Suppose initially I have a uniform magnetic field.

Now, in this formerly-uniform magnetic field, I place an un-magnetised cylindrical body of iron into this field of known dimension, with the axis of rotation aligned with the (formerly uniform) magnetic field.

How would one go about mathematically describing the magnetic field? How does this solution change as the cylinder is rotated about a different axis?

Note: This isn't a homework problem, but I'm guessing that there is a classic solution in ferromagnetism or magnetostatics to this problem. I'm not quite sure where to start looking - it's a bit beyond my freshman physics textbook that was gathering dust on my shelf. My vector calculus is a bit rusty, but I can probably work it out :)

The actual shape isn't too important, so if an object other than a cylinder helps illustrate the solution, then I am happy to use it.


I'll provide a simple example (other people who are more versed in EM than me will probably have more mathematically detailed answers). If you're viewing it as a boundary-value problem, then you can consider solving the magnetostatic Maxwell equations on a large box with a chunk of iron inside, with an input magnetic field flux coming in from two opposite ends of the box. Without the iron chunk, the field lines just run straight through the box from one end to the other, uniformly. With the iron chunk inside, the field lines do this:


And the scalar magnetic potential looks like this:


This was done assuming an iron bar ($\mu_r=4000$) embedded in a material with $\mu_r=500$ and input magnetic flux boundary conditions at the two ends. Obviously this is a bit different than an iron bar in air, but it's better for visualization purposes.

This completely ignores hysteresis and other ferromagnetic effects, which you sort of alluded to when you asked about rotating it around. For a cylinder of iron, however, the field will be symmetric with respect to rotations about the axis, and so any "memory effects" the iron bar has will make no difference when it is rotated. For anything else, I imagine it could be quite ugly, but other people might have answers.

For reference, the interior condition was magnetic flux conservation, $\nabla\cdot(\mu_0\mu_r{\bf H})=0$, 4 of the boundaries had the condition ${\bf n}\cdot{\bf B}=0$, and the other two boundaries had the condition ${\bf n}\cdot{\bf B}={\bf B}_n$ where ${\bf B}_n$ is the input flux. Just as a review, in magnetostatic conditions the ${\bf H}$-field is time-invariant and thus can be written as the gradient of a potential, ${\bf H}=-\nabla V$, very similar to electrostatics problems.

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