## 1 Introduction
For this purpose, they express \(\mathbf{J}^{ext}\) in terms of Green functions, generalizing the standard method of [32] to the case of a non-local potential. Other authors ([29] and references) define the extended current in a different way from References [27; 28], and take into account the possibility of adding to it a solenoidal component. The correct definition of the physical current is still an open question, also regarding the dissipation properties of the non-local part: should the latter be interpreted as a \"virtual\" current or as a real current with real dissipation? In this context, a detailed calculation and experimental verification of the predictions of Aharonov-Bohm extended electrodynamics would clearly be of special interest. In this work, we are concerned with the computation of the electromagnetic field generated by the non-local part of the current. This field is independent from any solenoidal component, and therefore the ambiguities mentioned above do not directly affect our results. It turns out that the radiation field generated by an oscillating dipole with a failure in local conservation (the most obvious example, apart from the quasi-static case examined in [9]) has very interesting features: namely, it contains an anomalous longitudinal electrical component with large strength and long range. For the frequency considered (10 GHz) we found that the strength of the longitudinal component at a distance between \(3\lambda\) and \(13\lambda\) is in the order of \(10^{2}\) to \(10^{3}\) times the standard transverse component. This factor must be weighted with a small factor that measures the importance of the non-local current in comparison to the standard current. According to [27], first principles calculations of conventional current density can give errors for current as large as 20% for molecular devices. However, most molecular devices do not carry currents large enough to generate macroscopic fields. An exception could be graphene [33]. Other materials which exhibit macroscopic quantization, large currents, and possibly non-local currents are, as mentioned, cuprate superconductors. The computation of the radiation field is technically very difficult due to the presence of double-retarded integrals and \"secondary sources\" \(\rho^{ext}\), \(\mathbf{J}^{ext}\) extended in space. So we had to resort to a complex integro-dipolar expansion and to long 6-dimensional Monte Carlo integrations, obtaining numerical results for some fixed values of the source parameters, chosen in view of plausible experimental situations. It is likely that in future developments, the finite-elements integration techniques currently used for the standard Maxwell equations can be extended to Aharonov-Bohm electrodynamics, but this extension is far from obvious because the familiar vector-analysis features of the Maxwell equations are strongly affected by the removal of the local charge conservation condition. Therefore any technique based on the usual properties of the divergence of \(\mathbf{E}\) and circuitation of \(\mathbf{B}\) must be reconsidered, and in a first approach, we deemed it safer to use only the retarded integral solutions, which, for the non-local part of the sources, can be written in terms of the potentials as (we set \(k=c^{-1}\)):  
\[\phi^{non-loc}=\frac{1}{4\pi}\int\frac{d^{3}y}{|\mathbf{x}-\mathbf{y}|}\left[- k^{2}\frac{\partial}{\partial t}\int\frac{d^{3}z}{|\mathbf{y}-\mathbf{z}|}I \left(t-k|\mathbf{y}-\mathbf{z}|,\mathbf{z}\right)\right]_{t\to t-k| \mathbf{x}-\mathbf{y}|} \tag{10}\]  
\[\mathbf{J}^{ext}=\mathbf{J}+\mathbf{J}^{non-loc}=\mathbf{J}+\frac{1}{4\pi c} \
abla\int d^{3}y\frac{I(t_{ret},\mathbf{y})}{|\mathbf{x}-\mathbf{y}|} \tag{2}\]0  
In the following, the suffix _non-loc_ will be omitted. The extra-source \(I(t,\mathbf{x})\) is represented by two opposite Gaussian peaks which can have spherical or ellipsoidal symmetry.