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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 $\overrightarrow{J}$).

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  • $\begingroup$ If you don’t know the currents (J) you can’t know the fields. It is not surprising the equations are underdetermined if you regard J as an unknown. $\endgroup$
    – Ben51
    Feb 10 at 16:55
  • $\begingroup$ @Ben51 No, I have assumed that the source charges $\rho$ and source currents J are known. $\endgroup$ Feb 10 at 16:58
  • $\begingroup$ @Ben51 Oh wait! Yeah, I got it now. Thanks! My bad : ) $\endgroup$ Feb 10 at 17:02
  • $\begingroup$ Related: Do Maxwell's Equations overdetermine the electric and magnetic fields? $\endgroup$
    – Qmechanic
    Feb 10 at 17:22
  • $\begingroup$ If you are going to talk about how charges move in response to an EM field you need some law to relate fields acting on charges to velocities of those charges. This is provided by the Lorentz force and Newton's Second Law (or the Schrodinger equation in QM). Newton's law (resp the Schrodinger equation) is clearly independent of Maxwell's equations and a necessary ingredient to derive something like Ohm's law from first principles (not to mention some statistical arguments). $\endgroup$ Feb 10 at 17:26
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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.

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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.

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