Why are magnetic field lines perpendicular to the surface of a ferromagnetic material? It is known that magnetic field lines become nearly perpendicular to the surface of a ferromagnetic material. The quantitative proof which uses the boundary condition requires that magnetization at the surface be zero.
Question: Why is magnetization at the surface of a ferromagnetic material zero? Also, can someone give a qualitative explanation for the approaching perpendicularness of magnetic field lines and the surface of a ferromagnetic material?
 A: They are not.
However, there are relations at the surface between two materials.
$$\vec{H_1}\wedge\vec{n}=\vec{H_2}\wedge\vec{n}$$
This comes from the Ampere's circuital law used on a very flat rectangle whose long edges are on either side of the domains frontier.
If $1$ designates the air and $2$ designates the ferromagnetic material, since $\vec{B}=\mu\vec{H}$, $$\frac{\vec{B_1}\wedge\vec{n}}{\mu_0}=\frac{\vec{B_2}\wedge\vec{n}}{\mu_{iron}}$$ , i.e. :
$$B_{air,t}=\frac{\mu_{0}}{\mu_{iron}}B_{iron,t}$$
Since $\mu_{iron}>>\mu_{0}$, the tangential component of the magnetic flux density in the air is very small.
As a complement, the other relation is $$\vec{B_1}.\vec{n}=\vec{B_2}.\vec{n}$$ (continuity of the normal component of the magnetic flux density) and comes from $div(\vec{B}=0)$.
A: As is well known divB is always null, and when there are not free currents curlH is also zero. Therefore both the normal components of B and the tangential components of H are continuous at the boundary of the sample. This allows us drawing a figure of the lines of force of H and B, showing the so-called law of the refraction of the magnetic field. This refraction is very pronounced when permeabilities are very different. In your example, if the permeability of the sample is very higher than the surrounding, the magnetic field B in the sample will be intense (field plot dense) and practically parallel to the boundary while outside the field is weak and perpendicular) to the border.
