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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.

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

You may look at, for exemple, section 6.3.1 of D. J. Griffiths Textbook "Introduction to Electrodynamics" for a discussion on this.

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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.

A good article is probably:

J.D. Jackson The nature of intrinsic magnetic dipole moments CERN 77-17

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