Missing equations in Maxwell's equations

Question

We have Maxwell's Equations (ignoring permittivity and permeability of free space)

$$ \nabla\cdot E=\rho\;;\;\nabla\times E=-\frac{\partial B}{\partial t} $$ $$ \nabla\cdot B=0\;;\;\nabla\times B=\frac{\partial E}{\partial t}+J $$ with $E$ and $B$ being the electric and magnetic fields, and $\rho$ and $J$ being the charge and current densities respectively.

Intuitively, if I put some negatively charged plate near some free electrons, they will be pushed away. From Maxwell's equations, there is nothing governing the dynamics of the charge and current densities that would describe this behavior.

So we can include the continuity equation $$ \frac{\partial \rho}{\partial t} = \nabla \cdot J$$

I was wondering if there are any additional equations that are missing. I mean, we have $E,B,\rho,$ and $J$, so there should be four differential equations for me to determine each one. However, I count five equations. Are these systems of equations over determined?

Answer 1

The continuity equation can actually be derived from Maxwell's equations: $$\nabla \cdot \mathbf{J} = \nabla \cdot \left( \frac{1}{\mu_0} \nabla \times \mathbf{B} - \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right) \\ = -\epsilon_0 \frac{\partial}{\partial t} (\nabla \cdot \mathbf{E}) \\ = -\frac{\partial \rho}{\partial t}$$ However, the Lorentz force law $\mathbf{F} = q(\mathbf{E}+ \mathbf{v} \times \mathbf{B})$ is an independent equation not contained within Maxwell's equations. Thus, these five equations, comprising Maxwell's equations and the Lorentz force law, govern almost all classical electrodynamics.

Answer 2

The Maxwell equations govern the time evolution of the electric and magnetic fields, not the motion of charged particles. The evolution of $\rho$ and $\mathbf J$ is governed by some generalization of Newton's second law with the Lorentz force equation

$$\mathbf F = q(\mathbf E + \mathbf v \times \mathbf B)$$

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To address your concern about an overdetermined system, $\mathbf E$ and $\mathbf B$ have three components each, so we need six equations to determine them uniquely. The Ampere and Faraday laws are vector equations and provide three each, while the Gauss' laws for $\mathbf E$ and $\mathbf B$ provide another two, for a total of eight.

Naively, the system appears to be overdetermined, but this is not so because these equations contain redundancy. In particular, Ampere's law and Gauss' law for $\mathbf E$ together imply the continuity equation

$$\nabla \cdot \mathbf J = \nabla \cdot \left( \frac{1}{\mu_0}\nabla \times \mathbf B - \epsilon_0 \frac{\partial}{\partial t} \mathbf E\right) = 0 - \epsilon_0\frac{\partial}{\partial t}\nabla \cdot \mathbf E = -\frac{\partial \rho}{\partial t}$$ $$\implies \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf J = 0$$

But this implies (exercise for the reader) that $$\frac{\partial}{\partial t} \left(\nabla \cdot \mathbf E - \frac{\rho}{\epsilon_0}\right) = 0$$ and $$\frac{\partial}{\partial t} (\nabla \cdot \mathbf B) = 0 $$

Meaning that if $\mathbf E$ and $\mathbf B$ obey the Gauss equations at some initial time, they will obey them forever. Therefore, one might say that the two Gauss equations tell you what configurations of $\mathbf E$ and $\mathbf B$ fields are allowed and $t=0$, and the remaining six equations tell you how to evolve them forward in time.

Answer 3

As far as I know, Maxwell's Equations describe everything there is to describe about Electric and Magnetic Fields (except perhaps for boundary conditions). If I'm not wrong, this is a consequence of Helmholtz's Theorem which states that for well-behaved vector fields (fields that fall off at least as $1/r^2$) the divergence and curl is all you require to completely specify the field.

If you specifically want to know how a charge interacts with the Electric and Magnetic fields, then you do need one more equation, the Lorentz Force Law:

$$\vec{F} = q \left(\vec{E} + \vec{v} \times \vec{B}\right).$$

So Maxwell's Equations and the Lorentz Force Law is all you need.

You actually don't need to include the continuity equation as a separate equation, it's (very conveniently!) already a part of it! Taking the divergence of the curl of $\vec{B}$, we can use the result from calculus that says the divergence of the curl is always zero, and so:

\begin{equation*} \begin{aligned} \vec{\nabla}\cdot \left(\vec{\nabla}\times \vec{B}\right) &= \vec{\nabla}\cdot \left(\mu_0 \vec{j} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}\right) \end{aligned}\\ 0 = \vec{\nabla}\cdot\vec{j} + \frac{\partial \rho}{\partial t} \end{equation*}

which is the continuity equation (the equation in your question has the sign wrong). What this means is that charge conservation is part of Maxwell's Equations, you don't need to specify anything more.

Interestingly, Maxwell's Equations are actually 8 equations: two scalar "constraint" equations (the divergences) and two vector "dynamical" equations with three components each (the curls), which need to be solved for 6 quantities: the vectors $\vec{E}$ and $\vec{B}$. So in this sense they are over-specified, but of course, all of them are needed! $\rho$ and $\vec{j}$ are treated as the sources of Electric and Magnetic Fields, they can change from system to system.

The redundancy is actually a lot more, since you could rewrite Maxwell's Equations in terms of the two potentials $\phi$ and $\vec{A}$, which are four parameters. But -- given that we also have Gauge Invariance -- we can actually set one of those parameters to zero, leading to a total of 3 independent parameters in all.