# 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.