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?