# A Charged Relativistic Engine Based on a Permanent Magnet
## 2 Force Considerations  
In this section we calculate the force that a macroscopic charged system induces on another charged system, this will be used later to calculate the momentum acquired in a two-system center of mass, eventually leading to a specific description of a relativistic motor based on a permanent magnet. Consider an electric field \(\vec{E}\) and magnetic flux density \(\vec{B}\) that is created by entity 1 and acts upon entity 2. Entity 2, which may be a charged body, contains nuclei and electrons, thus the Lorentz force \(\vec{F}_{21}\) generated by entity 1 and operating on entity 2 is obtained through the following formula:  
\[\vec{F}_{21}=\int d^{3}x_{2}\rho_{i2}(\vec{E}+\vec{v}_{i2}\times\vec{B})+\int d ^{3}x_{2}\rho_{c2}(\vec{E}+\vec{v}_{e2}\times\vec{B}). \tag{1}\]In the equation provided, we perform an integration \(\int d^{3}x_{2}\) across the entirety of the charged body 2. Within this context, \(\rho_{l2}\) represents the ion charge density and \(\rho_{\ell 2}\) denotes the electron charge density. Similarly, \(\vec{v}_{l2}\) and \(\vec{v}_{\ell 2}\), respectively, stand for the ion velocity field and the electron velocity field. The total charge density is the combined sum of the ion charge density and the charge density of free electrons, expressed as follows:  
\[\rho_{2}=\rho_{l2}+\rho_{\ell 2}. \tag{2}\]  
Thus, the electric terms in the above force expression add and we are left with the following:  
\[\vec{F}_{21}=\int d^{3}x_{2}\rho_{2}\vec{E}+\int d^{3}x_{2}\rho_{l2}\vec{v}_{l 2}\times\vec{B}+\int d^{3}x_{2}\rho_{\ell 2}\vec{v}_{\ell 2}\times\vec{B}. \tag{3}\]  
We assume that the ions are at rest we thus have the following: \(\vec{v}_{l2}=0\). This leads to the following expression:  
\[\vec{F}_{21}=\int d^{3}x_{2}\rho_{2}\vec{E}+\int d^{3}x_{2}\rho_{\ell 2}\vec{v}_{ \ell 2}\times\vec{B}. \tag{4}\]  
Introducing the current density \(\vec{J}_{2}=\rho_{\ell 2}\vec{v}_{\ell 2}\), we obtain the following:  
\[\vec{F}_{21}=\int d^{3}x_{2}\Big{(}\rho_{2}\vec{E}+\vec{J}_{2}\times\vec{B} \Big{)}. \tag{5}\]  
Now, let us examine the source that produces the fields. The electric and magnetic fields can be expressed using the vector \(\vec{A}\) and scalar \(\Phi\) potentials as follows [3]:  
\[\vec{E}=-\partial_{t}\vec{A}-\vec{\
abla}\Phi. \tag{6}\]  
\[\vec{B}=\vec{\
abla}\times\vec{A}. \tag{7}\]  
Here, \(\vec{\
abla}\) is the nabla operator of vector analysis, \(t\) represents time and \(\partial_{t}\) denotes a temporal partial derivative. Those expressions satisfy two out of the four Maxwell equations:  
\[\vec{\
abla}\cdot\vec{B}=0,\qquad\vec{\
abla}\times\vec{E}=-\frac{\partial \vec{B}}{\partial t}. \tag{8}\]  
The other Maxwell equations relate the displacement \(\vec{D}\) and magnetic \(\vec{H}\) fields to the free charge \(\rho_{f}\) and current \(\vec{J}_{f}\) densities:  
\[\vec{\
abla}\cdot\vec{D}=\rho_{f},\qquad\vec{\
abla}\times\vec{H}=\vec{J}_{f} +\frac{\partial\vec{D}}{\partial t}. \tag{9}\]  
The displacement and magnetic fields satisfy the relations:  
\[\vec{D}=\epsilon_{0}\vec{E}+\vec{P},\qquad\vec{H}=\frac{1}{\mu_{0}}\vec{B}- \vec{M}. \tag{10}\]  
In the above \(\epsilon_{0}\) is the vacuum susceptibility, and \(\mu_{0}\) is the vacuum permeability. \(\vec{P}\) is the polarization of the material under consideration, and \(\vec{M}\) is the magnetization of the same. Using the definitions  
\[\rho_{2}=\rho_{l2}+\rho_{\ell 2}. \tag{2}\]0  
Using the above definitions we may rewrite Equation (9) as follows:  
\[\rho_{2}=\rho_{l2}+\rho_{\ell 2}. \tag{2}\]1Notice that \(\vec{J}\) may contain a contribution of free current (a coil) and a magnetization current which can be generated by a permanent magnet.