Completeness of Maxwell's equations

Question

If we consider Maxwell's equations in the form:
$\nabla\cdot\overrightarrow{E}=\frac{\rho}{\epsilon_0}$
$\nabla\cdot\overrightarrow{B}=0$
$\nabla\times\overrightarrow{E}=-\frac{\partial\overrightarrow{B}}{\partial t}$
$\nabla\times\overrightarrow{B}=\mu_0\overrightarrow{J}+\mu_0\epsilon_0\frac{\partial\overrightarrow{E}}{\partial t}$
These are 1+1+3+3=8 scalar equations. These however relate the vectors $\overrightarrow{E},\overrightarrow{B} and \overrightarrow{J}$ which have 3$\cdot$3=9 scalar components. From what I have read, this anomaly is fixed by introducing so-called constitutive relations, which in this case can be, say, Ohm's law $\overrightarrow{J}=\sigma\overrightarrow{E}$, thus giving the 9 required equations. But this would imply that Ohm's law is independent from Maxwell's equations. I have read that Maxwell's equations uniquely determine the electromagnetic field, then shouldn't Ohm's law be contained somewhere in Maxwell's equations? I was of the opinion that from Maxwell's equations one could, in principle, derive an expression relating $\overrightarrow{J} and \overrightarrow{E}$ and for simplicity we could then choose only the first-order term, thus obtaining the linear Ohm's law. But this doesn't seem to be the case here. So are laws like Ohm's law not contained in Maxwell's equations, and thus these equations do not uniquely determine $\overrightarrow{E} and \overrightarrow{B}$? (Given, of course, $\rho$ and $\overrightarrow{J}$).

Answer 1

What you wrote as Maxwell's equations are valid for free charges $\rho$, free currents $J=\rho v$, in vacuum and in these there is no Ohm's law involved.

If instead of free charges/currents in vacuum you have macroscopic bulk material, fluid, gas, etc., your version of Maxwell's equations do not describe those. Instead we assume that we have some knowledge of the way the macroscopic currents/charges interact with the macroscopic $E$ and $B$ fields that are macroscopic averages of the microscopic fields. It turns out that then you need four (4), not two macroscopic fields, conventionally denoted as $E,D$ and $B,H$ and the relationships are either derived from microscopic physics (quantum and statistical mechanics + thermodynamics) or measured directly. One such macroscopic relationship is that of Ohm's law but there are others describing the macroscopic behavior of dielectrics $D=D(E)$ or magnetic matter $B=B(H)$, etc.

Answer 2

If I understand what you are saying correctly, $\overrightarrow{J}$ is a known quantity and so you only have $6$ scalar quantities to worry about - not $9$.

This is all slightly complicated by the fact that the equations don't just include $\overrightarrow B$ and $\overrightarrow E$, they also contain their time derivatives - $\frac{\partial \overrightarrow{ E}}{\partial t}$ and $\frac{\partial \overrightarrow{B}}{\partial t}$ - and a bunch of spatial derivatives in the curl and divergence. So, in fact, there are more than simply the $6$ scalar quantities implied by $\overrightarrow E$ and $\overrightarrow B$ - this is a problem involving differential equations, not simply linear algebra.

A useful way to think about what is going on here is that we have 6 differential equations (in the two vector equations involving the curl) and 2 constraint equations (in the two divergence equations). We then wish to uniquely determine the $6$ scalar quantities in $\overrightarrow E$ and $\overrightarrow B$.

Looking at the differential equations we have $6$ equations for $6$ unknowns which is not an issue. There is then only one question left: Given a suitable set of boundary conditions do we have enough information to uniquely determine $\overrightarrow E$ and $\overrightarrow B$ as a function of space and time.

I won't pretend to know how to prove that we do, however, there is a good answer here that explains it.

Answer 3

You may also include the continuity equation, which is a scalar equation that relates $\rho$ (which you already know) with $\vec{J}$: $$\vec{\nabla}\cdot\vec{J}+\partial_t\rho=0$$ Thus, you would add up to 9 equations (8 from Maxwell + the continuity equation) for your 9 scalar components of $\vec{E}$, $\vec{B}$ and $\vec{J}$. In case you didn't hear previously about the continuity equation, it simply relates the amount of charge that goes out of a volume (because of a current) with how much the density changes in that same volume. This is just a consequence of conservation of charge and you can find easy derivations of this formula on the internet.