )δ(  y − v t  ).  (3 b )The monopole modifies the conventional Maxwell equations in vacuum to ∇ ×   B  =  1 c ∂    E  ∂ t  +4 π    J  e c (3 c )  A new derivation of Dirac’s magnetic monopole strength 113 Figure 1.  The instantaneous positions of the electron and the monopole in a scattering event. ∇ ·   B  = 4 πρ m  (3 d  ) ∇ ×   E   =− 1 c ∂    B ∂ t  −  4 π    J  m c (3 e ) ∇ ·   E   = 4 πρ e .  (3  f  )Here,  ρ e  and    J  e  are the electric charge and electric current density respectively.A crucial modification of Faraday’s law will now be used: as the monopole moves from  y  =−∞ to + ∞ , the electric field produced along the ring L in figure 1 is |  E  |=−  12 π bc ∂ m ∂ t  −  4 π g vδ(  y − v t  ) 2 π bc (4)where   m  is the magnetic flux.The total impulse imparted to the electron in the  x  -direction is then P  x   =    ∞−∞ eE   d t   =−  e 2 π bc  m  + ∞−∞ +2 egbc .  (5)Clearly, the magnetic flux term   m  has the same value at  t   = ±∞  and thus cancels out. Itshould be noted that the evaluation of    m  needs a specific form of the force law; however,any physically sustainable theory would require a force to go to zero when the finite sourceis at infinity. At worse, on the other hand, the force could be a constant. This will decide thevalue of    m . This   m  will have the same value when the monopole is at  y  → −∞ as whenit is at  y  → ∞ . Thus, if initially the electron is at rest, then the minimum change in angularmomentum of the electron, which is quantized, is  bP  x  . The   m  term in equation (5) is zeroand thus we get2 egc = ¯ h  (6)which is Dirac’s result. 3. Discussion It isto benotedthat,inthederivationofequation(6),nospecificlawofforceis invokedtofind  m  and the corresponding term in equation (5) is zero by symmetry arguments. Moreover,since the vector potential does not appear explicitly anywhere, the concept of string is notnecessary. Since we have used integrated quantities and not trajectories under a specific
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