# Why don't we talk about the E & H fields instead of E & B fields?

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• deuteron
deuteron
TL;DR Summary
Mathematically, E & H fields are parallel to each other, then why do we take E & B for electromagnetic waves?
We have the following constitutive relations:
$$\vec D= \epsilon_0 \vec E +\vec P$$
$$\vec B=\mu_0\vec H + \vec M$$

And Maxwell's equations are:
$$\nabla\cdot\vec D = \rho$$
$$\nabla\cdot \vec B=0$$
$$\nabla\times\vec E=-\frac{\partial\vec B}{\partial t}$$
$$\nabla\times\vec H=\vec j +\frac{\partial\vec D}{\partial t}$$

then why do every book (e.g.: Jackson, Griffith's) mention ##E## and ##B## fields when talking about electromagnetic waves and not the ##E## and ##H## waves?

As far as I remember Griffiths disscusses this issue.

The formula ## B=\mu_o H+M ## comes from the pole theory of magnetostatics. The ## H ## has two contributors to it for sources=magnetic poles, where magnetic pole density ## \rho_m=-\nabla \cdot M ##, and currents in conductors. The ## H ## is something of a mathematical construction though, and does not represent an actual field. I think the same thing can be said for ## D ##. It seems somewhat coincidental that the two formulas are analogous to each other, but ## E ##, ##P##, ## B ##, and ## M ## are the physical observables.

deuteron and PhDeezNutz
Why don't we talk about the E & H fields instead of E & B fields?

We do. In fact, when looking at magnetic properties of steels, one of the most important aspects is the B vs H curve.

deuteron said:
TL;DR Summary: Mathematically, E & H fields are parallel to each other, then why do we take E & B for electromagnetic waves?
in what way are E and H parallel to each other? At least in simple media they are perpendicular.
deuteron said:
then why do every book (e.g.: Jackson, Griffith's) mention ##E## and ##B## fields when talking about electromagnetic waves and not the ##E## and ##H## waves?
That does not describe every book. For example, Field and Wave Electromagnetics by Cheng. I do think engineering-oriented texts are probably more likely to use E and H then physics texts. As an EE I usually use E and H unless there is a good reason not to.

Jason

deuteron said:
TL;DR Summary: Mathematically, E & H fields are parallel to each other, then why do we take E & B for electromagnetic waves?

We have the following constitutive relations:
$$\vec D= \epsilon_0 \vec E +\vec P$$
$$\vec B=\mu_0\vec H + \vec M$$

And Maxwell's equations are:
$$\nabla\cdot\vec D = \rho$$
$$\nabla\cdot \vec B=0$$
$$\nabla\times\vec E=-\frac{\partial\vec B}{\partial t}$$
$$\nabla\times\vec H=\vec j +\frac{\partial\vec D}{\partial t}$$

then why do every book (e.g.: Jackson, Griffith's) mention ##E## and ##B## fields when talking about electromagnetic waves and not the ##E## and ##H## waves?
That's because classical electromagnetism is a relativistic theory and in contradistinction to the physicists of the 19th century today we know so thanks of Einstein and particularly also Minkowski. It becomes very clear that the electromagnetic field in vacuum is defined by the vectors ##\vec{E}## and ##\vec{B}##, which together are the components of the antisymmetric field-strength tensor ##F_{\mu \nu}## in Minkowski space. Arguing with simple classical models of charged matter it becomes then clear that in the same way ##\vec{D}## and ##\vec{H}## belong together forming another antisymmetric four-tensor, ##H_{\mu \nu}##.

The trouble is that historically the physicists rather indeed grouped together ##\vec{E}## and ##\vec{H}## and took ##\vec{H}## as "the magnetic field" instead of ##\vec{B}##, which we now understand to be "the magnetic field". This lead to the idiosyncratic definition of ##\epsilon## and ##\mu##, i.e., writing ##\vec{D}=\epsilon \vec{E}## (which of course is just a free choice of definition) but then ##\vec{B}=\mu \vec{H}## instead of something like ##\vec{H}=\mu' \vec{B}##.

The Lorentz force is ##{\bf F} =q{\bf E+v\times B}##.
##{\bf D}## and ##{\bf H}## are mathematical constructs to help in finding ##\bf E## and ##\bf B##.

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