Additional comment: The geometry that they have found to be extremely useful is to wrap insulated wire windings around an iron cylinder and run a current through the windings to make an electromagnet. The magnetic fields for this case are quite straightforward and are very much standard textbook material. Had you asked about this case, you most likely would have gotten some very good and very detailed answers. In this electromagnet case, you do get two well defined poles on the end faces of the iron cylinder. ## \\ ## The electromagnet is an extremely interesting problem, and the magnetic fields can be precisely computed in two different ways: 1) By a magnetic pole method 2) By a magnetic surface current method, that really describes the underlying physics much better than the pole method.## \\ ## [Magnetic surface currents result on the iron core from a uniform magnetization throughout the iron. These surface currents go around the iron cylinder and are (approximately) 1000 x stronger than the current in the windings and in the same direction as the current in the windings, and they arise, not because of the close proximity to the current in the windings, but actually bacause of the uniform magnetization that occurs in the iron cylinder as a result of a uniform magnetic field that occurs from the windings. Without the iron core, these windings make a very well known solenoid geometry that generates a uniform magnetic field inside the cylinder along its axis. With the iron core, the magnetic field is stronger in the core by a factor of 1000 because of the magnetic field generated by the magnetic surface currents. The magnetic flux lines go through the iron core and emerge out one end , the north pole, and loop around and go back into the magnet at the south pole. ## \\ ## The equations of the pole method, (in particular the equation ## \vec{B}=\mu_o \vec{H}+\vec{M} ##), might lead one to believe that the magnetic field ## B ## in the iron core is created by the magnetization ## M ##, but in fact, the surface current method shows that the magnetic field in the iron core actually results from the surface currents that result from the magnetization ## M ## when it encounters the surface boundary.] ## \\ ## ## \\ ## Both methods get the exact same answer for the magnetic field. The pole method is mathematically simpler, but, in general, magnetic fields are caused by the motion of electrical charges (which are contained in the surface current method). The static pole method, in any case, is a very good mathematical shortcut. It has been mathematically proven that these two methods give identical results.