Why can't we directly relate the magnetic field ${\bf B}$ to the electric field ${\bf E}$ using the equation $\nabla \times {\bf B} =\mu_0 \sigma {\bf E}$?
In my understanding, a steady electric field ${\bf E}$ creates a current ${\bf I}$ in a conductor through ${\bf J} = \sigma {\bf E}$, and a current ${\bf I}$ generates a magnetic field ${\bf B}$ via Ampère’s Law. So, is it possible to use the relation $\nabla \times {\bf B} =\mu_0 \sigma {\bf E}$ to connect ${\bf B}$ and ${\bf E}$ directly?
In the scenario where the electric field is static ($\frac{\partial \vec{E}}{\partial t}=0$) and only present inside a conductor where Ohm's law $\vec{J}=\sigma \vec{E}$ holds, then yes you could replace substitute $\vec{J}=\sigma\vec{E}$ into the Maxwell equation $\nabla \times \vec{B}=\mu_0 \vec{J} + \frac{1}{c^2}\frac{\partial \vec{E}}{\partial t}$ and conclude $\nabla \times \vec{B}=\mu_0 \sigma \vec{E}$. This might be a useful step in some problems. However, this logic makes some assumptions about the electric field and the material which aren't always true. On the other hand, Maxwell's equations are always true for classical electrodynamics.
