# A Charged Relativistic Engine Based on a Permanent Magnet
## 4 Momentum Generated  
[PERSON]'s second law dictates that a system with a nonzero total force must have a modification of its cumulative linear momentum \(\vec{P}(t)\):  
\[\vec{F}_{T}^{[2]}=\frac{d\vec{P}}{dt} \tag{73}\]  
Assuming that \(\vec{P}(-\infty)=0\) and that there are null current or charge densities at \(t=-\infty\), we deduce from Equation (72) that  
\[\vec{P}(t)=\frac{\mu_{0}}{4\pi}\int\int d^{3}x_{1}d^{3}x_{2}\,\left[\frac{1}{ 2}(\rho_{2}\partial_{t}\rho_{1}-\rho_{1}\partial_{t}\rho_{2})\hat{R}-(\rho_{1 }\vec{J}_{2}+\rho_{2}\vec{J}_{1})R^{-1}\right] \tag{74}\]  
Comparing the momentum expression Equation (74) with the momentum gained by a non-charged relativistic motor described in Equation (68) of [17]  
\[\vec{P}_{mech}\cong\frac{\mu_{0}}{8\pi}\partial_{t}I_{1}(t)I_{2}(\frac{h}{c}) ^{2}\vec{K}_{122},\qquad\vec{K}_{122}=-\frac{1}{h^{2}}\oint\oint\hat{R}(d \vec{l}_{2}\cdot d\vec{l}_{1}) \tag{75}\]  
In the equation provided, \(h\) represents a typical scale of the system, while \(I_{1}(t)\) and \(I_{2}\) denote the currents flowing through two current loops, and \(d\vec{l}_{1},d\vec{l}_{2}\) signifies the line elements of the current loops. Several notable distinctions are apparent. Firstly, in the case of an uncharged motor, there is a factor of \(\left(\frac{h}{c}\right)^{2}\) present. Given that for any practical system, the scale \(h\) is typically of the order of one, this implies that the charged relativistic motor is substantially stronger than the uncharged motor by a factor of \(c^{2}\sim 10^{17}\), which is a significant difference. Secondly, it is observed that for the uncharged motor, the current must be consistently increased to maintain momentum in the same direction. However, this is not sustainable indefinitely. As a result, the uncharged motor functions akin to a piston engine, executing periodic motion backward and forward, and can only produce forward motion by interacting with an external system, such as the road. Conversely, for the charged relativistic motor, non-vanishing momentum is obtained even for stationary charge and current densities.  
\[\vec{P}(t)=-\frac{\mu_{0}}{4\pi}\int\int d^{3}x_{1}d^{3}x_{2}\,\,(\rho_{1}\vec {J}_{2}+\rho_{2}\vec{J}_{1})R^{-1} \tag{76}\]  
Therefore, the charged relativistic motor has the capability to generate forward momentum without necessitating an interaction with any external system apart from the electromagnetic field. This expression can be somewhat streamlined by employing the potential provided in Equation (13), resulting in the following:  
\[\vec{P}(t)=-\frac{1}{c^{2}}\left[\int d^{3}x_{2}\,\,\Phi_{1}\vec{J}_{2}+\int d ^{3}x_{1}\,\,\Phi_{2}\vec{J}_{1}\right] \tag{77}\]  
in which we are reminded that \(\frac{\mu_{0}}{4\pi k}=\mu_{0}e_{0}=\frac{1}{c^{2}}\). Another important point is that in a charged relativistic motor we do not require both subsystems to be charged, in fact, provided that \(\rho_{2}=0\):  
\[\vec{P}(t)=-\frac{\mu_{0}}{4\pi}\int\int d^{3}x_{1}d^{3}x_{2}\,\,\rho_{1}\vec{J }_{2}R^{-1}=-\frac{1}{c^{2}}\int d^{3}x_{2}\,\,\Phi_{1}\vec{J}_{2} \tag{78}\]  
Depending on a non-vanishing current density \(\vec{J}_{2}\), we emphasize that as in [17] the forward momentum gained by the material system will be balanced by a backward momentum earned by the electromagnetic field. We shall now direct our attention to the challenges of developing a charged relativistic motor based on a permanent magnet.