# Maxwell's equations in vaccuum

I'm trying to study classical electrodynamics on my own but I encounter problems all the time. I would like to ask the following two questions related to Maxwell's equations.

There are two sets of Maxwell's equations.

Maxwell's macroscopic equations:

$$\nabla\cdot\textbf{D}=\varrho$$, $$\nabla\times\textbf{E}=-\frac{\partial\textbf{B}}{\partial t}$$

$$\nabla\cdot\textbf{B}=0$$, $$\nabla\times\textbf{H}=\textbf{J}\texttt{+}\frac{\partial\textbf{D}}{\partial t}$$

Maxwell's equations in vacuum:

$$\nabla\cdot\textbf{E}=\frac{\varrho}{\varepsilon_0}$$, $$\nabla\times\textbf{E}=-\frac{\partial\textbf{B}}{\partial t}$$

$$\nabla\cdot\textbf{B}=0$$, $$c^2\nabla\times\textbf{B}=\frac{1}{\varepsilon_0}\textbf{J}{+}\frac{\partial\textbf{E}}{\partial t}$$

It is not clear to me when I can use the second set. How can it be called Maxwell's equations in vacuum when there is a charge density $$\varrho$$ and a current density J present?

And now to the second question. In isotropic media, the relationship between the electric displacement D and the electric field E is given by D=$$\varepsilon$$E, while the relationship between the magnetic field H and the magnetic induction B reads B=$$\mu$$H. But B is more fundamental than H, so why is the last relation usually written in this reversed manner?

• Surely in vacuum $\rho=0$ and $\vec J=0$ as well. Commented May 16, 2023 at 19:52
• Both sets of equations are "macroscopic". The first set is Maxwell's equations explicitly expanded so as to show the electric and magnetic fields in the context of dielectric media. Commented May 16, 2023 at 21:31
• This may help: physics.stackexchange.com/q/300741 Commented May 16, 2023 at 21:34

Vacuum is anywhere else but where $$\rho$$ and $$\mathbf J$$ are, and in vacuum there is no physical difference between $$\mathbf B$$ and $$\mathbf H$$ or between $$\mathbf E$$ and $$\mathbf D$$. Also, forget which field is more fundamental than the other. In a vacuum they are the same except for a dimension dependent multiplying constant factor (in Gaussian units the factor is $$1$$) and inside ponderable matter neither is more important than the other, in fact, all four fields (actually six if you include the charge and current) are mathematical averages taken over time and space of the two vacuum fields residing between the particles.

The presence of the fundamental constants ε0 and μ0 in the “vacuum” equations is a historical accident, arising from the traditionally accepted choice of fundamental units in which physical quantities are expressd. Those are the units of mass, length, time and charge, which are freely chosen by physicists, for convenience. Relativists often simplify their equations by choosing to define the unit of length so that the speed of light in a vacuum is c = 1 (astronomers have long done this by choosing a “year” as the unit of time and a “light year” as the unit of length). Similarly, instead of choosing "one Coulomb” of the unit of charge it can be chosen so that ε0 = μ0 = c = 1. It is then readily apparent that in the vacuum equations B and H are the same thing, as are D and E.

The second set of equations are not the vacuum Maxwell´s Equations. The vacuum is just when there are no sources, so $$\rho=0$$ and $$\vec{J}=0$$. The first set of equations are the Maxwell´s Equations in a dielectric media or material, and the other, when there are no such media.

With respect to your second question, i´ve been always told that $$\vec{H}$$ is more fundamental, so you define it in that way, just to get the same relation between $$\vec{P}$$ and $$\vec{E}$$ as for $$\vec{H}$$ and $$\vec{M}$$.

• Your first paragraph is not true. The second set of equations are the Maxwell Equations in every situation. Commented May 16, 2023 at 21:29
1. The second set of equations are the Maxwell equations. They are valid everywhere and anywhere in every single situation. The Maxwell equations in vacuum are, as you say, what you obtain when you set $$\rho = 0$$ and $$J = 0$$. You obtain the first set of equations from the second set of equations by observing that $$\rho = \rho_f + \rho_b$$ and $$J = J_f + J_b$$, i.e. splitting the sources into free and bound sources. Then, a few definitions later you obtain the first set of equations.

2. $$H$$ is more experimentally useful, so it takes precedence. You can define it the other way if you would like. Nothing physically changes.

There are two complementary concepts underlying Maxwell's equations. One is the presence of the sources in the field equations. They are the electric density $$\rho$$, and the current density $${\bf j}$$. Of course, wherever they are different from zero, we can be sure that there is something more than just the fields.

The other concept is the presence, in general, of a response of the material to the presence of the fields (in many cases, amenable to the polarization of the material).

The so-called Maxwell's equations in the vacuum refer to an ideal situation where we can (or want to) exclude any effect related to the response of the medium. Even if a real vacuum would be compatible only with $$\rho=0$$ and $${\bf j}=0$$, we use the term Maxwell's equations in the vacuum for the equations with non-zero sources whenever we can ignore any induced charge or current density.

About the second question, there is no real physical base for considering some fields more fundamental. Even ignoring the issue of the fundamental fields, one could be puzzled by the asymmetry between the way the dielectric constant and the magnetic permeability appear in the equation connecting $$\bf E$$ and $$\bf D$$ on the one side, and $$\bf H$$ and $$\bf B$$ on the other side, even in the vacuum. However, such an asymmetry is there for historical reasons and does not play any fundamental role. It could be easily removed by introducing a different constant like $$k=\frac{1}{\mu_0}$$. However, history plays a role in such a well-established area of Physics.

Jackson says (page 193, third edition): "We emphasize that the fundamental fields are E and B". "The derived fields, D and H, are introduced as a matter of convenience, to permit us to take into account in an average way the contributions to $$\varrho$$ and J of the atomic charges and currents."

• That is more of an opinion and historical tradition than anything else; in the ancient days of Maxwell everybody thought $H$ was fundamental not $B$. There is an advantage calling $B$ fundamental because if you average the microscopic field $\mathbf b$ that satisfies $div \mathbf b =0$ the averaged field will still satisfy $div \mathbf B =0$ where $Average[\mathbf b] = \mathbf B$. But in reality both macroscopic fields are equally fictitious or equally real and (non) fundamental in matter: $H$ is measured as a force while $B$ as a torque. Now, which is more fundamental? Commented May 17, 2023 at 1:30
• Let me add that in permeable matter, as Kelvin said, you can measure the $H$ field in a needle cavity along the local magnetic polarization while $B$ is a flat disk cavity normal to the local polarization; he used the term crevice not cavity. Since in vacuum there is no problem how to define "the magnetic field" Kelvin's crevice idea works because H is continuous tangentially and B is continuous normal to the surface of discontinuity. Neither is better than the other, they are equally important so that engineers prefer $H$ as more practical but physicists nowadays prefer $B$. Commented May 17, 2023 at 2:08