Mathematical Institute, University of Oxford, UK

Chains and rings of spherical magnets

Neodymium-iron-boron (NdFeB) magnets are now ubiquitous in low-temperature applications where high magnetic strength is required. But in addition to their essential role in hard drives and electric motors, NdFeB magnets have much more entertaining applications: collections of spherical NdFeB magnets have been sold as toys, and they can be used to construct complicated and interesting structures that are held together purely by magnetic attraction. The most basic structure that can be made from these magnets is a simple chain. To all appearances, such chains behave in a similar manner to elastic rods, but it is not immediately clear whether the equations that govern the deformation of a chain of magnets are the same as those that govern the deformation of an elastic rod.

In this talk, I will demonstrate that discrete-to-continuum asymptotic analysis can be used to derive a continuum equation for the mechanics of a chain of magnets based on the interactions between the magnetic dipoles. While an elastic rod simply has a local resistance to bending, we find that long-range interactions along a chain of magnets are also important, leading to a complicated expression for the energy associated with a given chain shape. This expression can be simplified in various situations, and I will present an analysis of a deformed circle of magnets; this leads to a simple expression for the vibrational modes of a ring of magnets that matches well with experimental results.