Electric field due to a time and space varying current density? Let us say I have a current density:
$$\vec J\equiv\vec  J(\vec r, t)$$
This will produce both a time varying magnetic field and a net charge density. Both these effects produce an electric field. If $\vec E_B$ is the electric field considering only the time varying magnetic field and $\vec E_{cd}$ is the electric field considering only the net charge density can we say in general:
$$\vec E=\vec E_B+\vec E_{cd}$$
If so can it be proven and if not why not?
 A: Too long for a comment:
Consider the Maxwell equations:
$$\nabla \cdot {\bf E}=\rho/\epsilon_0 \qquad \nabla \cdot {\bf B}=0$$
$$\nabla\times {\bf E}=-\dfrac{\partial {\bf B}}{\partial t} \qquad\nabla\times {\bf B}=\mu_0 {\bf J}+\dfrac{1}{c^2}\dfrac{\partial {\bf E}}{\partial t}$$
According to Heras (when commenting on a paper by Griffiths and Heald) the displacement current term $\epsilon_0 \frac{\partial {\bf E}}{\partial t}$ can be written in terms of the current density ${\bf J}$ as 
$$\epsilon_0 \frac{\partial {\bf E}}{\partial t} = \frac{{\bf J}}{3} + \frac{1}{4 \pi} \int d^3x'[{\bf F}({\bf J})]$$
where ${\bf F}({\bf J})$ depends upon ${\bf J}, \frac{\partial {\bf J}}{\partial t}, \frac{\partial^2 {\bf J}}{\partial t^2}$ and denotes quantities at the retarded time [I'm assuming it's OK to make use of Helmholtz, since although the rhs of the expression for $\nabla \times {\bf B}$ involves time, it does not involve time derivatives of ${\bf B}$ in which case you'd have to make use of the following generalisation of Helmholtz's Theorem]. Because of this, this means that we know $\nabla \cdot {\bf B}$ and $\nabla \times {\bf B}$. From Helmhotlz's Theorem this means we can write ${\bf B}$ in the form ${\bf B} = \nabla \Phi + \nabla \times {\bf A}$ and $\Phi, {\bf A}$ are determined uniquely. This gives ${\bf B}$ determined by the current density ${\bf J} (and it's time derivatives)$.
This now means we have $\nabla \cdot {\bf E}$ given in terms of the charge density and $\nabla \times {\bf E}$ given in terms of the time derivative of ${\bf B}$ (which is given in terms of the time derivative of the current density ${\bf J}$). It follows that since we know the divergence of ${\bf E}$ and the curl of ${\bf E}$ we can write ${\bf E}$ in terms of the gradient of some scalar function $\Phi'$ and the curl of a vector function ${\bf A'}$ as ${\bf E} = \nabla \Phi'+ \nabla \times {\bf A'}$, and it is seen that $\Phi'$ is related to the charge density while ${\bf A'}$ is related to the (time derivative) of the magnetic field. 
