When you take the Magneto-static limit, you can have constant currents J producing a constant B field. You drop the dB/dt, and dE/dt terms because it's a static problem with no changing fields. So the Maxwell's equations would just look like:
divE=4pi*rho, divB=0, curlE=0, curlB=4pi*J/c
I'm not sure what you mean "explain the context from the fundamental point of view".
If you have a magnetic medium, you have that H=B-4piM. For linear materials B=mu*H. H is produced by only "free" currents while B must take into account both free and bound currents. In general, you will induce magnetization so as to align the magnetic dipoles with the external field, so that the magnetic material "reinforces" the H field to create a much stronger B field (this means that mu>>1 for strongly magnetic materials). I think, then, the vectors should all be pointing in the same direction, at least for this simple case. I think you can get phenomenon like hysteresis where the magnetic dipoles take time to readjust if you change the external field, so in that case, the vectors would not point in the same direction, but could point in different directions. The details are a little complicated...I'll let someone with more expertise explain that.