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Magnetic Force Formulae for Magnets at Small Distances

Nikola Popovi´ c1, Dirk Praetorius2, and Anja Schl¨ omerkemper∗3

1BostonUniversity, DepartmentofMathematicsandStatisticsandCenterforBioDynamics, 111CummingtonStreet, Boston,

MA 02215, U.S.A.

2Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8-10, A-1040

Vienna, Austria

3Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany

In [7], the magnetic force on subregions of rigid magnetized bodies was studied as a discrete-to-continuum limit. The de-

rived force formula includes a new term, which depends on the underlying crystalline lattice structure L. It originates from

contributions of magnetic dipole-dipole interactions of dipole moments close to the boundary of the considered subregion.

Further studies of this new term have led to the question of how the magnetic force between two idealized magnets, which

are a distance ε > 0 apart, depends on ε as ε → 0. In this article, analytical aspects of this question are discussed, cf. [5],

where also numerical experiments are performed.

1Introduction

Ferromagnetic shape memory alloys have some potential as new micro-devices, cf. e.g. [1]. In order to construct such devices,

a better fundamental understanding of the dynamics of moving interfaces is of interest, cf. e.g. [3] for the study of a micro-

scale cantilever. In this context, the question arises which mathematical formula describes the force between two parts of a

magneto-elastic material best. There is a long list of related literature on this, cf. the references in [7].

To get started, we assume the magnetized material to be rigid. Several formulae are known for the magnetic force that is

exerted by one subbody on another one [6, 7]. While one can bring the formulae in a form such that the volume force densities

are the same, the surface force densities are different. Therefore there is some need for experiments that clarify which formula

is the most appropriate. Unfortunately, it is not possible or at least not obvious how to measure magnetic forces in the interior

of a magnetic body. To circumvent this, we suggest to consider the force between two magnetic bodies – instead of looking

at two subregions of one magnetic body. In particular, we discuss the force between two polygonal bodies A and B whose

boundaries have at least a set of positive surface measure in common. Even in this case one finds several different force

fourmulae.

The experimental idea which drove the analytical and numerical studies in [5] is the following: Take two cuboidal per-

manent magnets A,B that are a distance ε > 0 apart and measure the force F(sep,ε)as ε gets smaller. Then compare the

experimental results as ε → 0 with the different mathematical formulae. To prepare and motivate such experiments, we de-

rived the corresponding magnetic force formulae under these geometrical assumptions and performed numerical experiments.

In this article we discuss some analytical results from [5], to which we refer for details. For brevity we focus on polygonal

three-dimensional domains A and B with finitely many edges such that A ∩ B = ∅ and the surface measure of ∂A ∩ ∂B

is positive. Let mA: A → R3denote the magnetization corresponding to A which is a given Lipschitz-continuous vector

field and trivially extended to the entire space R3, i.e., mA = mAχA. By HAwe denote the magnetic field which is

generated by the magnetization in A and which is obtained from the magnetostatic Maxwell equations curlHA = 0 and

div(HA+ mA) = 0 in some suitable physical units. We adopt the same notation for B and denote with HA∪Bthe magnetic

field generated by m := mA+ mB.

2Formula derived from a classical formula for magnets not being in contact

Following the above mentioned experimental idea, we firstly move the body B a distance ε apart; for definiteness we move

B along the (1,0,0)-axis and set Bε= {x + ε(1,0,0)|x ∈ B}. Then we apply the classical, well-accepted magnetic force

formula for separated bodies (cf. e.g. [4])

?

Several numerical experiments are performed in [5]. There, we also prove that the limit F(sep,0)for ε → 0 exists and all forces

F(sep,ε)for ε ≥ 0 can be computed by closed formulae.

F(sep,ε)=

A

(mA· ∇)HBεdV.

∗Corresponding author: e-mail: schloem@mis.mpg.de, Phone: +493419959547, Fax: +493419959633

PAMM · Proc. Appl. Math. Mech. 5, 631–632 (2005) / DOI 10.1002/pamm.200510292

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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3Formula derived from a discrete setting of magnetic dipoles

In this section we consider two magnets A and B being in contact, i.e., the surface measure of ∂A∩∂B is positive. We focus

on a force formula which is obtained from a discrete setting of magnetic dipole moments, cf. [7]. Here we present a version

of the theorem generalized to polygonal domains which are in contact but not necessarily nested. In [5] we consider a further

magnetic force formula F(Brown), which was intensively studied in [2] and mathematically analyzed in [6].

Let L be a Bravais lattice describing the crystalline structure, e.g. L = Z3. For each x ∈

magnetic dipole moment m(?)(x) =

dipole moments in A ∩1

1

4π

i,j=1

x∈¯

Theorem 3.1 ([5]) Under the above assumptions on A, B, mAand mBthe limit lim

?

1

2

i,j,p=1

∂A∩∂B

where nAdenotes the outer normal to ∂A and

Sijkp:= −1

4π

?→∞

z∈Bδ∩1

for an arbitrary smooth function ϕ(δ): R3→ [0,1] with ϕ(z) = 1 if |z| <δ

The first two terms in (1) are also of interest themselves; we therefore set these terms equal to F(long)and refer to this

formula as the long range force. It turns out that F(sep,0)= F(long). The third term in (1), called short range force F(short), is

an additional surface term, which was firstly obtained in [6, 7].

1

?L, ? ∈ N we introduce a

i(x). The magnetic force between all

1

?3m(x) and denote the i-th component by m(?)

?L and those in B ∩1

3

?

?L is given by superposition of all dipole-dipole interactions [7, 5]

?

F(?)=

?

A∩1

?L

m(?)

i(x)

y∈B∩1

?L

∇∂i∂j|x − y|−1m(?)

j(y).

?→∞F(?)=: F(lim)exists and

?(mA· nA)(nA)kdΓ

F(lim)

k

=

A

(mA· ∇)(HA∪B)kdV +

?

1

2

?

∂A

?(mA− mB) · nA

(mA)i(mB)j(nA)pdΓ,

+

3

Sijkp

?

(1)

lim

δ→0lim

?

?L\{0}

?∂k∂i∂j(ϕ(δ)(z)|z|−1)?zp

1

?3.

(2)

2and ϕ(z) = 0 if |z| > δ.

4 Discussion and further development

Our analytical and numerical studies [5] are driven by two questions: (i) Which is the best force formula of those, which are

derived within the continuum theory? That is, we compare F(Brown)with F(sep,0)= F(long). (ii) Is the additional surface

force contribution F(short)of physical relevance? That is, we ask whether the discrete-to-continuum limit F(lim)is a good

formula to describe the magnetic force between two magnets being in contact.

In the numerical experiments in [5] we consider two idealized magnets of cuboidal shape with constant magnetization at

distance ε > 0 and ε = 0, respectively. We give here a first basic observation: To be specific, let A and B be two unit

cubes which touch at one face with normal (1,0,0). Moreover, let mA= (1,0,0) and mB= (1,0,0). Then F(long)

According to (2), we need to fix some underlying lattice structure to calculate the short range force term. We choose the cubic

lattice L = Z3and obtain F(short)

12.2%. This indicates that F(short)is of some relevance for the total force that one magnet exerts on the other. Though this

estimate of the force terms is based on quite some idealizations, it seems to be worth to perform similar real-life experiments.

In Theorem 3.1 the domains A and B are assumed to be in contact. The above observation together with the numerical

experiments for ε > 0 in [5] raise the following question: How does the short range contribution F(short)changes if the bodies

A and B are not in contact but a small distance ε > 0 apart? However, this requires new analytical techniques for the passage

from the discrete setting to the continuum.

1

≈ 5.5.

1

=1

2S1111|∂A ∩ ∂B| ≈ 0.67. Hence (F(lim)

1

− F(long)

1

)/F(long)

1

= F(short)

1

/F(long)

1

≈

References

[1] K. Bhattacharya and R. D. James, The material is the machine, Science 307, 53–54 (2005).

[2] W. F. Brown, Magnetoelastic Interactions, Springer-Verlag, Berlin (1966).

[3] R. D. James, Configurational Forces in Magnetism with Application to the Dynamics of a Small-Scale Ferromagnetic Shape Memory

Cantilever, Continuum Mech. Thermodyn. 14, 55–86 (2002).

[4] L. D. Landau, E. M. Lifschitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., Pergamon, Oxford (1984).

[5] N. Popovi´ c, D. Praetorius and A. Schl¨ omerkemper, Analysis and numerical simulation of magnetic forces between rigid polygonal

bodies, Preprint.

[6] A. Schl¨ omerkemper, Magnetic forces in discrete and continuous systems, Doctoral thesis, University of Leipzig (2002)

[7] A. Schl¨ omerkemper, Mathematical derivation of the continuum limit of the magnetic force between two parts of a rigid crystalline

material, Arch. Rational Mech. Anal. 176(2), 227–269 (2005).

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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