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On the Magnetic Dipole Energy Expression of an Arbitrary Current On the magnetic dipole energy expression of an arbitrary current distribution Keeyung Lee Department of Physics, Inha University, Incheon, 402-751, Korea (Dated: October 16, 2012) We show that the magnetic dipole energy term appearing in the expansion of the magnetic potential energy of a localized current distribution has the form U = +m · B which is wrong by a sign from the well known −m · B expression. Implication of this result in relation to the electric dipole energy −p·E, and the force and torque on the magnetic dipole based on this energy expression is also discussed. PACS numbers: 13.40 Em, 21.10. Ky I. Introduction netic potential energy and show that the expansion of Potential energy of a charge distribution in an external the magnetic energy of an arbitrary current distribution electric field or the potential energy of a current distribu- system results in the expression +m·B for the dipole en- tion in an external magnetic field is an important concept ergy term. Implication of this result in connection with in the electromagnetic theory. It is well known that in the force and torque on the dipole is also discussed. the energy expansion of a localized charge or of a local- ized current distribution, the dipole energy term becomes II. Magnetic Energy Expansion important. Before discussing the magnetic energy case, let us For an electrically neutral system, since the monopole consider first the expansion of electric potential energy term vanishes, electric dipole energy which has the form U = dV ρ(r)φ(r) of a static charge distribution with U = −p · E becomes the most important term in most chargeR density ρ(r) under the influence of an external cases. One elementary method of deriving this expression potential φ(r). It is straightforward to show that this is to consider the potential energy of two point charges ±q energy expression can be expanded as in an electric field due to the electrostatic potential φ(r). In that case, the potential energy can be written as U = +qφ(r + d) + (−q)φ(r), which gives the potential energy U = qφ(0) − p · E(0) + ... (1) expression −p·E by taking the first order approximation φ(r+d)= d·∇φ(r). This expression can also be derived in which q is the amount of total charge, p is the dipole by calculating the amount of work needed to rotate the moment, and E(0) is the electric field at an arbitrary dipole, if the torque expression τ = pE sin θ has already point inside the charge distribution. Therefore we find been considered. that the electric dipole energy of a sufficiently localized It is also well known that the magnetic dipole energy charge distribution system can be approximated as −p·E has the form U = −m · B, when the dipole is located Now let us consider the magnetic energy of a current inside an external magnetic field B. Elementary deriva- distribution with current density J under the influence tion of this expression can be done, first by calculating of an external magnetic field. When the vector potential the amount of work needed to rotate the dipole, if the which represents the magnetic field is A(r), the magnetic torque expression on the dipole τ = mB sin θ has already potential energy can be expressed as, been considered. However, compared with the electric dipole case, derivation of such torque expression is a rel- 3 atively difficult task. The derivation of torque expression U = dV J · A = dV JkAk (2) arXiv:1210.3925v1 [physics.class-ph] 15 Oct 2012 on a system with an arbitrary geometry is not straight- Z Z forward, and usually a square shaped loop current model kX=1 is assumed[1, 2]. Some textbooks directly consider the Now, for a sufficiently localized current distribution, this force on the rectangular or on the circular loop instead of energy expression can be expanded as follows, deriving the torque expression[3, 4]. Magnetic dipole en- ergy expression is then obtained by using the calculated force expression in the energy-force relation F = −∇U. 3 3 2 1 ∂ Ak Many advanced level textbooks also adopt such indirect A (r)= A (0)+r·[∇A ]0 + x x [ ]0 +... k k k 2 i j ∂x ∂x derivation approach, but with systems generalized to ar- Xi=1 Xj=1 i j bitrary geometry[5, 6]. However, derivation of the force (3) expression of an arbitrary current distribution itself is where the origin of the coordinate is chosen inside the not an easy task at all[7]. current distribution. Using such expansion, the magnetic In this work, we consider direct expansion of the mag- energy can be expressed as U = U0 + U1 + ..., where 2 is the only term left in the expansion for the particular 3 3 case of uniform external magnetic field. To show this point, let us note that the vector potential of the form U0 = dV J A = A dV J (4) 1 Z k k k Z k A = r × B produces a uniform magnetic field. Since X=1 X=1 2 k k the vector potential in this expression is of the first order ∂2A in the coordinate x , it is obvious that [ k ]0 is equal and i ∂xi∂xj to 0, which means that the quadrupole term, as well as 3 all remaining higher order terms vanish. We have shown that the dipole term in the expansion U1 = dV Jkr · [∇Ak]0 (5) Z of the magnetic potential energy is expressed as +m · B, kX=1 whereas the dipole term in the electric energy expansion which are the monopole and dipole terms respectively. has the −p · E form, which is an unexpected result con- For a stationary current system, the monopole term U0 sidering that the electric dipole and the magnetic dipole vanishes since JkdV = 0. Now, if the constant vector have so many similar properties. Although this result [∇Ak]0 is temporarilyR denoted as c, the dipole term U1 is rather unfamiliar, such ’wrong’ sign of the magnetic can be expressed as, dipole energy has already been discussed by D. J. Grif- fiths [4]. In that paper, the author derives the expression 3 U = +m · B using the the magnetic charge model and provides a rather lengthy explanation on why such oppo- U1 = U1 (6) k site sign result should be obtained. X=1 k The fact that the energy of a magnetic dipole under the influence of an external field should be expressed as +m· where U1k = dV Jk(r · c). U1k term can now be ex- pressed as, R B could be confusing to many students, since the energy of a magnetic dipole is usually given as −m · B in most textbooks[1, 5]. Such sign difference in magnetic dipole energy does not make any difference when the magnitude U1 = dV x (J · c) − dV [c × (r × J)] (7) k Z k Z k of force on the magnetic dipole only is of interest. But it results in the opposite direction of the force or torque on if the vector identity c × (r × J) = r(J · c) − J(r · c) the dipole, and a good understanding on this situation is used. It could be noted here that the integral of the may be important pedagogically. second term on the right hand side can be expressed as 2[c × m]k where m is the magnetic dipole moment which 1 III. Discussion on the Force and Torque on the is defined as m = 2 r × JdV . Magnetic Dipole Let us now considerR the first term on the right hand Before discussing the force and torque on the dipole, c side of Eq.(7). Since is a just constant vector, this let us make sure that +m · B is the right expression of term can be written as i ci xkJi. Therefore, using the magnetic dipole energy. To show that point, con- the relation dV (xkJi +PxiJk)R = 0 which is satisfied for sider situations where a current loop is placed in an uni- a stationaryR current system[8], this term can now be ex- form external magnetic field, in which magnetic dipole is r c pressed as − i ci dV xiJk, or, as − dV Jk( · ), which directed (1)parallel (2)anti-parallel and (3)perpendicular means that theP firstR term reduces to −RU1k. Therefore we to the magnetic field direction. Considering the super- get the expression, posed field of the external field and of the field due to the current loop and calculating the magnetic field en- ergy dV |B|2, it is obvious that the energy is increased U1 = −[c × m]k (9) k for caseR (1) and is decreased for case (2). It is also ob- vious that there is no change in the magnetic energy for in which c stands for [∇Ak]0. Therefore, using the Levi-Civita symbol and Einstein case (3). This example shows that the energy variation summation convention, we finally obtain the expression for each situation is consistent with the magnetic energy expression U =+m · B. for the magnetic dipole energy as U1 = −εkij (∂iAk)mj , or Now let us consider the force and torque on the mag- netic dipole. If one attempt to apply the F = −∇U relation between the force and potential energy using the U1 = εjik(∂iAk)mj (10) energy expression U =+m·B, the force on the magnetic dipole becomes F = −∇(m · B) which is wrong by a sign using the εkij = −εjik property . Now since εjik∂iAk = from the right expression[5]. We again do not get the Bj from the relation B = ∇× A, we find that the mag- right expression for the torque if U =+m · B is used in netic dipole energy can be written as +m·B(0), in which connection with the common expression τ(θ)= −dU/dθ, B(0) is the magnetic field at an arbitrary point inside the since we get the result τ(θ)=+mB sin θ, which shows current distribution.
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