A couple of simple magnetization problems

[diegzumillo]

Homework Statement

These are actually two problems that I'm merging into one because each of them seem to have conflicting solutions, and I want to clear this up.

Consider a long cylinder (very long) extending in the z direction with a constant magnetization $\vec{M}=M\hat{z}$. What are the H and B fields everywhere? Now drill a very small hole through it in the z direction, what are the H and B fields everywhere now?

Homework Equations

$\vec{j}=\nabla \times \vec{M}$
$\vec{K}=\vec{M}\times \hat{n}$
There are more, of course, but let me know if something crucial is missing.

The Attempt at a Solution

Without hole: there is no current density inside (curl of that constant magnetization is zero) but we do have surface current $\vec{K}=M\hat{\phi}$ because the walls are perpendicular to M. So with that it's easy to calculate B (it's zero outside and $\mu_0 M$ inside, like a solenoid). And to find H is also simple, using $\vec{H}=\vec{B}/\mu_0-\vec{M}$ we see that H is zero everywhere, inside and out.

Now with the drilled hole for some reason we have H and B fields not zero everywhere. Shouldn't it be the at least similar? Sure, there's a current in the central surface going in opposite direction as the outer current, but I woud still expect that outside the whole thing the fields to be zero. What am I missing?

 

[Goddar]

I don't see anything wrong with your reasoning: the only changes should be the inner surface current and B-field within the cavity created.
Where did you get that there should be non-zero fields outside? (in the "very long" cylinder approximation, of course)

 

[diegzumillo]

Oh! I feel dumb. I misread the second problem statement. It's not infinitely long, it's infinitely wide. That changes everything (we could really use a facepalm 'smiey')