I am approximating the magnetic field of a cylindrical permanent magnet using two fictitious magnetic monopoles with separation (magnetic length) of 2L. This model is analogous to an electric dipole, and so it seems like the direction of the magnetic field both inside and outside of the magnet are the same. This doesn't make sense to me since the magnetic field lines occur in "loops."

Is this just a limitation of this model? However, Wikipedia explains that the magnetic field due to a "magnetostatic dipole" looks like this (see image below), where field lines do in fact seem to occur in loops. I am quite confused.

enter image description here


1 Answer 1


That's not quite the correct approach. If the cylindrical permanent magnet has a constant magnetization directed along the axis of the cylinder, then the lateral surface will have an effective net bound surface current density. If the cylinder is long, $\mathbf{B}$ will be solenoid-like. If it's not long, $\mathbf{B}$ will be like what is produced by a current loop. In either case, you can calculate a net magnetic dipole moment that produces a dipole field that will well describe the magnetic field far from the cylinder.

The contribution of the end surfaces is to act as the source for the irrotational component in the Helmholtz decomposition of $\mathbf{H}$.

  • $\begingroup$ So say I wish to approximate fairly accurately the magnetic field close to a cylindrical magnet or radius 6mm and height 10mm, would it then be best to approximate the field using a coil of radius 6mm and height 10mm? I have attempted to derive the magnitude of the field on the mid-plane of the magnet using the Biot-Savart law, but I ended up with an integral that could only be numerically evaluated. Ultimately I wish to find the magnetic flux through a ring around the cylindrical magnet as a function of the radius of the ring. Is there an easier way to achieve this? $\endgroup$
    – Brian C.
    Oct 17, 2016 at 22:53
  • $\begingroup$ If you're interested in the field everywhere outside of the magnet, then I'm afraid that numerical integration is probably your only available tool. If you care about distances greater than the size of the magnet, though, you can probably do adequately by calculating the first few terms of the multipole expansion for the vector potential from the surface current, and then taking the curl. $\endgroup$ Oct 17, 2016 at 22:59

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