-\frac{1}{c^2}\frac{\partial \textbf{E}}{\partial t} =\mu_0 \textbf{J}.\end{aligned}$$ We note that the divergence of (14) together with (13) yield (1) back. Clearly, we have inferred other equivalent expressions for $\textbf{J}$ and $\rho$, namely, $\textbf{J}=\nabla\times \textbf{B}/\mu_0 -\epsilon_0\partial \textbf{E}/\partial t $ and $\rho=\epsilon_0\nabla\cdot \textbf{E}$, which satisfy the continuity equation (1). Of course, (13) and (14) must be completed with other two equations that specify the quantities $\nabla\cdot \textbf{B}$ and $\nabla\times \textbf{E}$ as dictated by the Helmholtz theorem [@22]. These other equations are not difficult to find. We quickly note that the fields $\textbf{E}$ and $\textbf{B}$ given in (12) imply the other two field equations $$\begin{aligned}
\nabla\cdot \textbf{B}=0, \\
\nabla\times \textbf{E} +\frac{\partial \textbf{B}}{\partial t} =0.\end{aligned}$$ The set formed by the first order equations (13)-(16) is equivalent to the set formed by the second order equations (8) together with the equation (6). The set of equations (13)-(16) is uniquely determined whenever we adopt boundary conditions for the fields $\textbf{E}$ and $\textbf{B}$ that are consistent with those of the potentials $\textbf{A}$ and $\Phi$.

In order to find the significance of the fields $\textbf{E}$ and $\textbf{B}$ we use (5) and (12) and obtain the retarded solutions of (13)-(16), $$\begin{aligned}
\textbf{E}=-\nabla \!\int\! \frac{[\rho]}{4\pi\epsilon_0R}d^3x' -\frac{\partial}{\partial t}\!\int\!\!\frac{[\textbf{J}]}{4\pi\epsilon_0 c^2 R}d^3x',\\
\textbf{B}=\nabla\times\!\int\!\!\frac{[\textbf{J}]}{4\pi\epsilon_0 c^2 R}d^3x'.\end{aligned}$$ 