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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 magnetized spheres are identical to the magnetic interactions between two point magnetic dipoles. I. INTRODUCTION Let’s consider two permanent magnets of arbitrary shape. If each magnet has a nonzero dipole moment, then the dipole moments of these magnets will dominate their interactions at separations that are large compared with their sizes, and each magnet may be treated as if it were a point magnetic dipole. Dipolar fields and forces are often used as the starting point for analytical and numerical approximations of the forces between permanent magnets of various shapes [1–10]. A uniformly magnetized sphere produces a magnetic field that is identical to its dipole field, not just at large 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 identical to those between two point dipoles, independent of their separation. Here we show this is indeed the case. This result has practical applications. Dipolar fields and forces have been used to approximate the interactions among assemblies of spherical nanoparticles [13] and magnetic microspheres [14]. Our results show that these approximations are, in reality, exact. In addition, small rare-earth magnetic spheres are used both in and out of the classroom to teach principles of mathematics, physics, chemistry, biology, and engineering [15–19]. Our results enable simple dipole interactions to be used to model the dynamical interactions between these magnets [20, 21]. Previous calculations of the force between two uniformly magnetized spheres have been carried out in three limiting geometries: (i) for magnetizations that are perpendicular to the line through the sphere centers [22], (ii) for parallel magnetizations that make an arbitrary angle with this line [23], and (iii) for configurations with one magnetization parallel to this line and the other in an arbitrary direction [24]. All three calculations yield forces that are identical to the force between two point magnetic dipoles [25–30]. In this paper, we show generally that the force between two uniformly magnetized spheres is identical to the force between two point magnetic dipoles, and show the same equivalence for the torque. Like the calculations for the three limiting geometries, our calculations exploit the equivalence of the field outside of a uniformly magnetized sphere and the field of an equivalent point dipole. Unlike the previous calculations, ours rely on simple symmetry arguments and pertain to spheres of arbitrary sizes, positions, magnetizations, and magnetic orientations. The force between two dipoles is noncentral. Namely, this force is not generally directed along the line through the dipoles. While other noncentral magnetic forces violate Newton’s third law [31–33], we show that the paired forces between magnet dipoles obeys this law. These forces therefore exert a net r × F torque on an isolated two-dipole system. This torque cancels the paired m × B torques, which are not equal and opposite. Thus angular momentum is conserved. II. POINT DIPOLE INTERACTIONS In this section we review the interactions between point magnetic dipoles. The magnetic field at position r produced by a point dipole m located at the origin is given by [34, 35] µ0 3m · r m B(m; r) = (1) r− 3 , 4π r5 r for r = |r| satisfying r > 0. This field can be obtained from the scalar potential µ0 m · r 4π r3 (2) B(m; r) = −∇ϕ(r). (3) ϕ(m; r) = via Because ∇ · B = 0, ϕ satisfies Laplace’s equation, ∇2 ϕ = 0. (4) We consider two dipoles, m1 and m2 , which are respectively located at positions r1 and r2 . From Eq. (1), the field produced by dipole mi is Bi (r) = B(mi ; r − ri ), (5) where i = 1, 2, and where r − ri is the position vector relative to dipole mi (Fig. 1). Accordingly, the field of 2 mi evaluated at the location of mj is Bi (rj ) = µ0 4π 3mi · rij mi rij − 3 5 rij rij r - r2 r - r1 ! , (6) F21 r M1 r12 a1 (7) Inserting Eq. (6) gives " # µ0 m i · m j (mi · rij )(mj · rij ) Uij = −3 . 3 5 4π rij rij (8) z r1 where rij = rj − ri is the position of mj relative to mi , and rij = |rj − ri |. The interaction energy between mj and the magnetic field of mi is Uij = −mj · Bi (rj ). M2 a2 F12 r2 y x FIG. 1. Diagram showing two uniformly magnetized spheres with positions r1 and r2 , radii a1 and a2 , magnetizations M1 and M2 , and paired non-central magnetic forces F12 and F21 . Shown also are an arbitrary position vector r and the relative position vectors r − r1 , r − r2 , and r12 = r2 − r1 . The same diagram applies for the forces between two point dipoles if spheres M1 and M2 are replaced by dipoles m1 and m2 at the same locations. The force of mi on mj follows from Fij = −∇j Uij , (9) where ∇j is the gradient with respect to rj . Inserting Eq. (8) yields 3µ0 Fij = (mi · rij ) mj + (mj · rij ) mi 5 4πrij (mi · rij ) (mj · rij ) (10) r + (mi · mj ) rij − 5 ij . 2 rij is generally unequal to zero. Therefore, unlike the paired A A forces F12 and F21 , the paired torques τ12 and τ21 are not generally equal and opposite. The second contribution to the torque arises from the force of mi on mj , τijB = rj × Fij , (16) B B = r12 × F12 τ12 + τ21 (17) for which the sum The first two terms in the square brackets are respectively parallel to mj and mi . Consequently, Fij is not central, namely, it is not generally parallel to the vector rij between the dipoles. Equations (8) and (10) imply that is also generally unequal to zero. Equations (15) and (17) and a little algebra reveal that the net torque on an isolated pair of dipoles vanishes identically; U21 = U12 (11) F21 = −F12 , (12) A A therefore cancels the and τ21 The torque supplied by τ12 B B torque supplied by τ12 and τ21 , and angular momentum is conserved. Thus, an isolated dipole-dipole system does not spontaneously rotate, and there is no exchange between mechanical and electromagnetic momentum. Such exchanges have been the subject of considerable study [32, 37–40]. and confirming that Newton’s third law applies to the magnetic force between point magnetic dipoles, and ensuring that linear momentum is conserved in an isolated twodipole system. We now investigate the torque of mi on mj , which has two contributions [36], τij = τijA + τijB . (13) The first arises from mj residing in the field of mi , τijA = mj × Bi (rj ). The sum of paired torques 3µ0 h A A τ12 + τ21 = (m1 · r12 ) m2 × r12 5 4πr12 + (m2 · r12 ) m1 × r12 (14) i (15) A A B B τ12 + τ21 + τ12 + τ21 = 0. III. (18) FORCE BETWEEN SPHERES We now come to the crux of this paper, the force between two uniformly magnetized spheres. We present four separate arguments that show that the force between uniformly magnetized spheres is identical to the force between point dipoles. While each proof is sufficient to show this equivalence, each utilizes different concepts from mechanics and electromagnetism, and each has pedagogical value. As seen in Fig. 1, we take sphere i to have position ri , radius ai , magnetization Mi , and total dipole moment mi = 4 3 πa Mi . 3 i (19) 3 (a) F12 m1 m2 Fig. 2(c): Newton’s third law gives the force F21 = −F12 of dipole 2 on sphere 1. This force is produced by the field B2 of dipole 2. (b) F12 M1 m2 m2 B. F21 (d) M1 Fig. 2(d): Sphere 2 produces the same field B2 , and therefore exerts the same force F21 on sphere 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) M1 Fig. 2(b): Sphere 1 produces the same field B1 , and therefore exerts the same force F12 on dipole 2. M2 F21 Direct integration The force between two uniformly magnetized spheres can be determined by integrating the energy density −Mj · Bi (r) associated with sphere j sitting in the magnetic field Bi (r), giving the total interaction energy Z Z Uij = − Mj · Bi dV = −Mj · Bi dV. (20) j (e) M1 F12 M2 FIG. 2. Diagram illustrating the five steps of the Newton’s third law argument showing the force between two uniformly magnetized spheres is identical to the force between two point dipoles. Its magnetic field is given by Bi = 2µ0 Mi /3 inside the sphere (for |r − ri | < ai ) and is given by Eq. (5) outside the sphere [11, 12]. We treat Mi as spatially uniform and constant in time, neglecting any demagnetization by external fields. This assumption is appropriate for high coercivity materials [41, 42]. j Here, the integral is over the volume of sphere j and the second equality exploits the uniformity of Mj . If all sources of a magnetic field lie outside a particular sphere, 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, Z 4 (21) Bi dV = πa3j Bi (rj ). 3 j Equations (19), (20), and (21) give Uij = −mj · Bi (rj ), which replicates Eq. (7). Thus, the energy of interaction between the two spheres is identical to the energy of interaction between two point dipoles. The associated force Fij = −∇j Uij of sphere i on sphere j is therefore identical to the force between two point dipoles, and obeys Newton’s third law. C. A. Newton’s third law A five-step argument involving Newton’s third law shows that the force between two spheres with uniform magnetizations M1 and M2 is identical to the force between two point dipoles with corresponding equivalent magnetic moments m1 and m2 , located at the same positions as the spheres. Figure 2 illustrates this argument. Fig. 2(a): As in the preceding discussion, F12 represents the force of dipole 1 on dipole 2. This force is produced by the field B1 of dipole 1. Field energy We can also show the force equivalence by integrating the magnetic energy density B 2 /2µ0 over all space, giving the total magnetic energy of interaction Z 1 U (r1 , r2 ) = (B1 + B2 )2 dV 2µ0 Z 1 B12 + 2 B1 · B2 + B22 dV. (22) = 2µ0 Because the magnetic energy of a single dipole does not depend on R its location in space, the self-energy integrals (2µ0 )−1 Bi2 dV do not depend on ri . Therefore the force 4 Fij = −∇j U on sphere j depends only on the interaction energy Z 1 Uint = B1 · B2 dV µ0 Z Z Z 1 = + + ∇ϕ1 · ∇ϕ2 dV, (23) µ0 1 2 outside where we have inserted Eq. (3), and we have separated the integral over all space into integrals over sphere 1, sphere 2, and the region outside of both spheres. For the integral over sphere 1, we use ∇2 ϕ2 = 0 and the divergence theorem to write Z Z ∇ϕ1 · ∇ϕ2 dV = ∇ · (ϕ1 ∇ϕ2 ) dV 1 Z1 ϕ1 ∇ϕ2 · n̂1 dA, (24) = S1 where S1 denotes the integral over the surface of sphere 1, and n̂1 is the unit vector directed normally outward from the sphere’s surface. The quantity ϕ1 ∇ϕ2 is continuous across this surface. The surface values of ϕ1 and ϕ2 are identical to the potentials of point dipoles, and so the integral over sphere 1 is the same as it would be if the spheres were replaced by equivalent point dipoles. Similarly, the integral over sphere 2 is the same as for equivalent point dipoles. Since the fields outside of both spheres match the fields of equivalent point dipoles, all three integrals Eq. (23) are the same for point dipoles as for uniformly magnetized spheres. Therefore, the force Fij = −∇j Uint must also be the same. Invoking Eqs. (19) and (21) gives τijA = mj × Bi (rj ), which replicates Eq. (14). Thus, τijA between two uniformly magnetized spheres is identical to the torque between two point magnetic dipoles. The second contribution arises from the force of sphere i on sphere j, which can be obtained from the interaction energy density uij = −Mj · Bi according to Z (26) Fij = − ∇uij dV. j This gives the same force as Fij = −∇j Uij , with Uij given by Eq. (7). The torque follows by integrating the torque density r × (−∇uij ) over the volume of sphere j; Z B τij = − r × ∇uij (r)dV. (27) j Rewriting using the position r′ = r − rj relative to the center of sphere j gives Z τijB = − (r′ + rj ) × ∇uij (r)dV j Z Z = − r′ × ∇′ Ψ(r′ )dV ′ − rj × ∇uij (r)dV, (28) j j ′ ′ where Ψ(r ) = uij (r + rj ). Converting the first integral into a surface integral gives Z Z (29) r′ × ∇′ ΨdV ′ = − ∇′ × (Ψr′ )dV ′ j j Z n̂j × r′ ΨdA′ . (30) =− Sj D. Stress tensor The total force on an object can be calculated by integrating the Maxwell stress tensor over an arbitrary surface surrounding the object [44, 45]. The magnetic stress tensor depends only on the magnetic field. The field outside a uniformly magnetized sphere is the same as the field of a point dipole located at its center. The total field produced by and outside of two uniformly magnetized spheres, and the associated stress tensor, is the same as that of two point dipoles. Therefore, the force between spheres must be the same as the force between point dipoles. IV. TORQUE BETWEEN SPHERES Here we calculate the torque τij of sphere i on sphere j. This torque has two contributions, as before. The first arises from sphere j residing in the field of sphere i, and can be obtained by integrating the torque density Mj × Bi over the volume of sphere j, giving Z Z (25) τijA = Mj × Bi dV = Mj × Bi dV. j j Since r′ is parallel to the unit normal vector n̂j , their cross product is zero and the integral vanishes. Combining Eqs. (26) and (28) gives τijB = rj × Fij , (31) which is identical to Eq. (16). Thus, the torque associated with the force on a uniformly magnetized sphere in a dipole field is the same as the torque on the corresponding point dipole. These results ensure that the net torque on an isolated two-sphere system is zero, as seen earlier for point dipoles. V. SUMMARY We have demonstrated that the forces and torques between two uniformly magnetized spheres are identical to the forces and torques between two point magnetic dipoles. This equivalence extends to uniformly polarized spheres and point electric dipoles. This is because the fields, forces, and torques of electric dipoles have the same mathematical forms as their magnetic counterparts, and the field outside of a uniformly polarized sphere is identical to the electric dipole field. 5 We gratefully acknowledge support from NSF Grant No. 1332265, discussions with W. Farrell Edwards on the applicability of Newton’s third law to magnetostatic interactions, correspondence with David Vokoun about calculations of the force between magnet spheres in special cases, correspondence with Marco Beleggia about the potential theory argument, and discussions with Kirk McDonald and David Griffiths regarding the point-sphere equivalence. We are especially grateful to David Griffiths for suggesting the stress tensor argument to us, and to Kirk McDonald for pointing out Ref. [43], which simplifies some of the calculations. [1] G. Akoun and J. Yonnet, IEEE Trans. Magn. 20, 1962 (1984). [2] A. Kruusing, J. Magn. Magn. Mater. 234, 545 (2001). [3] D. 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Typing “Zen Magnets” into the YouTube search field at https://www.youtube.com identifies over 90,000 videos describing various (accessed Mar. 23, 2016). As of August 22, 2014, the most popular of these had a total view count exceeding 145 million (Ref. [16], Appendix D). Several countries, including the United States, have banned the sale of 5-mm diameter nickel-coated neodymium magnets marketed as a desk toys under the trade names BuckyBalls, Zen Magnets, Neoballs, etc. following reports of intestinal injuries from ingestion of these magnets. But 2.5-mm magnets may be still be purchased at http://micromagnets.com (accessed Feb. 16, 2016), and magnet spheres of various sizes, including 5-mm spheres, may still be purchased from industrial suppliers (http://www.kjmagnetics.com/, http://www.alibaba.com, http://www.magnet4less.com, accessed Feb. 16, 2016). Boyd F. Edwards and John M. Edwards, “Dynamical interactions between two uniformly magnetized spheres,” to be published (2016). 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