PHYSICAL REVIEW APPLIED 4, 064007 (2015)

Smooth Teeth: Why Multipoles Are Perfect Gears
Johannes Schönke
Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904 0495, Japan(Received 18 September 2015; revised manuscript received 29 October 2015; published 22 December 2015)
A type of gear is proposed based on the interaction of individual multipoles. The underlying principle
relies on previously unknown continuous degenerate ground states for pairs of interacting multipoles which
are free to rotate around specific axes. These special rotation axes, in turn, form a one-parameter family of
possible configurations. This allows for the construction of magnetic bevel gears with any desired
inclination angle between the in- and output axes. Further, the design of gear systems with more than two
multipoles is possible and facilitates tailored applications. Ultimately, an analogy between multipoles andmechanical gears is revealed. In contrast to the mechanical case, the multipole “teeth” mesh smoothly. As
an illustrative application, the example of a quadrupole-dipole interaction is then used to construct a 1∶2
gear ratio.
DOI: 10.1103/PhysRevApplied.4.064007

I. INTRODUCTION
A very useful device in a chemistry lab is the magnetic
stirrer, but magnetic couplings like this one are found not
only in scientific equipment—even the newest milk frothers
use them to create the perfect topping for your coffee. On asmaller scale, magnetic mechanisms and couplings are usedin microfluidic systems to serve as pumps, valves, ormixers [1,2], and on the even smaller scales of the nanoworld, molecular machines are, in fact, restricted toelectronic or magnetic interactions [3]. What if we could
use such elementary structures like dipoles or quadrupoles
individually to transmit rotary motion in a very flexible
way? Today’s industrial magnetic gear designs are based onarrays of many permanent magnets which mimic the teethof classical mechanical gears (for a review, see Ref. [4] andthe next paragraph). Here we propose gears fundamentallydifferent from these designs, based on the interaction ofindividually rotating multipoles. This approach allows us toconstruct gears with inclined axes and leads further toconfigurations like the one depicted in Video 1, which
consists of three dipoles that form a smooth coupling.
Finally, we uncover an analogy between multipoles andmechanical gears and demonstrate that a quadrupole and adipole can be used to construct a gear ratio of 1∶2 (seeVideo 3). The findings in this paper initiate questions fortheoretical physics about the necessary conditions forwhich smooth couplings between multipoles are possible.
At the same time, the constructions reported here canalready be used in many different applications, a few ofthem we mention above.
Although the mechanical gear was invented in antiquity,
the idea of a magnetic gear is only a hundred years old [5].
Magnetic gears have many advantages: being free ofcontact, they are not subject to mechanical wear, needno lubrication, possess inherent overload protection, are2331-7019=15=4(6)=064007(9)noiseless, are highly reliable, and operate with reduced
maintenance. Moreover, the input and output can bephysically isolated. With the appearance of strong magnets
based on alloys of rare-earth elements, the interest in
magnetic gears grew because of increased torque transmission capabilities [6]. In the last decade, innovative
designs for the magnetic gear topology further increased
the possible torque densities. Today, magnetic gears cancompete with mechanical realizations in industrially relevant applications. Most of the designs are coaxial [7–16]
and include intricate constructions like planetary [17] andcycloid gears [18]. Biaxial systems can also be found[19,20]. Specifically related to the current work is thediscussion of the undesirable cogging torque [13,14,16,17],
namely, periodic variation of the transmitted torque during

VIDEO 1. (Animated version online.) Three interacting dipoles(red arrows) form a smooth coupling capable of driving a paddle
boat. If the central dipole is rotated, the other two dipoles at thepaddles rotate accordingly (the motion is symbolized by yellow
arrows in the nonanimated version). Up to an overall scaling insize, the positioning of the dipoles is crucial. Only in this specifictriangle geometry does the smooth coupling work (Fig. 5).

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JOHANNES SCHÖNKE

(a)

(b)

(c)

FIG. 1. Definition of the angles which describe the rotation of
the dipoles (red). The upper panel introduces the two angles θ andφ which set the positions of the rotation axes (light gray). The
rotation axes are restricted to lie in the x-y plane. The angles θ andφ (and, therefore, the rotations axes) are fixed for a given
configuration. The lower panel introduces the angles α and βwhich correspond to the 2 degrees of freedom the system
possesses. α and β describe the actual orientation of the twodipoles during rotation.

(d)revolution induced by the geometry and magnetization
pattern of the gear. So far, the only realization capable ofavoiding cogging torque is the coaxial gear with a 1:1 ratio,
generally referred to as the “synchronous axial coupling,”
first thoroughly analyzed in Refs. [21,22].
II. TWO INTERACTING DIPOLES
We begin with the following problem statement: place a
set of multipoles in space and allow them to rotate around
predefined axes. Under which conditions (regarding positioning and orientation of the axes) do these multipoles
possess a continuous ground state (CGS) in the sense thatthe lowest possible energy configuration is degenerate?
Once such a state is found, we can rotate one of themultipoles, and the others will follow without any counterforce; a smooth coupling is created without any cogging
torque.
If the system consists of dipoles only, the interactionenergy reads for a total number of N dipoles
EN ¼

N
X
mi · mj jrij j2 − 3ðmi · rij Þðmj · rij Þi<jjrij j5;

ð1Þ

where m1 ; …; mN are the variable dipole moments, and rij
denotes the fixed relative position vector between dipole iand j.
First, we consider the case of two dipoles separated by afixed distance d that are allowed to rotate only around fixedaxes orthogonal to their dipole moment; see Fig. 1. In this

FIG. 2. (a) The trivial example for a continuous ground statecreated by two dipoles which are allowed to rotate about acommon axis. The (red, green, blue) colored arrows illustrate thedipole motion in the ground state for three different times. The
light gray arrows define the prescribed rotation axes, and the rings
guide the eye regarding the path the colored arrows take whilerotating. The small upward arrow in the middle is normal to thehorizontal plane. (b) Example of a continuous ground statecreated by two dipoles which are allowed to rotate about two
orthogonal axes. (c) The same configuration as in (b) but viewed
from the top along the normal vector of the horizontal plane inwhich the rotation axes are lying. (d) Schematic of the view
from the top defining the positions of the rotation axes via theangles θ and φ.

case, there is an obvious trivial solution to the posed
problem: the two rotation axes are aligned, and the twodipoles are antiparallel [Fig. 2(a)]. This configuration isprecisely the previously mentioned synchronous axial
coupling realized with two dipoles. The principle behind
this solution is not related to the nontrivial findings
described below because it does not rely on dipole
interactions. This system describes a continuous state
simply because it exhibits an obvious rotational symmetry
about the one and only rotation axis.

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SMOOTH TEETH: WHY MULTIPOLES ARE PERFECT GEARS

E2

E2

1

1

A

0
-1
2π

0

B

β

2π

π
π
0 0

α

-1

C
FIG. 4. Degenerate energy landscape (3) for two interacting
dipoles (m1 ¼ m2 ¼ d ¼ 1) obeying Eq. (2). The system is in its
ground state if α − β ¼ π. Only in the trivial case of aligned
rotation axes (θ ¼ φ ¼ 0) this means the two dipoles are
antiparallel in the ground state.

D

so that they form a CGS? It turns out that there are infinitely
many. Referring to the definition of the angles θ and φwhich fix the rotation axes of the two dipoles [cf. Figs. 1
and 2(d)], for every choice of θ there exists a φ which
creates a CGS. The condition can be expressed as thenonlinear relation (see Appendix A)

E
2
A
B

C
D

3/2

cos θ cos φ − 2 sin θ sin φ ¼ 1:

E

1/2

0
0

1/2

3/2

2

FIG. 3. Configurations of continuous ground states created by
two dipoles which are allowed to rotate about fixed axes in acommon plane. The schematic view in panels A–E is introducedin Fig. 2(d). Panel A shows the trivial example from Fig. 2(a).
Panel B shows the orthogonal axes example from Fig. 2(b). In
panel C, the left axis is orthogonal, and the right axis is at an angleof 30° to the connecting line. In panel D, both axes are parallel
and at an angle of approximately 54.74° to the connecting line.
Panel E is again the trivial example rotated by 180°. At thebottom, the graph of one branch of relation (2) between the angles
θ and ϕ (marked red in panels A–E) which set the rotation axes isshown [cf. Fig. 2(d)]. The positions of the different configurations
are marked on the graph.

It should be noted that all input and output axes of thegears mentioned in the Introduction are parallel; moreover,most of them are, in fact, coaxial. We now consider two
dipoles and seek inclined rotation axes lying in a common
plane with the objective of creating a CGS. For an example
showing that such states exist, see Figs. 2(b) and 2(d). In
this example, the rotation axes are orthogonal. The obvious
question is how many configurations exist for two dipolesð2Þ

In view of Eq. (2), every angle between the two rotation
axes can be realized. In Fig. 3, we show some notableconfigurations in the schematic view introduced in Fig. 2(d)together with the graph of one branch of the relation
between θ and φ implied by Eq. (2). The second branch(obtained by reflecting the depicted branch over the lineθ ¼ φ) does not contain new configurations up to symmetries. The explicit formula for the depicted branch is
given in Appendix D. For the continuum of possible
rotation axes described by Eq. (2), the energy is independent of θ and φ and has the form (see Appendix A)
E2 ¼

m1 m2
cosðα − βÞ;d3ð3Þ

where m1 and m2 are the magnitudes of the moments, and dis the distance between the dipoles. Therefore, the dynamics of two rotating dipoles is identical for all configurations
obeying Eq. (2) since the Hamiltonian has the same form.
Figure 4 shows the energy landscape (3).
Note that if the restriction to specific rotation axes isdropped and free orientations in 3D space are allowed, so
far, only one finite arrangement with a CGS is previously
known: the case of eight dipoles at the corners of a cube[23]. Indeed, the discovery of the dipole cube provides theinitial motivation for seeking other CGSs since it can beargued that this arrangement is a unique peculiarity. We

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FIG. 5. Construction of a system with three interacting dipoles
which form a continuous state. This construction begins with theconfiguration shown in Fig. 3, panel D and then the dipole D3 isadded. The dipole pair D2 D3 interacts in the same way as D1 D2 ,while D1 D3 forms theptrivial
continuum with aligned axes. The
ﬃﬃﬃ
angle ψ ¼ 2 arcsinð1= 3Þ ≈ 70.53° and the distances D1 D2 andD2 D3 are equal. The paddle boat in Video 1 uses this construction.

stress that the restriction to specific rotation axes in thecurrent paper constitutes a fundamentally different class of
systems compared to the free orientations. The best
example is the dipole-dipole case: put two dipoles next
to each other and let them orient freely in 3D space, they
will align their moments coaxial head to tail and form a
discrete (noncontinuous) ground state. For free orientations, not any one of the proposed mechanisms in this paper
will work. For engineering applications, the restriction torotation axes has an even greater practical significancebecause the primary concern is with the transmission of
torques from one axis to one or several other axes. But thedipole cube [23] is also a solution for this class of problems
if we choose the rotation axes as the respective volume
diagonals of the cube: because even if the dipoles can be
freely oriented in the cube, they actually force each otherinto planes orthogonal to the respective volume diagonal.
III. THREE INTERACTING DIPOLES
We next use the results for two dipoles to construct asystem of three interacting dipoles which form a CGS and,
therefore, act as gears which transmit mechanical torques.
For this three dipole system to be a continuous state, every
pair of dipoles needs to form such a state. For theconfiguration shown in Fig. 3, panel D, it is possible to
add another dipole at the position shown in Fig. 5, while
keeping the continuous state. The additional dipole D3interacts with D2 exactly in the same way as the pair D1 D2does, whereas the pair D1 D3 forms the trivial continuumwith aligned axes. The configuration in Fig. 5 conveys thenotion underlying the construction of the paddle boat inVideo 1. From symmetry arguments, one might assume thatfor the system to work, the dipoles D1 and D3 need to be

VIDEO 2. (Animated version online.) The behavior for thesystem of three dipoles using the paddle boat construction fromVideo 1 as an example. The ratio of the driver dipole moment
magnitude to the paddle dipole moment magnitude is at thecritical value μc (top panel) or below μc (lower panels). The sizeof the driver dipole (right) is proportional to its dipole moment
magnitude. The inclination angle between the paddle dipoles(left) is constant during rotation.

oriented parallel (like in Video 1). This configuration
corresponds to the maximum energy state for the pair
D1 D3 , which is unstable. That raises the question how theoverall system can be in its stable ground state. The idea is
to increase m2 to make this particular configuration stable(with mi we denote the magnitude of the dipole moment miof Di ). For the stability analysis of the possible equilibria,
we assume m1 ¼ m3 and define the magnitude rationμ ≔ m2 =m1 . For a critical ratio
pﬃﬃﬃ
3 3
μc ¼
≈ 1.299;
ð4Þ
4
the system undergoes a pitchfork bifurcation. If μ ≥ μc , thedipoles D1 and D3 are parallel. Interesting enough, for μ <μc the two stable states are both CGS (an “additional”
twofold degeneracy). In these states, the dipoles D1 and D3
are inclined with respect to their common axis and maintainthis inclination during the rotation in the continuous stateanalogous to a rigid body rotation; see Video 2. Denoting

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SMOOTH TEETH: WHY MULTIPOLES ARE PERFECT GEARS

FIG. 6. Bifurcation diagram for the system of three dipoles
showing the stable (solid lines) and unstable (dotted line)
inclination angles δ between dipole D1 and D3 as a functionof the dipole moment ratio μ ≔ m2 =m1 ; see Eq. (5). At μc , apitchfork bifurcation occurs.

the inclination angle with δ, we can characterize the threestates which are stable for at least some values of μ as(see Appendix B)
δ0 ¼ 0;δI;II ¼ 2 arccos

μ
μc

ð5Þ

and show them in Fig. 6. With the decrease of μ, theangles δI;II grow in magnitude until D1 and D3 reach their
trivial antiparallel ground state when μ is zero (i.e., D2
vanishes). From the Hessian matrix for the energy of thesystem, we obtain the eigenvalues (EVs) for the threestates
3
2
3
2
0
0
7
6
7
6
EVI;II ¼ 4 6s2 5; ð6Þ
EV0 ¼ 4 6s 5;
2ð1 − s2 Þ2ðs − 1ÞFIG. 7. Different ways to construct a point quadrupole. Left:
Four monopoles of strength σ describing a square of area A are
brought together (A → 0) while σ is increased such that the
product σA stays finite. Middle: Two antiparallel dipoles arebrought together. Right: Four dipoles are brought together; thisarrangement is used as a schematic visualization in Video 3.

between a quadrupole and a dipole, where the geometry
is assumed as before: the rotation axes of the two objects lie
in a common plane; the orientation of the quadrupole withrespect to its rotation axis is analogous to the dipole in thesense that if one considers the quadrupole made up of fourmonopoles, these monopoles lie in the plane orthogonal to
the rotation axis (Fig. 7). The interaction energy between aquadrupole at the origin and a dipole at position r reads(see Appendix C)
EQD ¼

An interesting generalization of the findings for two
dipoles acting as gears involves higher multipole moments
and combinations thereof. We consider the interaction

ð7Þ

where Q is the quadrupole moment tensor, and m is the
dipole moment. If we introduce the coordinate systemanalogous to Fig. 1, we again find a φ for every θ which
creates a CGS. The respective nonlinear condition is (seeAppendix C)
2 cos θ cos φ − 3 sin θ sin φ ¼ 2:

with s ≔ μ=μc . The first zero EV in all three states reflects
again the degeneracy of any state of the system, and thecorresponding eigenvector is ð1; 1; 1ÞT in the spacespanned by the angles α, β, γ which describe the actual
orientations of the three dipoles. The second EV in all
three states has the eigenvector ð−1; 2; −1ÞT describing
the joint rotation of D1 and D3 against the “natural”
rotation. The system is always stable against perturbationsin this direction since the EV is always positive (forμ > 0). The third EV in all three states has the eigenvector
ð−1; 0; 1ÞT and describes the relative rotation between
D1 and D3 . For μ > μc, perturbations in this direction
are stable for δ0. At μ ¼ μc , the pitchfork bifurcation
occurs, so that for μ < μc, perturbations in this direction
are unstable for δ0 and stable for δI;II ; see Fig. 6. Ananimation of the system for different values of μ is given
in Video 2.
IV. THE QUADRUPOLE-DIPOLE SYSTEM

2jrj2 ðmT QrÞ − 5ðm · rÞðrT QrÞ
;
2jrj7

ð8Þ

2π

dipole-dipole

3/2 π
quadrupole-dipole

ϕ
π

1/2 π

0
0

1/2 π

π

θ

3/2 π

2π

FIG. 8. The graph of one branch of relation (8) is shown in red,
where the angles θ and ϕ define the rotation axes of thequadrupole-dipole system (see Figs. 2 and 3 for further explanations). As a comparison, the relation (2) for the dipole-dipole
case is shown in black. The configurations from Fig. 3 are marked
with filled circles.

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JOHANNES SCHÖNKE

EQD

EQD

1

1
0
-1
2π

0
2π

π

β

π
0 0

VIDEO 3. (Animated version online.) Example of a continuous
ground state created by a quadrupole and a dipole (shown in red)which are allowed to rotate about two orthogonal axes (shown ingray). The quadrupole (left) is symbolized by four dipoles. The
mechanical bevel gears are added to illustrate an analogy to amechanical system with a gear ratio of 1∶2.

In contrast to the dipole-dipole case, the quadrupole-dipole
system yields a gear ratio of 1∶2. If a dipole is thought of asa gear with one “tooth” at the north pole and one “notch” atthe south pole, a quadrupole is a gear with two teeth andtwo notches. When the dipole is rotated by a full cycle, thequadrupole rotates only half a cycle. An example of thequadrupole-dipole system is depicted in Video 3, whereas
the graph of one branch of relation (8) is shown in Fig. 8.
The explicit form of the interaction energy for the configurations obeying Eq. (8) reads (see Appendix C)
EQD ¼ −qmsin θ cosð2α − βÞ;d4ð9Þ

where q and m are the magnitudes of the quadrupole anddipole moment, respectively, and d is the distance between
the two objects. The energy landscape (9) is depicted inFig. 9. The form of the energy (9) reflects the interesting
effect that a quadrupole and a dipole do not “see” eachother if the dipole is positioned anywhere along the rotation
axis of the quadrupole, because this case corresponds toθ ¼ 0; π for which EQD ¼ 0. There is no such phenomenon
in the dipole-dipole case, where E2 does not depend on θ or
φ. As with the dipole systems discussed before, it is now
possible to construct systems with more than one dipole orquadrupole. Again, for these systems to be in a continuous
state, each pair of multipoles has to form such a state. The
possibilities are even more diverse now since we canconstruct systems where certain quadrupole-dipole pairs
do not see each other (cf. above). Generally, it is an
intriguing puzzle to find a CGS for an increasing numberof multipoles. A systematic strategy for doing so has not yet
been proposed. One of many possible extensions is theinclusion of static (nonrotating) multipoles.

α-1FIG. 9. Degenerate energy landscape (9) for the interaction of aquadrupole and a dipoles (q ¼ m ¼ d ¼ 1, θ ¼ π=2) giving rise
to a continuous ground state for 2α ¼ β.

V. CONCLUDING REMARKS
Although the main purpose of this paper is the introduction of a gear concept based on a symmetry for
multipole interactions, we comment on some technicalaspects regarding the actual realization of the corresponding machines. The analysis relies on the assumption of ideal
point multipoles. For the case of dipole systems, we canmake use of the fact that a homogeneously magnetized
sphere creates an outer field identical to a point dipole at thecenter of the sphere. Whether there exists an analogue
principle for higher-order moments is an interesting question by itself: is there a shape with a specific magnetization
pattern which creates an outer field identical to a point
quadrupole?
Regarding characteristics such as torque densities, thetheoretical analysis for a concrete application is straightforward since the expressions for the interaction energies arereadily given in analytical form. The characteristics should
qualitatively resemble those in Refs. [21,22].
Concerning the direct applicability in an actual machine,
it is noted that we are already working on an experimental
realization of a car drive based on the three-dipole system inFig. 5, which is also the basis of the design in Video 1. The
system is realized with inch-sized neodymium magnetic
spheres and 3D printer technology. Engineering characteristics, like, for example, the transmittable torque densities,
off-axis torques (which will bend the axes), or frictional
losses at the bearings, can be investigated once theprototype for the drive is finished. The results will bereported in a forthcoming publication.
As indicated in the Introduction, other potential applications can be found in the area of microfluidic systems,
where especially the possibility of transmitting torques intoa physically isolated domain is an advantage of magnetic
mechanisms over mechanical ones. On the length scales ofnanotechnology, we can even think about the constructionof the proposed mechanisms with electric multipoles. All

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SMOOTH TEETH: WHY MULTIPOLES ARE PERFECT GEARS
the findings in this paper are, of course, valid for magnetic
as well as electric systems.
ACKNOWLEDGMENTS
The author is very grateful to Eliot Fried and Ingo
Rehberg for many inspiring discussions and kind support.
APPENDIX A: THE ENERGY FORTWO DIPOLES

orientation of the dipoles with α, β, γ. Further, we assume
m1 ¼ m3 and define the dipole moment ratio μ ≔ m2 =m1 .
Without loss of generality, we can set m1;3 ¼ d13 ¼ 1. It
pﬃﬃﬃ
then follows from geometry that d12 ¼ d23 ¼ 3=2. Since
all three pairwise interactions of the system form adegenerate continuous state, we can use Eq. (A6) to write
down the energy E3 of this system asE3 ¼ k½cosðα − βÞ þ cosðβ − γÞ þ cosðα − γÞ;From Fig. 1, we infer the components of the two-dipole
moment vectors to be0
1− cos α sin θ
BC
m1 ¼ m1 @ cos α cos θ A;
ðA1ÞB
m2 ¼ m2 @− cos β sin φ

1C
cos β cos φ A;

ðA2Þ

sin β
with m1 and m2 being the magnitudes of the moments. We
assume the distance between the dipoles to be d and theirpositions to be ð0; 0; 0ÞT and ðd; 0; 0ÞT , respectively. Now
we can introduce Eqs. (A1) and (A2) into Eq. (1) and get
E2 ¼

m1 m2
½cos α cos βðcos θ cos φ − 2 sin θ sin φÞd3þ sin α sin β:ðA3Þ

If we treat θ and φ as fixed parameters, then E2 is a function
E2 ðα; βÞ. The only way to create a degeneracy in the energy
landscape E2 ðα; βÞ is to set the term in round brackets in
Eq. (A3) to unity, i.e.,cos θ cos φ − 2 sin θ sin φ ¼ 1;

k sinðα − βÞ þ k sinðγ − βÞE2 ¼¼There are four families of degenerate equilibria (states with
gradE3 ¼ 0). We discuss only the two families which arestable for at least some values of μ. They are covered by theansatzα ¼ β þ π þ δ=2;

m1 m2
½cos α cos β þ sin α sin β
d3

ðA5Þ

m1 m2
cosðα − βÞ:
d3

ðA6Þ

Because of the degeneracy, the Hessian of E2 has a zero
eigenvalue in the entire space spanned by α and β, thecorresponding eigenvector being ð1; 1ÞT .
APPENDIX B: STABILITY ANALYSIS FOR
THE SYSTEM OF THREE DIPOLES
Referring to Fig. 5, we denote the distance between
dipole Di and Dj with dij , the magnitudes of the dipole
moments with m1;2;3 , and the angles describing the actualγ ¼ β þ π − δ=2:

ðB3ÞThe angle δ describes the inclination of D1 and D3 withrespect to their common axis as discussed in the main text.
Introducing Eq. (B3) into Eq. (B2) and equating to zero, wefind that the third component of Eq. (B2) is always zero,and the first and second component result in the same
condition


δ
δ
sin
2 cos − k ¼ 0:
ðB4Þ
2
2
The three relevant solutions to Eq. (B4) are the states
δ0 ¼ 0;ðA4Þ

which is exactly the condition (2). Because only if Eq. (A4)holds, Eq. (A3) can be written as

ðB1Þ

pﬃﬃﬃ
with k ≔ 8μ=ð3 3Þ. The respective gradient reads
0
1− sinðα − γÞ − k sinðα − βÞBCðB2ÞgradE3 ¼ @ sinðα − γÞ − k sinðγ − βÞ A:

sin α
0

PHYS. REV. APPLIED 4, 064007 (2015)k
δI;II ¼ 2 arccos :2

ðB5Þ

The fourth solution δ ¼ π is always unstable and notconsidered here. For the state δ0 , the dipoles D1 and D3
are parallel, whereas for the states δI;II they are inclined. The
solutions δI;II do exist only below the critical value ofpﬃﬃﬃ
μc ¼ 3 3=4. For the stability analysis, we compute theeigenvalues and vectors of the Hessian matrix of E3 for thethree states δ0 and δI;II using Eqs. (B3) and (B5). The result
is given in the main text.
APPENDIX C: THE ENERGY FOR
THE QUADRUPOLE-DIPOLE CASE
There are several ways to construct a point quadrupolefrom a limit process for the positioning of point dipoles ormonopoles. Figure 7 illustrates some constructions. For thederivation of the quadrupole moment tensor, we use themonopole construction (Fig. 7 left), and the coordinate

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JOHANNES SCHÖNKE
system as introduced in Fig. 1. We assume the uppermonopole position in Fig. 7 left to be described with the left
dipole moment vector in Fig. 1 and the other threemonopoles to lie in the plane orthogonal to the rotation
axis. Then we can write down the monopole positions(counting from the upper monopole in the clockwise
direction 1, 2, 3, 4)
0B
p1 ¼ ϵ @10C
cos α cos θ A;sin α

B
p2 ¼ ϵ @− cos α sin θ− sin α sin θ

1C
sin α cos θ A;

− cos αðC1Þp3 ¼ −p1 and p4 ¼ −p2 . The factor ϵ contains the lengthscale. The symmetric traceless quadrupole moment tensor
is commonly defined asX
Qij ¼
ck ð3pki pkj − δij jpk j2 Þ;ðC2Þkwhere pki is the component i of pk , ck is the monopole’s
strength, and δij is the Kronecker delta. Now we introduce
the positions (C1) together with c1;3 ¼ σ and c2;4 ¼ −σinto Eq. (C2) and consider the limit σϵ2 → q=6 (cf. Fig. 7).
q describes the magnitude of the quadrupole moment,
and without loss of generality, the factor 1=6 is added toreduce the numerical prefactors in the final result. We thenarrive at the quadrupole tensor components
Qxx ¼ −a sin2 θ;

Qxy ¼ a sin θ cos θ;
Qxz ¼ −Qyz ¼ −2q sin α cos α cos θ;ðC3Þ

where a ≔ qðsin2 α − cos2 αÞ. Now the potential at a
spatial position r due to the quadrupole at the origin canbe expressed as
1 r Qr:2 jrj5T

EQD ¼

2jrj ðm QrÞ − 5ðm · rÞðr QrÞ:
2jrj7

Introducing Eq. (C3) andC
cos β cos φ A

ðC6Þ

sin β
together with r ¼ ðd; 0; 0ÞT into Eq. (C5), we finally obtainqm
sin θ½G cosð2αÞ cos β − sinð2αÞ sin β;
d4
3
ðC7Þ
where G ≔ cos θ cos φ − sin θ sin φ:2
EQD ¼ −

Again, like in the dipole-dipole case, the only way to createa degeneracy in the energy landscape EQD ðα; βÞ is to set
G ¼ 1, i.e.,
2 cos θ cos φ − 3 sin θ sin φ ¼ 2;

ðC8Þwhich is exactly the condition for degeneracy in the maintext. Because only if Eq. (C8) holds, the energy can be
written as
EQD ¼ −qmsin θ cosð2α − βÞ:
d4

ðC9Þ

APPENDIX D: EXPLICIT FORMULAS FOR
THE RELATIONS BETWEEN θ AND φ
The relation between θ and φ for the dipole-dipole casecos θ cos φ − 2 sin θ sin φ ¼ 1

ðD1Þ

for the branch shown in Fig. 3 of the main text. For thedefinition of the function atan2, see [24]. The relation
between θ and φ for the quadrupole-dipole case
2 cos θ cos φ − 3 sin θ sin φ ¼ 2

T

ðC5Þ

ðD3Þ

has the explicit form
pﬃﬃﬃ
pﬃﬃﬃ
φ ¼ atan2½2 sin θð 5 cos θ − 3Þ; 3 5sin2 θ þ 4 cos θ
ðD4Þ

ðC4Þ

With the corresponding force field of the quadrupole
F ¼ −gradϕ, we can write the interaction energy between
the quadrupole at the origin and a dipole with dipole
moment m at position r as EQD ¼ −m · F ¼ m · gradϕresulting in
T

1

pﬃﬃﬃ
pﬃﬃﬃ
φ ¼ atan2½sin θð 3 cos θ − 2Þ; 2 3sin2 θ þ cos θ ðD2ÞQzz ¼ a;2

Bm ¼ m@− cos β sin φhas the explicit form

Qyy ¼ −a cos2 θ;

ϕ¼

0

for the branch shown in Fig. 8 of the main text.

[1] Ala’aldeen Al-Halhouli, Stefanie Demming, Andreas
Waldschik, and Stephanus Büttgenbach, Implementationof synchronous micromotor in developing integrated microfluidic systems, Micromachines 5, 442 (2014).
[2] Wesley T. E. van den Beld, Natalia L. Cadena, Johan Bomer,
Eddy L. de Weerd, Leon Abelmann, Albert van den Berg,

064007-8

SMOOTH TEETH: WHY MULTIPOLES ARE PERFECT GEARS
and Jan C. T. Eijkel, Bidirectional microfluidic pumping
using an array of magnetic Janus microspheres rotatingaround magnetic disks, Lab Chip 15, 2872 (2015).
[3] Mark Peplow, The tiniest Lego: A tale of nanoscale motors,
rotors, switches and pumps, Nature (London) 525, 18(2015).
[4] P. M. Tlali, R.-J. Wang, and S. Gerber,, Magnetic gear
technologies: A review, in Proceedings of International
Conference on Electrical Machines (ICEM), 2014 (IEEE,
Berlin, 2014), pp. 544–550.
[5] C. G. Armstrong, Power-transmitting device, U.S. Patent
No. 687,292 (Nov. 26, 1901).
[6] K. Tsurumoto and S. Kikuchi, A new magnetic gear using
permanent magnet, IEEE Trans. Magn. 23, 3622 (1987).
[7] K. Atallah and D. Howe, A novel high-performancemagnetic gear, IEEE Trans. Magn. 37, 2844 (2001).
[8] Kais Atallah, Stuart D. Calverley, and David Howe,
High-performance magnetic gears, J. Magn. Magn. Mater.
272–276, E1727 (2004).
[9] S. Mezani, K. Atallah, and D. Howe, A high-performanceaxial-field magnetic gear, J. Appl. Phys. 99, 08R303(2006).
[10] K. T. Chau, Dong Zhang, J. Z. Jiang, and Linni Jian,
Transient analysis of coaxial magnetic gears using finite
element comodeling, J. Appl. Phys. 103, 07F101 (2008).
[11] A. Ya. Krasilnikov and A. A. Krasilnikov, Torque determination for a cylindrical magnetic clutch, Russ. Eng. Res. 29,544 (2009).
[12] Xinhua Liu, K. T. Chau, J. Z. Jiang, and Chuang Yu, Design
and analysis of interior-magnet outer-rotor concentric magnetic gears, J. Appl. Phys. 105, 07F101 (2009).
[13] Linni Jian, K. T. Chau, Yu Gong, J. Z. Jiang, Chuang Yu,
and Wenlong Li, Comparison of coaxial magnetic gears

PHYS. REV. APPLIED 4, 064007 (2015)

with different topologies, IEEE Trans. Magn. 45, 4526
(2009).
[14] Linni Jian and K. T. Chau, A coaxial magnetic gear with
Halbach permanent-magnet arrays, IEEE Trans. Energy
Convers. 25, 319 (2010).
[15] Kenji Nakamura, Michinari Fukuoka, and Osamu
Ichinokura, Performance improvement of magnetic gearand efficiency comparison with conventional mechanicalgear, J. Appl. Phys. 115, 044301 (2014).
[16] Ping Zheng, Jingang Bai, Jia Lin, Zhenxing Fu, Zhiyi Song,and Fei Lin, Design and transient behavior of magnetic
gears, J. Appl. Phys. 115, 17E706 (2014).
[17] Cheng-Chi Huang, Mi-Ching Tsai, D. G. Dorrell, and
Bor-Jeng Lin, Development of a magnetic planetary gearbox, IEEE Trans. Magn. 44, 403 (2008).
[18] F. T. Jorgensen, T. O. Andersen, and P. O. Rasmussen, The
cycloid permanent magnetic gear, IEEE Trans. Ind. Appl.
44, 1659 (2008).
[19] J. F. Charpentier and G. Lemarquand, Mechanical behaviorof axially magnetized permanent-magnet gears, IEEE Trans.
Magn. 37, 1110 (2001).
[20] F. T. Jorgensen, T. O. Andersen, and P. O. Rasmussen, Twodimensional model of a permanent magnet spur gear, inProceedings of the Industry Applications Conference, 2005
(IEEE, New York, 2005), Vol. 1, pp. 261–265.
[21] E. P. Furlani, Formulas for the force and torque of axial
couplings, IEEE Trans. Magn. 29, 2295 (1993).
[22] E. P. Furlani, Analysis and optimization of synchronous
magnetic couplings, J. Appl. Phys. 79, 4692 (1996).
[23] Johannes Schönke, Tobias M. Schneider, and Ingo Rehberg,
Infinite geometric frustration in a cubic dipole cluster, Phys.
Rev. B 91, 020410 (2015).
[24] https://en.wikipedia.org/wiki/Atan2.

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