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Dancing paperclips and the geometric influence on magnetization: A surprising result
1.In hindsight, I can remember a related phenomenon that I experienced as a graduate student. We used to bring in nonscience students to briefly encounter a very strong magnetic field from an old cyclotron. I can distinctly remember that this tendency to align in the field is so strong (for ferromagnetic objects) that it was virtually impossible to hold a long iron bolt perpendicular to the field.
2.We use a magnetic demonstration system available from Pasco Scientific, item EM-8644A. This demonstration consists of a variable gap magnet and various accessories for demonstrating magnetic forces and eddy currents.
3.Although no connection is made to the experiments discussed here, this problem is discussed by D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, Upper Saddle River, NJ, 1999). Problem 6.21, pp. 281–282.
4.See, for example, Ref. 3, pp. 264–265.
5.We are assuming that the coordinate axes are aligned with the principal axes of the object. In general the demagnetizing effect is described by a demagnetizing factor tensor and , , and are the principal values of this tensor. See L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskiĭ, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, New York, 1984).
7.See, for example, J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 193–194.
8.Technically speaking, demagnetizing factors are only defined for objects that have uniform internal fields.
10.The magnetization can still be expressed as where is no longer constant but a field-dependent proportionality factor. Therefore, the results for a ferromagnetic sphere can be obtained from the results for a non-ferromagnetic sphere by replacing by . See E. M. Pugh and E. W. Pugh, Principles of Electricity and Magnetism (Addison-Wesley, Reading, MA, 1960), pp. 289–292.
11.Recall that the magnetic susceptibility is defined by a linear relation between the magnetization and the applied field ; for paramagnetism and for diamagnetism. The magnetic permeability of an object is related to its susceptibility by . The quantity is the relative permeability.
12.Note that this strong preference to magnetize along the axis of a long ferromagnetic rod will make it difficult to hold such an object perpendicular to a strong external field. See comment in Ref. 1.
13.R. E. Rosensweig, Ferrohydrodynamics (Cambridge U.P., New York, 1985), p. 98.
14.Note that this expression for the energy does not include the work done by the sources against the induced electromotive forces. For a brief discussion of this point, see Ref. 7, p. 216.
15.Our definition of is such that when is increasing, the object is rotating in the negative direction.
16.The corresponding result for a dielectric ellipsoid in an applied electric field is similar. See V. V. Batygin and I. N. Toptygin, Problems in Electrodynamics (Academic, London, 1964), pp.
16.The corresponding result for a dielectric ellipsoid in an applied electric field is similar. See V. V. Batygin and I. N. Toptygin, Problems in Electrodynamics (Academic, London), pp. 44–and.
17.The estimates in this section are only order-of-magnitude estimates and should not be taken too seriously. Nevertheless, these estimates indicate that a fairly strong uniform field would be necessary to observe the alignment of a typical diamagnetic cylinder.
18.Due to the presence of the horizontal surface on which the paperclips lie, the angle is confined to lie between and .
19.This dimensionless number is proportional to the Cowling number (which is related to the Alfvén or Kármán numbers) that arises when studying magnetohydrodynamics. See for example, K. Ražnjević, Physical Quantities and the Units of the International System (SI) (Begell House, New York, 1995), p. 185.
20.The minima at are not “functional” minima in the sense that Eq. (16) has minima at these point. Instead, these minima occur because the angle is restricted to lie between and . That is, we are dealing with “boundary value” minima.
21.The manufacturer responded to our request for information on the composition of their paperclips by saying it was proprietary information.
22.The field strength at which spontaneous standing takes place depends on whether the paperclips have been used and whether they are sitting on a flat smooth surface. Old paperclips on a rough surface will spontaneous stand at a lower field strength than new paperclips on a perfectly smooth surface.
23.Spontaneous standing for disks is more difficult to achieve than for paperclips. Presumably, this difficulty is due to the fact that the disks lie much more flatly against the table than do the paperclips.
24.This larger difference between and can be traced to the effective dimension of the object. A long thin rod is essentially one dimensional while a flat plate is two dimensional.
26.L. Sun, Y. Hao, C.-L. Chien, and P. C. Searson, “Tuning the properties of magnetic nanowires,” IBM J. Res. Dev.0018-8646 49, 79–102 (2005).
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