# 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 Commented Mar 7, 2017 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. :)
Commented Mar 7, 2017 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 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.

There is no reason at all why you can't draw lines of H-field and in some cases you should. It is not always the case that they look like B-field lines. They certainly don't when there are media with a permanent magnetisation or non-linear permeabilities (which covers most permanent magnets and ferromagnetic materials).

Electromagnetism needs both the H-field and the B-field. Indeed the Ampere-Maxwell law needs to have H-field on the LHS, whilst the solenoidal law reflecting the lack of monopoles must be written in terms of the B-field divergence (since there are sources of H-field).

The E-field and B-field are probably considered more fundamental because it is these fields that lead to the observational manifestations encoded by the Lorentz force law.

Symmetry is restored to the situation if you include the D-field and also allow the possibility of magnetic charges and currents. In such a situation, the E- and H-fields would be directly analogous, as would the B- and D-fields.

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

• There are no magnetic charges - but only if we define the fields in the particular way that we do. i.e. The Lorentz force law. There are "charges" (poles) for H-field. Commented Dec 17, 2020 at 9:01
• @ProfRob I'd say this is still an open question. This is why there is ongoing research. Apart from that I do not see the necessity of this comment, except maybe that "hypothetical" would be preferred over "virtual" as the latter has a well defined meaning within QED and high energy physics. Commented Jan 11, 2021 at 7:38