# Maxwell equations in absence of magnetic field

I think you are looking at this the right way but it is probably easier to think in terms of fields than potentials. Plugging $\newcommand{b}{\mathbf{B}}\renewcommand{e}{\mathbf{E}}\renewcommand{ed}{\dot{\e}}\newcommand{j}{\mathbf{j}}\renewcommand{z}{\mathbf{0}} \b=\z$ into$ \nabla \times \b = \ed + \j$ we get $\ed=-\j$. Then $\e = \e_0+\int_{t_0}^t -\j dt'$. We can now check if this definition of $\e$, together with $\b=\z$, satisfies Maxwell's equations. The ones concerning $\b$ are satisfied by construction. Checking Gauss's law, we find $$\nabla \cdot \e = \nabla \cdot \e_0 + \int_{t_0}^t -\nabla \cdot \j\, dt'=\rho_0 + \int_{t_0}^t \dot{\rho}\, dt'=\rho.$$

So Gauss's law checks out.

Now lets check the last equation. $$\z=-\dot{\b}=\nabla \times \e = \nabla \times \e_0 + \int_{t_0}^t -\nabla \times \j\, dt'.$$ If the rightmost side is to be zero for all $t$, then we must have that $\nabla \times \e_0=\z$ and then for all $t$, $\nabla \times \j=0$. The first equation tells us that $\e_0$ must be conservative, and the second tells us that $\j$ must be irrotational for all time.

In summary, we have found that $\b$ is zero then it is necessary to have $\ed = -\j$, so that $\e=-\int \j\, dt$, and then we found it is also necessary for $\j$ to be irrotational. Moreover, these two conditions are sufficient since you can construct a solution.

So in conclusion you can find a $\b=\z$ solution precisely when $\j$ is irrotational, in which case the solution is $\e=-\int \j\, dt$.

As shown here section 18.2, it is possible to have configurations in which current density is non-zero but the magnetic field is zero. My understanding is that it is perfectly legitimate to have a time-varying Electric field and null magnetic field at all times. The simplest case is *a variable current source $j(r,t)$ eminating radially from a source*. Since $j(r,t)$ has spherical symmetry, $B=0$, however $E(r,t)$ varies in both space and time.

I think you have overlooked that a sentence like

a change in electric field induces a magnetic field and vice-versa.

is true in the vacuum, i.e. it is not valid in a region where non-zero charge density and current exist.