How can we prove the boundary conditions of the magnetic field $\vec{B}$ that the tangential component of the magnetic field changes when magnetic field lines travel from one medium to another?
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
2
The normal component of the B-field at an interface is always continuous, since $$\oint {\bf B}\cdot d{\bf S} = 0$$ and so $$B_{1\perp} = B_{2\perp}\ .$$
If there are no (non-induced) surface currents however, the component of the H-field parallel to the interface is the same on either side, which means the parallel component of the B-field will change according to $$\frac{B_{1\parallel}}{\mu_1} = \frac{B_{2\parallel}}{\mu_2}\ .$$
At the microscopic level, the change in $\mu$ means there are induced magnetic dipoles that line up in such a way that they enhance $B_{2\parallel}$ but leave $B_{2\perp}$ unchanged. Another way of thinking about this is that the magnetisation induces a surface current density, but the B-field produced by a surface current is only in the direction parallel to the plane of the surface.
-
$\begingroup$ Can u please explain? I am new to this chapter. Magnetic dipole develop inside the material so shouldn't they change the normal component of the field and not the parallel component just like in electric field where the field gets reduced by a dielectric due to dipoles aligned in the field direction. $\endgroup$– shahrOZeCommented Jun 30, 2020 at 20:36
-
$\begingroup$ How does the parallel field only changes? Can u describe what's happening on the atomic scale as though what is really happening? I can't get an intuition about it. $\endgroup$– shahrOZeCommented Jul 1, 2020 at 5:01
$\begingroup$
$\endgroup$
Magnetization produces a bound current on the boundary, $$\mathbf{K}_b=\mathbf{M}\times\mathbf{\hat n}.$$ This current produces a B-field perpendicular to both the surface normal and the current, as you can easily confirm using Ampere's Law. Since there is no normal component of the B-field due to the magnetization, the normal component of the total B-field must be continuous on the boundary.
