If I understand correctly, the electrostatic approximation assumes that all charges are stationary (i.e the charge density is constant in time, and current density is zero). The magnetostatic approximation assumes the current density is constant in time. **Are these the correct definitions?**

I have also seen electrostatics being characterized by an electric field which is constant in time. **Is this supposed to be some obvious consequence of the above definition?**

Looking at Maxwell's equations, the divergence of $E$ is fixed, but the curl of $E$ might vary in time if $B$ varies in time. Even in empty space (an electrostatic situation by defualt) we have the wave solution to Maxwell's equations, in which both $E$ and $B$ vary in time.

So I'm guessing most authors are just sloppy when it comes to nomenclature, and when they say electrostatics, they really mean **quasi-electrostatics**, i.e electrostatics in the quasistatic approximation where the deplacement current $j_D$ is neglected from Ampere's law.

In quasi-electrostatics, we definitely can't have the wave solution in empty space anymore. But is it still possible that the $B$ field varies in time? The magnetic field equations reduce to $curl(B)=0$ and $div(B)=0$. **How do we know it's not possible to find some time-varying $B$ field which always has zero curl and zero divergence?**

## Best Answer

This are Maxwell Equations: $$ \begin{align} \nabla\cdot\mathbf E = \frac{\rho}{\epsilon_0}, \quad\quad &\nabla\cdot\mathbf B = 0\\ \nabla\times\mathbf E = -\frac{\partial\mathbf B}{\partial t}, \quad\quad &\nabla\times\mathbf B = \mu_0\left(\mathbf J + \epsilon_0\frac{\partial\mathbf E}{\partial t}\right)\\ \end{align} $$

In the electrostatic case: $\nabla\times\mathbf E = 0$.

In the magnetostatic case: $\nabla\times\mathbf B = \mu_0\mathbf J$.

So, In the static cases, we do not have the other field changing. That is, in the electrostatic case, magnetic fields cannot change. In the magnetostatic case, eletric fields cannot change.

An electric or magnetic field is static when it is uncoupled. Not to be confused with spatially-constant field, or non-time-varying field. On normal conditions (Maxwell Equations), the sources which generates the electric field are $\mathbf B$ and $\rho$. That is, the electric field depends on what the magnetic field is doing, that is, they are coupled.The whole point of the static approximations is to neglect field coupling. In the electrostatic case, a magnetic field can never generate an electric field. Meaning, the only sources of electric fields are $\rho$. The same is valid for the magnetostatic case: Only $\mathbf J$ generates $\mathbf B$, that is, $\mathbf E$ cannot produce $\mathbf B$.