# A couple of simple magnetization problems

## 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?