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.

## Best Answer

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 density is $\frac{1}{2}\epsilon_0 | \vec{E} |^2$ while the magnetic field energy is $\frac{1}{2}\frac{1}{\mu_0} |\vec{B}|^2$. 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 example, section 6.3.1 of D. J. Griffiths Textbook "Introduction to Electrodynamics" for a discussion on this.