## 5 Results of the Retarded Integrals for the Anomalous Longitudinal Field. Discussion  
### Dependence on the Distance  
The double-retarded integrals, computed numerically as described in the previous sections, give a longitudinal (i.e., radial) component of the electric field of the order of \(10^{2}\) to \(10^{3}\) times greater than the standard transverse component. The computation is done along a radial line forming an angle \(\theta=45^{\circ}\) with respect to the oscillation axis of the dipole. Initially, we take the oscillation frequency equal to \(10^{10}\) Hz and the field is computed at the instant \(t=0\). In order to check that the two field components \(E_{T}\), \(E_{L}\) oscillate in the wave zone with a wavelength \(\lambda=3\) cm, corresponding to the chosen frequency, we have computed these components at distance \(r=10\) cm and have then increased the distance in steps of \(\lambda/4\). The results (Table 1) actually show an oscillating behavior, as expected, within the uncertainties. All values in the table are normalized to \(E_{T}^{c,rms}(10)\), which is the r.m.s. value of the transverse field generated at distance \(r=10\) cm by a standard \"completed\" oscillating dipole, as described in Section 4. The values of \(E_{T}\) obtained for the anomalous source are of the same magnitude order as those of the standard source and are not reported in the table.  
With a further increase in the distance we then pass to \(r=25\) cm (\(10\) cm \(+5\lambda\)) and \(r=40\) cm (\(10\) cm \(+10\lambda\)). Like for \(r=10\) cm, we find values of \(E_{L}\) close to the maxima of the oscillation but the oscillation amplitude appears to have increased: we have, respectively:  
\[\frac{E_{L}(25)}{E_{T}^{c,rms}(10)}\simeq 2.2\times 10^{3};\qquad\qquad\frac{E_{L }(40)}{E_{T}^{c,rms}(10)}\simeq 3.4\times 10^{3} \tag{64}\]  
The raw data (non normalized) of the contributions to \(E_{L}\) coming from the various integration regions (Table 1 of the Appendix A) show that the secondary charge which generates the large values of \(E_{L}\) at \(r=25\) cm and \(r=40\) cm is located farther away from the dipole, in comparison to the secondary charge generating \(E_{L}\) at \(r=10\) cm. In other words, as we move farther away from the dipole, the longitudinal field increases because it is generated by a larger portion of the \"cloud\" of secondary charge. It is not easy, however, to understand intuitively exactly how the different portions of the cloud contribute to the field, because we are not in a stationary state but everything oscillates at a high frequency, including the charge cloud itself, and the phase \(\Omega=\omega(t-k|\mathbf{x}-\mathbf{y}|-k|\mathbf{y}-\mathbf{z}|)\) in the integrals (19), (20) produces double-retarded interference effects.  
From the present data, it is not possible to assess the behavior of the longitudinal field at greater distances, because the uncertainties in the integrals in the regions with distance above approx. 100 cm are too large. We expect, of course, an eventual decrease of \(E_{L}\).  
Notice that while the \(E_{T}\) component of a standard e.m. radiation field must decrease steadily at \(1/r\) in order to maintain the Poynting flux constant, the \(E_{L}\) component does not contribute to this flux. However, it is not yet clear, in our opinion, what the correct expressions for the e.m. densities of energy and momentum are in the extended Aharonov-Bohm electrodynamics, even though this issue has been addressed in some of the cited works.