Magnetostatics of the uniformly polarized torus

Marco Beleggia, Marc De Graef, Yonko Millev

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    We provide an exhaustive description of the magnetostatics of the uniformly polarized torus and its derivative self-intersecting (spindle) shapes. In the process, two complementary approaches have been implemented, position-space analysis of the Laplace equation with inhomogeneous boundary conditions and a Fourier-space analysis, starting from the determination of the shape amplitude of this topologically non-trivial body. The stray field and the demagnetization tensor have been determined as rapidly converging series of toroidal functions. The single independent demagnetization-tensor eigenvalue has been determined as a function of the unique aspect ratio α of the torus. Throughout the range of values of the ratio, corresponding to a multiply connected torus proper, the axial demagnetization factor Nz remains close to one half. There is no breach of smoothness of Nz(α) at the topological crossover to a simply connected tight torus (α=1). However, Nz is a non-monotonic function of the aspect ratio, such that substantially different pairs of corresponding tori would still have the same demagnetization factor. This property does not occur in a simply connected body of the same continuous axial symmetry. Several self-suggesting practical applications are outlined, deriving from the acquired quantitative insight.
    Original languageEnglish
    JournalRoyal Society of London. Proceedings. Mathematical, Physical and Engineering Sciences
    Volume465
    Issue number2112
    Pages (from-to)3581-3604
    ISSN1364-5021
    DOIs
    Publication statusPublished - 2009

    Cite this

    @article{37388d92e0de4bfdaaab66cba516f8d1,
    title = "Magnetostatics of the uniformly polarized torus",
    abstract = "We provide an exhaustive description of the magnetostatics of the uniformly polarized torus and its derivative self-intersecting (spindle) shapes. In the process, two complementary approaches have been implemented, position-space analysis of the Laplace equation with inhomogeneous boundary conditions and a Fourier-space analysis, starting from the determination of the shape amplitude of this topologically non-trivial body. The stray field and the demagnetization tensor have been determined as rapidly converging series of toroidal functions. The single independent demagnetization-tensor eigenvalue has been determined as a function of the unique aspect ratio α of the torus. Throughout the range of values of the ratio, corresponding to a multiply connected torus proper, the axial demagnetization factor Nz remains close to one half. There is no breach of smoothness of Nz(α) at the topological crossover to a simply connected tight torus (α=1). However, Nz is a non-monotonic function of the aspect ratio, such that substantially different pairs of corresponding tori would still have the same demagnetization factor. This property does not occur in a simply connected body of the same continuous axial symmetry. Several self-suggesting practical applications are outlined, deriving from the acquired quantitative insight.",
    author = "Marco Beleggia and {De Graef}, Marc and Yonko Millev",
    year = "2009",
    doi = "10.1098/rspa.2009.0355",
    language = "English",
    volume = "465",
    pages = "3581--3604",
    journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
    issn = "1364-5021",
    publisher = "The/Royal Society",
    number = "2112",

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    Magnetostatics of the uniformly polarized torus. / Beleggia, Marco; De Graef, Marc; Millev, Yonko.

    In: Royal Society of London. Proceedings. Mathematical, Physical and Engineering Sciences, Vol. 465, No. 2112, 2009, p. 3581-3604.

    Research output: Contribution to journalJournal articleResearchpeer-review

    TY - JOUR

    T1 - Magnetostatics of the uniformly polarized torus

    AU - Beleggia, Marco

    AU - De Graef, Marc

    AU - Millev, Yonko

    PY - 2009

    Y1 - 2009

    N2 - We provide an exhaustive description of the magnetostatics of the uniformly polarized torus and its derivative self-intersecting (spindle) shapes. In the process, two complementary approaches have been implemented, position-space analysis of the Laplace equation with inhomogeneous boundary conditions and a Fourier-space analysis, starting from the determination of the shape amplitude of this topologically non-trivial body. The stray field and the demagnetization tensor have been determined as rapidly converging series of toroidal functions. The single independent demagnetization-tensor eigenvalue has been determined as a function of the unique aspect ratio α of the torus. Throughout the range of values of the ratio, corresponding to a multiply connected torus proper, the axial demagnetization factor Nz remains close to one half. There is no breach of smoothness of Nz(α) at the topological crossover to a simply connected tight torus (α=1). However, Nz is a non-monotonic function of the aspect ratio, such that substantially different pairs of corresponding tori would still have the same demagnetization factor. This property does not occur in a simply connected body of the same continuous axial symmetry. Several self-suggesting practical applications are outlined, deriving from the acquired quantitative insight.

    AB - We provide an exhaustive description of the magnetostatics of the uniformly polarized torus and its derivative self-intersecting (spindle) shapes. In the process, two complementary approaches have been implemented, position-space analysis of the Laplace equation with inhomogeneous boundary conditions and a Fourier-space analysis, starting from the determination of the shape amplitude of this topologically non-trivial body. The stray field and the demagnetization tensor have been determined as rapidly converging series of toroidal functions. The single independent demagnetization-tensor eigenvalue has been determined as a function of the unique aspect ratio α of the torus. Throughout the range of values of the ratio, corresponding to a multiply connected torus proper, the axial demagnetization factor Nz remains close to one half. There is no breach of smoothness of Nz(α) at the topological crossover to a simply connected tight torus (α=1). However, Nz is a non-monotonic function of the aspect ratio, such that substantially different pairs of corresponding tori would still have the same demagnetization factor. This property does not occur in a simply connected body of the same continuous axial symmetry. Several self-suggesting practical applications are outlined, deriving from the acquired quantitative insight.

    U2 - 10.1098/rspa.2009.0355

    DO - 10.1098/rspa.2009.0355

    M3 - Journal article

    VL - 465

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    JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

    JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

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