## 2 Oscillating Dipolar Source and Integral Expressions for the Radiation Field  
In many papers on extended Maxwell equations it is noticed that, unlike the standard Maxwell equations, they admit wave solutions with a longitudinal electric component [3; 5; 8; 10]. Some authors cite experimental evidence reportedly showing the existence of electromagnetic waves with non-transverse components [34; 35; 36]. Such evidence is scarce, compared to the immense body of precision measurements and technological applications of transverse electromagnetic waves [37; 38]. This implies, however, that the potential practical interest for such propagation modes is large, in case their existence is confirmed. It is immediately seen how the prediction of longitudinal electromagnetic waves emerges from the extended Maxwell equations. The first Maxwell equation in vacuum states that \(\
abla\cdot\mathbf{E}=0\), so for a plane wave \(\mathbf{E}=\mathbf{E}_{0}e^{i(\mathbf{k}\mathbf{x}-\omega t)}\) (or locally) one obtains the transversality condition \(\mathbf{E}_{0}\cdot\mathbf{k}=0\), where \(\mathbf{k}\) defines the propagation direction of the wave. The first equation of the extended Aharonov-Bohm theory in vacuum is instead  
\[\
abla\cdot\mathbf{E}=-\frac{1}{c}\frac{\partial S}{\partial t} \tag{12}\]  
where \(S\) is a scalar field which satisfies the equation  
\[\frac{1}{c^{2}}\frac{\partial^{2}S}{\partial t^{2}}-\
abla^{2}S=I=\partial_{t }p+\
abla\cdot\mathbf{J} \tag{13}\]  
The \"extra-current\" \(I\) is non zero at the points where the local conservation of charge fails. If the charge is locally conserved everywhere, then the \(S\) field is completely decoupled from matter. In this case, even in the extended theory no longitudinal components should be expected.  
Equation (13) can be solved for \(S\), obtaining the first extended Maxwell equation in vacuum with a non-local source term:  
\[\
abla\cdot\mathbf{E}=-\frac{1}{c}\frac{\partial}{\partial t}\int d^{3}y\frac {I\left(t_{ret.}\mathbf{y}\right)}{|\mathbf{x}-\mathbf{y}|} \tag{14}\]  
This shows that the divergence of \(\mathbf{E}\) in vacuum is equal to a term that we can call \"secondary charge density\" or \"cloud charge\", generated in the surrounding space by the local non-conservation of the \"primary current'.' Therefore, in a wave solution in vacuum the electric field can have a longitudinal component.  
In order to find the concrete predictions of the theory and assess the feasibility of an experimental check, it is necessary to compute exactly the longitudinal electric radiation field \(E_{L}\) generated by an appropriate source, compare its magnitude order with that of the transverse field \(E_{T}\) and make sure that it does not vanish for some reason not apparent from the general form of the equations. Symmetry can play a crucial role here. We have previously proven in [9], for instance, that in the case of a quasi-stationary extra-source \(I\) representing a Josephson weak link with local non-conservation, the anomalous magnetic field generated by \(I\) is zero and there is indeed an observable effect because the corresponding Biot-Savart field is missing. This happens, however, for a source \(I\) with spherical symmetry; otherwise the anomalous field partially replaces the missing Biot-Savart field.