Magnetic levitation from negative permeability materials

Physics Letters A 376 (2012) 2739–2742

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Magnetic levitation from negative permeability materials Mark W. Coffey Department of Physics, Colorado School of Mines, Golden, CO 80401, United States

a r t i c l e

i n f o

Article history: Received 24 June 2011 Received in revised form 30 November 2011 Accepted 25 July 2012 Available online 31 July 2012 Communicated by A.R. Bishop

a b s t r a c t As left-handed materials and metamaterials are becoming more prevalent, we examine the effect of negative permeability upon levitation force. We first consider two half spaces of differing permeability and a point magnetic source, so that the method of images may be employed. We determine that the resulting force may be larger than for conventional magnetic materials. We then illustrate the inclusion of a finite sample thickness. © 2012 Elsevier B.V. All rights reserved.

Keywords: Metamaterial Left-handed material Permeability Levitation force Method of images Hankel transformation

The experimental realization of metamaterials and left-handed materials [1] has stimulated much research in the last decade [2,3]. With metamaterials, typically formed from a repeating structure, negative permittivity or permeability may be obtained. When both  and μ are negative, the material is said to be left-handed. In this Letter, we are interested in the effect of μ < 0 upon the phenomenon of magnetic levitation. As concerns a magnetic half-space and point magnetic dipole, we find that the levitation force can be larger than with conventional materials. We then extend the levitation arrangement to include a metamaterial of finite thickness. Model equations for materials providing a negative permeability include

μ f ω2 =1− , μ0 ω2 − ω02 + 2i γ0 ω

(1.1)

with filling factor f , magnetic plasma frequency ω0 , and dissipation coefficient γ0 [4]. Eq. (1.1) is based upon the effective medium point of view. The right side there does not approach 1 as ω → ∞—this model necessarily breaks down at sufficiently large frequencies. Another model permeability is the Drude–Lorentz contribution

f ω02 μD L =1− . 2 μ0 ω − ω02 + i γ0 ω

(1.2)

We note that simultaneous negative μ and  requires frequency dispersion in order to avoid having negative electromagnetic field

energy density [1]. In the presence of dispersion the energy density is given by



U=

1 ∂(ω) 2

∂ω

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.07.024

 ∂(μω) 2 H . ∂ω

(1.3)

So a necessary condition is that at least one of the derivatives here must be positive. For illustration, we consider two half spaces of permeabilities μ1 and μ2 and a point dipole of moment m in the upper half space along z = d where μ = μ1 . Using the method of images we have for the moment of the image dipole [5]

m =

μ1 − μ2 m. μ1 + μ2

(1.4)

The image dipole is located at an equal distance within the lower half space from the interface as the source dipole. In terms of the effective media description, delivering average effective μ and  , the unit cell would need to be much smaller than d. Otherwise, the metamaterial cannot be treated as a continuous media. There is a possibility of negative permittivity also enhancing the levitation force from electric fields. In this analogy with an electric dipole as a source, for instance, the method of images could again be applied. However, we focus on the magnetic effect here. The potential energy of the source magnetic dipole-image dipole interaction is given by



E-mail address: [email protected].

E2 +

V =



 ·m   )(r · m  ) μ1 m (r · m , −3 3 5 8π r r

(1.5)

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M.W. Coffey / Physics Letters A 376 (2012) 2739–2742

Fig. 1. Vertical force plotted versus

m = 0.162 G cm3 . Then with the magnet at height 0.15 cm, the levitation force is 9.7 dynes, corresponding to the weight from a 10 mg mass. We very briefly mention some experimental possibilities and tradeoffs. It is possible to use an array of dielectrics (TiO2 ) giving both negative μ and  , but then there is a so-far unquantified spatial dispersion [8]. Apparently NIST’s is one of the few such dielectric-based metamaterials that provides isotropic permeability. Other groups typically use an array of metallic rods and split rings leading to anisotropic properties. We note in passing that in the presence of ac electromagnetic fields, the light pressure force is entirely negligible for a macroscopic dipole. This force is the photon momentum h¯ k times scattering rate γ . With k = 2π /λ, λ = 6000 Å, and γ = 1/(1.6 × 10−8 ) s−1 , this force is ∼ 7 × 10−20 N, much less than the levitation force. We note that the levitation arrangement is related to models of magnetic force microscopy (MFM) [9]. A typical MFM tip may be treated as acting effectively somewhere between a dipole and monopole.

μ2 /μ0 .

where r = |r |. Then taking F = −(∂ V /∂ z)z=d to find the vertical force component we have

F=

3μ1 (1 + cos2 θ)mm

(2d)4

4π

=

3μ1m2 (1 + cos2 θ) (μ1 − μ2 )

(2d)4

4π

(μ1 + μ2 )

. (1.6)

Here θ is measured from the normal to the interface at z = 0. From Eq. (1.6) we may take various special cases. We may, for instance, consider having a superconducting half space. This is the case of a perfect diamagnet (Meissner effect) [6,7], μ1 = μ0 , μ2 = 0:

Fs =

3μ0m2 2π (2d)4

(1.7)

.

This maximum force from a superconductor corresponds to θ = 0. Taking the situation of one half space being vacuum, μ1 = μ0 ,

F=

3μ0m2 (1 + cos2 θ) (μ0 − μ2 )

(μ0 + μ2 )

(2d)4

4π

,

(1.8)

the force is attractive for μ2 > μ0 and repulsive for μ2 < μ0 . Fig. 1 shows a plot of the maximum force F / F s versus μ2 /μ0 obtained for θ = 0. In general μ = μ(ω) is complex valued, and should Re μ ≈ Re μ2 , the contribution of the imaginary part becomes more important. As an example, when μ2 = −3μ0 , we have

F =−

6μ0 m2 2π (2d)4

= −2F s ,

(1.9)

being twice as large in magnitude as even the superconducting case. It may be worth mentioning that a levitation force measurement could be used when other quantities (height, dipole moment and orientation) are known to find μeff . For an experiment with a spherical permanent magnet, one can use the result that its field is equivalent to that of a point dipole of the same moment located at the center of the sphere. To obtain an approximate relation between levitation height z0 , dipole moment M, and permanent magnet mass m0 , we can consider for instance case (1.7), so that

z0 =

1 2



3μ0 2π m 0 g

1/4 m

1/2

.

(1.10)

Some representative numbers are the following values. Rewriting (1.7) for CGS units, F s = 3m2 /16d4 , we take the dipole moment

Extension to sample of finite thickness We next illustrate how a finite thickness of metamaterial may be included for calculation of the levitation force. The technique is to apply 2D Fourier transformation, taking advantage of the cylindrical symmetry. The applied field in the region above a slab of thickness t is assumed due to a point magnetic dipole of moment  = m zˆ (see Fig. 2). m  and magWe continue to assume a linear relation between B  . The permeability enters the magnetic flux calculation netic field H through the continuity boundary condition on the tangential com. ponent of H Let the metamaterial slab occupy the region −t  z  0. Using the geometrical arrangement shown in Fig. 2, where the point dipole is located a distance a above the top surface, the governing partial differential equations for the vertical component of magnetic induction are 2 ∇ 2 B z (ρ , z  0) = μ1mδ( z − a)∇2D δ2D (ρ ),

(2.1a)

∇ 2 B z (ρ , z) = 0 in −t  z  0,

(2.1b)

∇ 2 B z (ρ , z) = 0 for −∞ < z  −t .

(2.1c)

In Eq. (2.1a), δ is the Dirac delta function, taken in the sense of distribution theory. Here (ρ , z) are cylindrical coordinates and the z-axis goes through the point dipole. In the following, we use the abbreviation C ≡ μ1 m/4π , 0  k < ∞ is the wavenumber, and J n is the Bessel function of the first kind of order n. For z  0, we have B z (ρ , z) = B 1z (ρ , z) + B 2z (ρ , z),  1 is the applied magnetic induction. In particular, we where B have

B 1z (ρ , z) = C

[2( z − a)2 − ρ 2 ] , [ρ 2 + ( z − a)2 ]5/2

(2.2a)

and the Hankel representations

∞ B 2z (ρ , z  0) = C

A 1 (k)e −kz J 0 (kρ )k dk,

(2.2b)

0

B z (ρ , −t  z  0)

∞ =C





k J 0 (kρ ) A c (k) cosh(γ z) + A s (k) sinh(γ z) dk,

(2.3)

0

∞ A 2 (k)ekz J 0 (kρ )k dk.

B z (ρ , z  −t ) = C 0

(2.4)

M.W. Coffey / Physics Letters A 376 (2012) 2739–2742

2741

In Eqs. (2.2)–(2.4), the boundary conditions B z → 0 as z → ±∞ and the axisymmetry have been applied. We recall that the vertical component of applied induction, Eq. (2.2a), for a point dipole with vertically oriented moment, has the following Hankel representation:

∞ B 1z (ρ , z) = C

e −|z−a|k J 0 (ρ k)k2 dk.

(2.5)

0

The wavenumber-dependent coefficients A 1 , A 2 , A c , and A s are determined by the continuity boundary conditions on B z and H ρ .  = 0 gives With rotational symmetry, the condition ∇ · B

∂ ∂ Bz (ρ B ρ ) = −ρ . ∂ρ ∂z

(2.6)

Thus, together with (ρ /k) J 1 (kρ ), we find

B ρ (ρ , z) = −

1



ρ

the

integration

rule



ρ J 0 (kρ ) dρ =

Fig. 2. Geometry of boundary value problem with point dipole.

 · B 2 (ρ = 0, z = a) gives The interaction energy U (a) = −(1/2)m the levitation force F z (a) = −∂ U (a)/∂ a. The 1/2 factor occurs due to the self-interaction [10]. Therefore, we have U (a) = −

2

A 1 (k)e −ka k dk

0

∂ Bz ρ dρ . ∂z

 Cm  =− 1 − (μ1 /μ2 )2 2

(2.7)

∞

Then

H ρ 2 (ρ , z) =

C

μ1

∞

× A 1 (k)e −kz J 1 (kρ )k dk,

(2.8a)

∞  C

μ2

0





(2.8b)

F z (a) = −Cm 1 − (μ1 /μ2 )2

× C

μ1

0

∞ A 2 (k)ekz J 1 (kρ )k dk.

F z (a) → −Cm

(2.9a)

μ1 As, μ2

A c cosh kt − A s sinh kt = A 2 e

− A c sinh kt + A s cosh kt =

(2.10a)

,

μ2 k A 2 e −kt . μ1

ke −ka [1 − (μ1 /μ2 )2 ] sinh kt 2(μ1 /μ2 ) cosh kt + [(μ1 /μ2 )2 + 1] sinh kt

=−

(2.10b)

These equations follow from the orthogonality of the Bessel functions J 0 (kρ ) and J 1 (kρ ) on [0, ∞) and their algebraic solution is given by

,

(2.11a)

2k(μ1 /μ2 )ek(t −a)

, (2.11b) 2(μ1 /μ2 ) cosh kt + [(μ1 /μ2 )2 + 1] sinh kt (sinh kt + (μ1 /μ2 ) cosh kt ) , A c (k) = 2ke −ka 2(μ1 /μ2 ) cosh kt + [(μ1 /μ2 )2 + 1] sinh kt (2.11c) (cosh kt + (μ1 /μ2 ) sinh kt ) . A s (k) = 2ke −ka 2(μ1 /μ2 ) cosh kt + [(μ1 /μ2 )2 + 1] sinh kt (2.11d)

The factor of sinh kt in the numerator of A 1 (k) for the induced field indicates that properly B 2z vanishes when the metamaterial slab is no longer present, t → 0.

k3 dk.

[1 − (μ1 /μ2 )2 ] 2μ1 /μ2 + (μ1 /μ2 )2 + 1

∞

e −2ka k3 dk

(2.14)

0

[1 − (μ1 /μ2 )2 ] 3 = −Cm [1 + (μ1 /μ2 )]2 8a4

(2.9b) −kt

2(μ1 /μ2 ) cosh kt + [(μ1 /μ2 )2 + 1] sinh kt

For t → ∞ in (2.13) we have

0

ke −ka + A 1 = A c ,

e −2ka sinh kt

(2.13)

(2.8c)

Applying the two continuity conditions at the two surfaces z = 0 and z = −t gives the four equations



∞

0

H ρ (ρ , z  −t ) =

A 2 (k) =

k2 dk.

Then

and

A 1 (k) =

2(μ1 /μ2 ) cosh kt + [(μ1 /μ2 )2 + 1] sinh kt

(2.12)

A c (k) sinh(kz) + A s (k) cosh(kz) J 1 (kρ )k dk,

ke −ka − A 1 =

e −2ka sinh kt

0

H ρ (ρ , −t  z  0)

=

∞

Cm

μ1m2 3 (1 − μ1 /μ2 ) , 4π 8a4 (1 + μ1 /μ2 )

(2.15)

in agreement with the result of using the image method. We very briefly describe how the above procedure could be generalized for arbitrary dipole orientation. For magnetostatic con = μH  = −μ∇Φ , where the scalar potential for ditions we have B the dipole is given by

Φ(x1 ) =

 · x1 m 4π |x1 |3

(2.16)

,

 = (mρ , m z ). Then we find with x1 = (x, y , z − a) = (ρ , z − a) and m for the applied induction

 1z (ρ , z) = − B

μ1 {−3mρ (z − a) + m z [ρ 2 − 2(z − a)2 ]} . 4π [ρ 2 + ( z − a)2 ]5/2

(2.17)

The additional Hankel representation needed is

∞

e −|z−a|k J 1 (ρ k)k2 dk =

0

Due to a new integration

3ρ |a − z|

[ρ + ( z − a)2 ]5/2 2

.

(2.18)

2742



ρ J 1 (kρ ) dρ =

M.W. Coffey / Physics Letters A 376 (2012) 2739–2742

π ρ 2 k



J 1 (kρ )H0 (kρ ) − J 0 (kρ )H1 (kρ ) ,

(2.19)

where Hn is the Struve function, the solution becomes more involved. Some further details are given elsewhere [11]. In summary, we have considered the influence of negative permeability upon magnetic levitation and found that the effect could be enhanced over conventional materials. The force from a negative-μ sample may be either repulsive or attractive. There are experimental possibilities to realize this effect. Necessary conditions are a dispersive metamaterial and nonzero frequency. Acknowledgements This work was supported in part by the Electromagnetics Division of NIST Boulder. Useful discussions with J. Baker-Jarvis and S. Kim are gratefully acknowledged.

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