## 4 Benchmark Values of \(E_{t}\), \(E_{l}\) from a Conserved Source  
The numerical solution of the extended Maxwell equations found through the double-retarded integrals described in the previous Section, the raw results of which (only for \(E_{L}\)) are given in the Appendix A, gives the components of the electric field in CGS units, referred to a source equal to 1 in the same units. From this solution, we can see that \(E_{L}\gg E_{T}\), and this certainly signals that something interesting occurs, compared to the usual propagation of \(E_{T}\), which only occurs in the Maxwell theory with locally conserved sources. The absolute value of the fields, however, provides little information in itself, and we need some benchmark. For this purpose, we shall now compute the field generated at the same position (\(r=10\) cm, \(\theta=45^{\circ}\)) by a standard oscillating dipole with the same frequency and amplitude. By standard, we mean that its current is locally conserved. A textbook formula for this case is  
\[E_{T}=\frac{q\wp\sin\theta}{c^{2}r} \tag{53}\]  
and yields an amplitude \(E_{T}\simeq q\cdot 0.8\times 10^{-7}\) (CGS units), supposing an harmonic oscillation with amplitude \(a=2.5\times 10^{-7}\) cm, \(\omega=2\pi\times 10^{10}\) Hz. Since \(E_{L}\) is of the order of \(q\cdot 10^{-4}\) (see raw data in Table 11 of the Appendix A), this shows that the anomalous longitudinal field \(E_{L}\) of an oscillating dipole with \"full\" strong tunnelling (i.e., one in which _all_ charge oscillates between \(-\mathbf{a}\) and \(\mathbf{a}\) without an intermediate current) is about 2 or 3 orders of magnitude larger than the regular transverse field \(E_{T}\) of a corresponding conserved source. In order to obtain a more precise estimate of the benchmark transverse field, we shall next compute it from the standard solution of the Maxwell equations with a source which is exactly equal to the source (15) \"completed\" with a current that ensures local conservation. This also makes the entire computation self-contained and yields a consistency check for the formalism employed. After writing the time derivative of the charge density \(\rho\) in (15), we set it equal, by definition, to \(-\
abla\mathbf{J}^{c}\) and in this way obtain the conserved current density \(\mathbf{J}^{c}\). It is straightforward to check that from the condition  
\[\frac{\partial\rho^{c}}{\partial t}=\frac{\partial}{\partial t}q\cos(\omega t) \left[\delta^{3}(\mathbf{x}-\mathbf{a})-\delta^{3}(\mathbf{x}+\mathbf{a}) \right]\equiv-\frac{\partial}{\partial x_{3}}J_{3}^{c} \tag{54}\]  
one has  
\[J_{3}^{c}=-q\omega\sin(\omega t)\left[\theta(x_{3}+a)+\theta(-x_{3}+a)-1 \right]\delta(x_{1})\delta(x_{2}) \tag{55}\]  
The standard Maxwell equations in Lorenz gauge in CGS units for the potentials \(\phi^{c}\), \(\mathbf{A}^{c}\) are (the subscript \(c\) stays for \"conserved\")  
\[\frac{1}{c^{2}}\frac{\partial^{2}\phi^{c}}{\partial t^{2}}-\
abla^{2}\phi^{c} =4\pi\rho^{c} \tag{56}\]  
\[\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}^{c}}{\partial t^{2}}-\
abla^{2} \mathbf{A}^{c}=\frac{4\pi}{c}\mathbf{J}^{c} \tag{57}\]  
and their solutions (\(k=c^{-1}\))  
\[\phi^{c}(\mathbf{x},t) =\int d^{3}y\frac{1}{|\mathbf{x}-\mathbf{y}|}\rho^{c}(\mathbf{y},t-k|\mathbf{x}-\mathbf{y}|) \tag{58}\] \[\mathbf{A}^{c}(\mathbf{x},t) =\int d^{3}y\frac{1}{|\mathbf{x}-\mathbf{y}|}\mathbf{J}^{c}( \mathbf{y},t-k|\mathbf{x}-\mathbf{y}|) \tag{59}\]  
The integral for \(\phi^{c}\) gives  
\[\phi^{c}=\frac{q}{|\mathbf{x}-\mathbf{a}|}\cos\left[\omega(t-k|\mathbf{x}- \mathbf{a}|)\right]-\frac{q}{|\mathbf{x}+\mathbf{a}|}\cos\left[\omega(t-k| \mathbf{x}+\mathbf{a}|)\right] \tag{60}\]  
The corresponding contribution to the electric field is obtained from \(-\
abla\phi\).