# Why aren't Maxwell's equations overdetermined? [duplicate]

Consider the four differential equations in the table given on wikipedia here and assume there is no charge distribution at any point in time, and thus also no current. If there is no charge, then the four equations reduce to the following:

$$\nabla\cdot E = 0$$
$$\nabla\cdot B = 0$$
$$\frac{\partial B}{\partial t} = -\nabla\times E$$
$$\frac{\partial E}{\partial t} = c^2\nabla\times B$$

The last two equations tell us how both the magnetic and electric fields change over time respectively, thus given some initial magnetic and electric fields, one should be able to determine any future state of both field. This makes the first two equations seem redundant to me and thus the system seems over determined. However they are clearly necessary, so I must be missing something. Are the first two equations simply initial conditions?

The first two Maxwell equations describe static electric and magnetic fields. From these equations we learn the geometric properties of such fields, and the nature of the lines of force these fields produce. The first one (when there is charge present)

$$\nabla \cdot \vec E = \rho$$

leads us to determine the form of the electric field for any kind of charge distribution. This is extremely important for the study of electrostatics. Furthermore, this equation can be used to derive the Poisson equation,

$$\nabla^2 V = -\rho$$

which allows us to determine the electrostatic potential $$V$$ for various charge distributions. We can also use the above Maxwell equation to derive Coulomb’s law (though this law is not necessarily a direct result of this equation only). The Poisson equation is also a very powerful tool in the study of electrostatics. This equation also has powerful applications in semiconductor physics.

The second equation you mention,

$$\nabla \cdot \vec B = 0$$

tells us something very important, which is that magnetic monopoles do not exist. The mathematical implication of this equation is that there must exist magnetic vector potential $$\vec A$$ where

$$\vec B = \nabla \times \vec A$$

This is a powerful mathematical result. This magnetic vector potential is ubiquitous in classical electrodynamics and quantum electrodynamics.