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Let $\mathbf{E}(r,t),\mathbf{B}(r,t)$ be two vector fields (in $\mathbb{R}^3$), s.t. they satisfy fot $t=0$ the equations:

  1. $\nabla \cdot \mathbf{B}(r,0)=0.$

  2. $\nabla \cdot \mathbf{E}(r,0)=\frac{\rho(r,0)}{\epsilon_0}.$

The question now is:

What properties have $\mathbf{E}$ and $\mathbf{B}$ got in order to satisfy equation 1 and 2 for all $t>0$?

I think $\mathbf{B}$ must be independent of $t$, i.e. $\mathbf{B}(r,t)=\mathbf{B}(r,0)$ for all $t>0$. But what about $\mathbf{E}$?

Doesn't the answer depend on the function $\rho(r,t)$?

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If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Apr 14 '13 at 8:45

All four of the Maxwell equations, including the divergence ones, $$\nabla\cdot\mathbf{E}=\rho/\epsilon_0\textrm{ and }\nabla\cdot\mathbf{B}=0,$$ are kinematic and dynamical conditions that must be imposed on electric and magnetic fields for them to be physical. That is, Maxwell's equations are the "properties $\mathbf{E}$ and $\mathbf{B}$ must have" to represent the fields we observe in nature.

You might then ask, "well, how do we find fields that satisfy them?" This is indeed a problem and it is the central problem of classical electromagnetism. It boils down to solving a set of (partial differential, linear, coupled) equations, and there are various techniques to do that. Maybe a look in Griffiths will help with that.

And, as Lubos says, you do not need the magnetic field to be time-independent for Maxwell's equations to hold.

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You really need all four Maxwell equations to get an idea of what information is needed to characterize the electric and magnetic fields.

Consider this rearrangement of the four Maxwell equations:

$$\begin{align*} \frac{\partial \mathbf{E}}{\partial t} &= -\frac{\mathbf{j}}{\epsilon_0} + \frac{\nabla \times \mathbf{B}}{\mu_0 \epsilon_0}\\ \frac{\partial \mathbf{B}}{\partial t} &= -\nabla \times \mathbf{E} \\ \nabla \cdot \mathbf{E} - \frac{\rho}{\epsilon_0}&= 0 \\ \nabla \cdot \mathbf{B} &= 0 \end{align*}$$

Any given pair of electric and magnetic fields must obey the last two equations as constraints at all times. The first two equations govern the time evolution of the fields. The only way the time derivative of the magnetic field is zero (and hence, the magnetic field doesn't depend on time) is if the curl of the electric field is zero at all times. If there is no current density either, then neither the electric nor magnetic fields change.

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The property that $E,B$ have to have to obey Maxwell's equations is that they have to obey Maxwell's equations. The OP's idea that one may fomulate this condition or property in a simpler way is completely unfounded (if we could write the equations in a simpler way, we would), much like the opinion that the magnetic field has to be static. It doesn't have to be static. Indeed, in the real world, most of the magnetic fields are time-dependent and they still obey Maxwell's equations. Electromagnetic waves are a clear example of electromagnetic fields that are time-dependent.

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