Permanent magnet on the surface of a hollow sphere

[baggiano]

Hello

I would like to calculate the magnetic flux density given by a permanent magnet which is placed about the surface of a iron hollow sphere (assume mur = Inf). Actually, I am interested in the flux density just outside the hollow sphere.

The idea is to replace the magnet with equivalent volume/surface current densities (or with equivalent volume/surface charge densities) and subsequently apply the method of images. As far as I understand this is quite classical if the magnet is placed above a ferromagnetic infinite plate. However, as the magnet is placed above the hollow sphere I am struggling to figure out how to find the image sources (position and value). I have found a nice example in the Griffiths book (example 3.2) but I am not sure I can directly apply one to one the theory.

Any suggestion? Hints?

Thanks in advance and have fun!

 

[eljose79]

Hello,

Thank you for your question. Calculating the magnetic flux density in this scenario can be approached using the method of images, as you mentioned. However, since the magnet is placed above a hollow sphere instead of a flat plate, the image sources will be different.

One way to approach this problem would be to first consider the hollow sphere as a collection of smaller, infinitesimal current loops. Each of these loops will act as an individual magnetic dipole, with its own magnetic moment and corresponding magnetic field. Then, using the method of images, we can find the equivalent current loops that would produce the same magnetic field as the original magnet.

To do this, we can use the concept of multipole expansion. The magnetic field at a point outside the hollow sphere can be expressed as a sum of contributions from each of the current loops. By setting the magnetic field produced by the equivalent current loops equal to the field produced by the original magnet, we can solve for the position and value of the image sources.

I would suggest looking into multipole expansion and the method of images for more information and guidance on this approach. I hope this helps and good luck with your calculations!