Interactions between uniformly magnetized spheres Boyd F. Edwards, D. M. Riffe, Jeong-Young Ji, and William A. Booth Department of Physics Utah State University Logan, UT 84322 (Dated: May 27, 2016) We demonstrate the magnetic interactions (both forces and torques) between two uniformly mag- netized spheres are identical to the magnetic interactions between two point magnetic dipoles. I. INTRODUCTION magnetized sphere and the field of an equivalent point dipole. Unlike the previous calculations, ours rely on Let’s consider two permanent magnets of arbitrary simple symmetry arguments and pertain to spheres of shape. If each magnet has a nonzero dipole moment, arbitrary sizes, positions, magnetizations, and magnetic then the dipole moments of these magnets will dominate orientations. their interactions at separations that are large compared The force between two dipoles is noncentral. Namely, with their sizes, and each magnet may be treated as if it this force is not generally directed along the line through were a point magnetic dipole. Dipolar fields and forces the dipoles. While other noncentral magnetic forces vio- are often used as the starting point for analytical and nu- late Newton’s third law [31–33], we show that the paired merical approximations of the forces between permanent forces between magnet dipoles obeys this law. These magnets of various shapes [1–10]. forces therefore exert a net r × F torque on an isolated two-dipole system. This torque cancels the paired m×B A uniformly magnetized sphere produces a magnetic torques, which are not equal and opposite. Thus angular field that is identical to its dipole field, not just at large momentum is conserved. distances, but everywhere outside of the sphere [11, 12]. One is thus naturally lead to ask whether the forces and torques between two uniformly magnetized spheres are II. POINT DIPOLE INTERACTIONS identical to those between two point dipoles, independent of their separation. Here we show this is indeed the case. In this section we review the interactions between point This result has practical applications. Dipolar fields magnetic dipoles. The magnetic field at position r pro- and forces have been used to approximate the interac- duced by a point dipole m located at the origin is given tions among assemblies of spherical nanoparticles [13] by [34, 35] and magnetic microspheres [14]. Our results show that these approximations are, in reality, exact. In addition, µ0 3m · r m B(m; r)= r − , (1) small rare-earth magnetic spheres are used both in and 4π r5 r3 out of the classroom to teach principles of mathemat- ics, physics, chemistry, biology, and engineering [15–19]. for r = |r| satisfying r > 0. This field can be obtained Our results enable simple dipole interactions to be used from the scalar potential to model the dynamical interactions between these mag- nets [20, 21]. µ0 m · r ϕ(m; r)= 3 (2) Previous calculations of the force between two uni- 4π r formly magnetized spheres have been carried out in three via limiting geometries: (i) for magnetizations that are per- pendicular to the line through the sphere centers [22], B(m; r)= −∇ϕ(r). (3) (ii) for parallel magnetizations that make an arbitrary angle with this line [23], and (iii) for configurations with Because ∇ · B = 0, ϕ satisfies Laplace’s equation, one magnetization parallel to this line and the other in 2 an arbitrary direction [24]. All three calculations yield ∇ ϕ =0. (4) forces that are identical to the force between two point We consider two dipoles, m1 and m2, which are re- magnetic dipoles [25–30]. spectively located at positions r1 and r2. From Eq. (1), In this paper, we show generally that the force be- the field produced by dipole mi is tween two uniformly magnetized spheres is identical to the force between two point magnetic dipoles, and show Bi(r)= B(mi; r − ri), (5) the same equivalence for the torque. Like the calcula- tions for the three limiting geometries, our calculations where i = 1, 2, and where r − ri is the position vector exploit the equivalence of the field outside of a uniformly relative to dipole mi (Fig. 1). Accordingly, the field of 2 r r r r mi evaluated at the location of mj is - 1 - 2 F 21 r M M 2 µ0 3mi · rij mi 1 Bi(rj )= 5 rij − 3 , (6) r 4π rij rij ! a 12 F a 1 z 12 2 r r 1 2 where rij = rj − ri is the position of mj relative to mi, and rij = |rj − ri|. y The interaction energy between mj and the magnetic x field of mi is FIG. 1. Diagram showing two uniformly magnetized spheres Uij = −mj · Bi(rj ). (7) with positions r1 and r2, radii a1 and a2, magnetizations M1 and M2, and paired non-central magnetic forces F12 and F21. Inserting Eq. (6) gives Shown also are an arbitrary position vector r and the relative position vectors r − r1, r − r2, and r12 = r2 − r1. The same µ0 mi · mj (mi · rij )(mj · rij ) diagram applies for the forces between two point dipoles if Uij = 3 − 3 5 . (8) spheres M1 and M2 are replaced by dipoles m1 and m2 at 4π " rij rij # the same locations. The force of mi on mj follows from ∇ is generally unequal to zero. Therefore, unlike the paired Fij = − jUij , (9) A A forces F12 and F21, the paired torques τ12 and τ21 are where ∇j is the gradient with respect to rj . Inserting not generally equal and opposite. Eq. (8) yields The second contribution to the torque arises from the force of mi on mj , 0 3µ B Fij = 5 (mi · rij ) mj + (mj · rij ) mi τ = r × F , (16) 4πr ij j ij ij (m · r ) (m · r ) for which the sum + (m · m ) r − 5 i ij j ij r . (10) i j ij r2 ij τ B τ B ij 12 + 21 = r12 × F12 (17) The first two terms in the square brackets are respec- is also generally unequal to zero. tively parallel to mj and mi. Consequently, Fij is not Equations (15) and (17) and a little algebra reveal that central, namely, it is not generally parallel to the vector the net torque on an isolated pair of dipoles vanishes rij between the dipoles. identically; Equations (8) and (10) imply that A A B B τ12 + τ21 + τ12 + τ21 =0. (18) U21 = U12 (11) A A The torque supplied by τ12 and τ21 therefore cancels the torque supplied by τ B and τ B , and angular momentum and 12 21 is conserved. Thus, an isolated dipole-dipole system does F21 = −F12, (12) not spontaneously rotate, and there is no exchange be- tween mechanical and electromagnetic momentum. Such confirming that Newton’s third law applies to the mag- exchanges have been the subject of considerable study netic force between point magnetic dipoles, and ensuring [32, 37–40]. that linear momentum is conserved in an isolated two- dipole system. III. FORCE BETWEEN SPHERES We now investigate the torque of mi on mj, which has two contributions [36], We now come to the crux of this paper, the force be- τ τ A τ B ij = ij + ij . (13) tween two uniformly magnetized spheres. We present four separate arguments that show that the force be- The first arises from mj residing in the field of mi, tween uniformly magnetized spheres is identical to the force between point dipoles. While each proof is sufficient τ A = m × B (r ). (14) ij j i j to show this equivalence, each utilizes different concepts The sum of paired torques from mechanics and electromagnetism, and each has ped- agogical value. τ A τ A 3µ0 As seen in Fig. 1, we take sphere i to have position ri, 12 + 21 = 5 (m1 · r12) m2 × r12 4πr12 radius ai, magnetization Mi, and total dipole moment h 4 + (m2 · r12) m1 × r12 (15) m = πa3 M . (19) i 3 i i i 3 (a) Fig. 2(b): Sphere 1 produces the same field B1, and m therefore exerts the same force F12 on dipole 2. m F 2 1 12 Fig. 2(c): Newton’s third law gives the force F21 = −F12 of dipole 2 on sphere 1. This force is pro- duced by the field B2 of dipole 2. (b) Fig. 2(d): Sphere 2 produces the same field B2, and therefore exerts the same force F21 on sphere 1. m F 2 M 12 1 Fig. 2(e): To complete the argument, we again apply Newton’s third law to show that the force F12 = −F21 of sphere 1 on sphere 2 is identical to the force of dipole 1 on dipole 2 (Fig. 2a). (c) m M 2 1 F B. Direct integration 21 The force between two uniformly magnetized spheres can be determined by integrating the energy density (d) −Mj · Bi(r) associated with sphere j sitting in the mag- netic field Bi(r), giving the total interaction energy M M 2 1 F 21 Uij = − Mj · Bi dV = −Mj · Bi dV. (20) Zj Zj Here, the integral is over the volume of sphere j and (e) the second equality exploits the uniformity of Mj. If all M sources of a magnetic field lie outside a particular sphere, M F 2 1 12 then the spatial average of the field over the sphere is given by the value of the field at the sphere center [43]. That is, FIG. 2. Diagram illustrating the five steps of the Newton’s 4 3 Bi dV = πaj Bi(rj ). (21) third law argument showing the force between two uniformly j 3 magnetized spheres is identical to the force between two point Z dipoles.