# The directions of $\vec{B}$ and $\vec{H}$

I have just started reading about magnetism and I cannot understand the sense behind the definition of the magnetic field vector $$\vec H$$. $$\vec H$$ acts like electric field as if magnetic monopole exists; so its direction inside the bar magnet is same as that of electric dipole.

1) But what good is this? Isn't $$\vec H$$ the magnetising field? So why does it make sense to define it this way? 2) According to wikipedia, the vector $$\vec B$$ (magnetic induction) is defined as the vector field necessary to make the Lorentz force law correctly describe the motion of a charged particle. But according to my textbook, magnetic induction is also called magnetic flux density which is the number of field lines passing through area vector, $$\vec A$$ (kept normal to the field lines). Which definitions should be applied when?

3) I checked out Direction of H and B inside and outside a bar magnet and I cannot understand statements like "the fields perpendicular to the interface just inside and outside the magnet"... Could you please explain?

4) And according to this image, shouldn't the magnetic induction inside the magnet given by $$\vec{B} =μ_0(\vec{H} + \vec{M})$$ be zero? • 2) Wiki's definition based on Lorentz force law is better. Just my opinion... – velut luna Mar 25 '18 at 18:49

In vacuum there is only one magnetic field and there is no argument regarding its meaning; there the notational difference between $$\textbf{B}$$ and $$\textbf{H}$$ is strictly a question of convenient units. Kelvin's view is that there are two different magnetic fields inside magnetizable matter and they can be defined by noting that the $$\textbf{B}$$ field is surface "thing"or a flux density, the mathematician would call it a 2-vector, while the $$\textbf{H}$$ field is a line thing (1-vector). As a consequence, inside a magnetized matter the two fields can be defined by creating crevasses of the correct shape.
Take an infinitesimal but thin and long needle-like cavity that is parallel with the local $$\textbf{M}$$ dipole magnetization density, then the field inside the empty cavity is the $$\textbf{H}$$. Now take another infinitesimal but flat and short cylindrical cavity whose flat top is locally perpendicular to $$\textbf{M}$$, now the field inside the cavity is $$\textbf{B}$$. Remember, these cavities have vacuum inside.
Mathematically the reason for the dependence of the fields on the shape of the infinitesimal cavity is that magnetizable matter is a collection of dipoles and not free magnetic charges. A dipole field has $$1/r^3$$ dependence and consequently the integral representing the total field of a dipole distribution is only semi-convergent (not convergent) when a small infinitesimal volume is excluded around a fixed point to calculate its value.
The terminology of $$\textbf{H}$$ being the magnetizing field is a correct one experimentally for a toroid with tightly wound coil around it so that there are no fields outside the toroid. In this case $$\textbf{H}$$ depends only on the current in the coil via Ampere's law $$\textrm{curl} \textbf{H} = \textbf{J}$$, and furthermore for a given ferromagnetic material one can also write that $$\textbf{M} = \textbf{M(H)}$$ and you can also see the source of the terminology. This fails already for a toroid with an air gap although most practical calculations may ignore the complications if the gap is very small compared to other linear dimensions of the toroid.