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A difference between solenoid and magnetic spin
Journal of Magnetism and Magnetic Materials 43 (1984) 59-60 North-Holland, A m s t e r d a m
59
A DIFFERENCE BETWEEN SOLENOID AND MAGNETIC SPIN
E. COMAY Department of Physics, Tel-Aviv University, TeI-Aviv 69978, Israel Received 15 August 1983; in revised form 5 January 1984
The interactions of a circular ring through which an electric current flows with a magnetic monopole and that of a magnetic spin with a monopole are discussed. An inconsistency of the calculation of energy is shown.
Two experiments will describe a difference between a solenoid and a magnetic spin. Units where c = 1 are used in this work. The first experimental device consists of a circular ring and a magnetic monopole (see fig. 1). Since the solenoid is a superposition of rings, it is enough to examine a single one. Along the ring flows an electric current I which is kept constant in time by a special regulator. Let B and B1 denote the fields of the monopole and that of the ring, respectively. A virtual displacement of the monopole along the axis of the ring changes its mechanical energy. The change is [1] dWM = F. dr
through the ring. The work of the electric field associated with this change is dWE= Vldt
= -Id~ gId~ -
= 2 ¢ r g l a 2 / ( a 2 + r12)3/2 dr.
n= g(r-
r,)/lr-
rtl 3
= - grad( g / l r - rll ). (1)
(2)
The change in the interaction energy of the magnetic fields is obtained as follows. The field of the monopole is
=gBl.dr = 2~rgIaZ/(a 2 + rlZ) 3/2 d r .
~-~ I d t
-
(3)
The energy of the interaction part of the fields is
The displaced monopole changes the magnetic flux 1 fBt. W,~T=Tg~
B dSr
1 = 4---~f - g r a d ( g / ] r - rll ) • Bt d3r = 11-4~f[(div
B,)(g/lr- rl[ )
- div( g B t / l r - r~ I)] d3r Fig. 1. A circular ring of radius a through which an electric current 1 flows, generates a magnetic field. A magnetic monopole g is located at r~ on the axis of the ring. The origin is at the center of the ring.
= 0.
(4)
The third line is obtained after using vector analysis and the last line is obtained from
0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
60
E. Comay / Difference between solenoid and magnetic spin
Maxwell's equation for B~, from the Gauss theorem and from the fact that at infinity the second integrand behaves like r -4. The interaction energy vanishes identically and therefore it does not change when the monopole is displaced. It is seen that the sum of eqs. (1), (2) and (4) balance. In the second experiment, a neutron replaces the solenoid. The two particles are far apart and the classical limit [2] is valid. An examination of this configuration shows that the situation here is not the same as in the first example. In this case, the monopole, after being subjected to the same virtual displacement, gains mechanical energy like that of the first experiment. However, the self-energy of the magnetic spin does not change. The potential energy of the system does not change either. This is due to the fact that the potential energy of an electromagnetic system is the energy of the interaction part of the fields [3-5]. It has been shown in eq. (4) that this quantity is identically zero and therefore it does not change.
The question is, can the energy which balances the mechanical energy gained by the monopole come from somewhere else?
Acknowledgement I wish to acknowledge the hospitality of the University of Michigan where this work has been done.
References [1] J.D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley, New York, 1975) p. 178. [2] L.I. Schiff, Quantum Mechanics, 2nd ed. (McGraw-Hill, New York, 1955) pp. 25, 26. [3] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (Oxford, 1962) p. 85. [41 J.B. Marion, Classical Electromagnetic Radiation (New York, 1965) p. 115. [5] D.E. Soper, Classical Field Theory (John Wiley, New York, 1976) pp. 120, 128.
59
A DIFFERENCE BETWEEN SOLENOID AND MAGNETIC SPIN
E. COMAY Department of Physics, Tel-Aviv University, TeI-Aviv 69978, Israel Received 15 August 1983; in revised form 5 January 1984
The interactions of a circular ring through which an electric current flows with a magnetic monopole and that of a magnetic spin with a monopole are discussed. An inconsistency of the calculation of energy is shown.
Two experiments will describe a difference between a solenoid and a magnetic spin. Units where c = 1 are used in this work. The first experimental device consists of a circular ring and a magnetic monopole (see fig. 1). Since the solenoid is a superposition of rings, it is enough to examine a single one. Along the ring flows an electric current I which is kept constant in time by a special regulator. Let B and B1 denote the fields of the monopole and that of the ring, respectively. A virtual displacement of the monopole along the axis of the ring changes its mechanical energy. The change is [1] dWM = F. dr
through the ring. The work of the electric field associated with this change is dWE= Vldt
= -Id~ gId~ -
= 2 ¢ r g l a 2 / ( a 2 + r12)3/2 dr.
n= g(r-
r,)/lr-
rtl 3
= - grad( g / l r - rll ). (1)
(2)
The change in the interaction energy of the magnetic fields is obtained as follows. The field of the monopole is
=gBl.dr = 2~rgIaZ/(a 2 + rlZ) 3/2 d r .
~-~ I d t
-
(3)
The energy of the interaction part of the fields is
The displaced monopole changes the magnetic flux 1 fBt. W,~T=Tg~
B dSr
1 = 4---~f - g r a d ( g / ] r - rll ) • Bt d3r = 11-4~f[(div
B,)(g/lr- rl[ )
- div( g B t / l r - r~ I)] d3r Fig. 1. A circular ring of radius a through which an electric current 1 flows, generates a magnetic field. A magnetic monopole g is located at r~ on the axis of the ring. The origin is at the center of the ring.
= 0.
(4)
The third line is obtained after using vector analysis and the last line is obtained from
0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
60
E. Comay / Difference between solenoid and magnetic spin
Maxwell's equation for B~, from the Gauss theorem and from the fact that at infinity the second integrand behaves like r -4. The interaction energy vanishes identically and therefore it does not change when the monopole is displaced. It is seen that the sum of eqs. (1), (2) and (4) balance. In the second experiment, a neutron replaces the solenoid. The two particles are far apart and the classical limit [2] is valid. An examination of this configuration shows that the situation here is not the same as in the first example. In this case, the monopole, after being subjected to the same virtual displacement, gains mechanical energy like that of the first experiment. However, the self-energy of the magnetic spin does not change. The potential energy of the system does not change either. This is due to the fact that the potential energy of an electromagnetic system is the energy of the interaction part of the fields [3-5]. It has been shown in eq. (4) that this quantity is identically zero and therefore it does not change.
The question is, can the energy which balances the mechanical energy gained by the monopole come from somewhere else?
Acknowledgement I wish to acknowledge the hospitality of the University of Michigan where this work has been done.
References [1] J.D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley, New York, 1975) p. 178. [2] L.I. Schiff, Quantum Mechanics, 2nd ed. (McGraw-Hill, New York, 1955) pp. 25, 26. [3] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (Oxford, 1962) p. 85. [41 J.B. Marion, Classical Electromagnetic Radiation (New York, 1965) p. 115. [5] D.E. Soper, Classical Field Theory (John Wiley, New York, 1976) pp. 120, 128.