Electric and Magnetic Field Intensity Analogy

The electric field lines denote the electric field intensity $\vec{E}$ at every point so why doesn't magnetic field lines denote the magnetic field intensity $\vec{H}$ at every point but the magnetic flux density $\vec{B}$?

I understand that basically it is the same thing and it actually the lines do denote $\vec{H}$ as well because:

$$\vec{B}=\mu \vec{H}$$

but I haven't seen a single source mentioning that the magnetic field lines express $\vec{H}$.

Also the electric flux is is given by:

$$\Phi = \iint_{S}\vec{E}\cdot d\vec{S}$$

So why isn't the magnetic flux given by:

$$\Phi = \iint_{S}\vec{H}\cdot d\vec{S}$$

Again it is the same thing and it is only a matter of semantics but I have yet to come across a source using the latter expression for the magnetic flux.

• I know $D$ as electric flux density, making your confusion slightly less asymmetric – mikuszefski Mar 7 '17 at 11:52
• @mikuszefski I know that and I use it in my equations for symmetry. It is nice that someone else mentioned symmetry too. :) – Adam Mar 7 '17 at 12:11

There is a historical confusion about which $\vec{B}$ or $\vec{H}$ deserves to be called "magnetic field" (the magnetic counterpart of $\vec{E}$). This caused the $\vec{H}$ field to be called the "magnetic field" and not $\vec{B}$.
However, it turns out that $\vec{B}$ is the one that should be called "magnetic field", it is the one appearing in Maxwell's equations in a vacuum side by side with $\vec{E}$. The electric field energy is $\frac{1}{2}\epsilon_0\vec{E}$ while the magnetic field energy is $\frac{1}{2}\frac{1}{\mu_0}\vec{B}$. And so on...
$\vec{H}$, on the other hand, is defined in the context of magnetic fields in matter, being actually an "auxiliary field".
I agree with Lucas Francisco that $H$ is an auxiliary field, but in vacuum one could, e.g., write $$\frac {1}{ 2} \mu_0 H^2,$$ which, for symmetry reasons, looks much better with respect to the electric field. Moreover, looking at the units and considering the fact that equivalent to $E$ one may define a scalar potential for $H$ (introducing virtual magnetic charges), I believe it is absolutely OK to call $H$ magnetic field. Nevertheless, one has to keep in mind that $B$ and the nature of magnetism is fundamentally different in the sense that there are no magnetic charges. This difference breaks the symmetry.