On the Pole Method of Magnetostatics and Permanent Magnets

[Charles Link]

The pole method of magnetostatics is presented in many E&M textbooks, particularly the older ones, to do computations in magnetostatics and even to try to explain permanent magnets. An equation that arises in the pole method is B=H+4*pi*M (c.g.s. units), where H consists of contributions from magnetic poles via the inverse square law plus any contributions from currents in conductors via Biot-Savart's law. In the pole method, any magnetic surface currents are completely ignored. It was very puzzling how any magnetic theory that used static poles instead of moving electrical charges could possibly work. In the pole method, the magnetic field is considered to come in two types-an H field, and a B field. After much review of the E&M subject, I recently performed some calculations that show/prove the pole method actually follows as a result of the surface currents, and that the computations of the pole method are in precise agreement with the B field from magnetic surface current calculations. The H of the pole method in the material is shown by these calculations to be a (subtractive) correction to the 4*pi*M of the surface currents for non-infinite cylinder geometries. Thereby, the H of the pole method is often misinterpreted, and it is the B and not the H that causes the magnetization M in materials and maintains the M. The equation B=H+4*pi*M is initially derived from the surface currents in the absence of currents in conductors, where H is the contribution from the poles. Outside of the material, B=H so that the H can be considered as an actual magnetic field. Inside the material, the H is however simply a (subtractive) correction term, and thereby H does not represent a magnetic field and is simply a mathematical construction. The H from currents in conductors is included as an add-on to the B=H+4*pi*M equation. These concepts are discussed in depth in a paper that I recently wrote-up. Additional computations are also discussed in the paper and a graph of M vs. B is presented, from which a typical hysteresis curve of M vs. H is generated by overlaying the line B=H+4*pi*M and allowing H to vary. Here is a link to the paper that I recently wrote up. https://www.overleaf.com/read/kdhnbkpypxfk I welcome any feedback.

 

[Charles Link]

Additional comment: The magnetic field in a permanent magnet is generated by magnetic surface currents that arise at the material boundary as the magnetization changes abruptly to zero. These surface currents can be explained in a two-dimensional analogy where the squares of the checkerboard each represent a single atom that has an electron orbiting in the same direction (e.g. counterclockwise) and thereby there is a current circulating on each square. The currents in adjacent squares precisely cancel, and the net effect will be a current circulating on the outer edge of the checkerboard. The paper (with the above link) contains a complete discussion of magnetism and permanent magnets. Additional puzzle arises in magnetic computations that is discussed in this paper: Starting with B=H+4*pi*M and taking the divergence of both sides of this equation, div B=0 so that div H=-4*pi*div M. Solving for H, the H can be computed using the inverse square law with -div M=magnetic charge density. The question is, where is the contribution to H from the currents in conductors in this calculation? And the answer, as is described in the paper, is that it appears in the solution of the homogeneous differential equation div H=0 that needs to be included to have a complete answer for H. Anyway, I am hoping some of the viewers may find the paper of interest.