# [Physics] Do Maxwell’s equations imply that still charges produce electrostatic fields and no magnetic fields

electromagnetismelectrostatics

Suppose we have a charge distribution $\rho$, whose current density is J$=\vec{0}$ everywhere; the continuity equation implies $\frac{\partial \rho}{\partial t}=0$, i.e., the charges don't move and the density is always the same. We'd expect such a distribution to produce a static electric field and no magnetic field. If we plug our variables into Maxwell's equations we get $$\nabla\cdot \textbf{E}=4\pi \rho$$ $$\nabla \cdot \textbf{B}=0$$ $$\nabla \times \textbf E= -\frac{1}{c}\frac{\partial \textbf{B}}{\partial t}$$ $$\nabla \times \textbf B= \frac{1}{c}\frac{\partial \textbf{E}}{\partial t}$$

But how does one go from this to $\bf B=0$ and $\frac{\partial \bf E}{\partial t}=0$?

You can see the consistency of the static fields by setting the time derivatives in your equations to zero. Then there is only a static divergence source for $$\vec{E}$$, meaning just an electrostatic field. However, it is possible to add the time-dependent fields of one or more propagating waves in top of that.