# A Charged Relativistic Engine Based on a Permanent Magnet
## 3 Retardation Effects  
Consider the charge density \(\rho(\vec{x}^{\prime},t_{ret})=\rho(\vec{x}^{\prime},t-\frac{R}{c})\), if \(\frac{R}{c}\) is small but not null we can derive a Taylor series around \(t\):  
\[\rho(\vec{x}^{\prime},t_{ret})=\rho(\vec{x}^{\prime},t-\frac{R}{c})=\sum_{n=0} ^{\infty}\frac{\partial_{t}^{n}\rho(\vec{x}^{\prime},t)}{n!}(-\frac{R}{c})^{n}. \tag{33}\]  
In the above, \(\partial_{t}^{n}\rho(\vec{x}^{\prime},t)\) represents the \(n\)th-order partial time derivative of \(\rho(\vec{x}^{\prime},t)\). A similar expansion for the current density gives the following expression:\[\vec{f}(\vec{x}^{\prime},t_{ret})=\vec{f}(\vec{x}^{\prime},t-\frac{R}{c})=\sum_{n=0 }^{\infty}\frac{\partial_{t}^{n}\vec{f}(\vec{x}^{\prime},t)}{n!}(-\frac{R}{c})^ {n}. \tag{34}\]  
In the above, \(\partial_{t}^{n}\vec{f}(\vec{x}^{\prime},t)\) represents the \(n\)th order partial time derivative of \(\vec{f}(\vec{x}^{\prime},t)\). These expansions are valid only within a certain interval around \(t\), which depends on the functions involved. This interval is defined by the convergence radius \(T_{max}\), which may vary for each function. Therefore, Equations (33) and (34) are valid only within the domain \([t-T_{max},t+T_{max}]\). As we expand in the delay time \(\frac{\mathbb{E}}{c}\), the expansion remains valid only within a limited range. \[R<R_{max}\equiv c\ T_{max}. \tag{35}\]  
This implies that we are essentially working with a near-field approximation. However, since \(c\) (the speed of light) is significantly large, \(R_{max}\) will be quite extensive for most systems. Now, by substituting Equation (33) into Equation (13), we obtain the following:  
\[\Phi_{1}(\vec{x}_{2}) = k\int d^{3}x_{1}\frac{\rho_{1}(t_{ret})}{R}=k\sum_{n=0}^{\infty }\frac{1}{n!}\int d^{3}x_{1}\partial_{t}^{n}\rho_{1}(\vec{x}^{\prime},t)\frac {1}{R}(-\frac{R}{c})^{n} \tag{36}\] \[= k\sum_{n=0}^{\infty}\frac{1}{n!}\biggl{(}-\frac{1}{c}\biggr{)}^ {n}\int d^{3}x_{1}\partial_{t}^{n}\rho_{1}(\vec{x}_{1},t)R^{n-1}\]  
Similarly, by substituting Equation (34) into Equation (14), we arrive at the following expression:  
\[\vec{A}_{1}(\vec{x}_{2}) = \frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{f}_{1}(t_{ret})}{R }=\frac{\mu_{0}}{4\pi}\sum_{n=0}^{\infty}\frac{1}{n!}\int d^{3}x_{1}\partial_ {t}^{n}\vec{f}_{1}(\vec{x}^{\prime},t)\frac{1}{R}(-\frac{R}{c})^{n} \tag{37}\] \[= \frac{\mu_{0}}{4\pi}\sum_{n=0}^{\infty}\frac{1}{n!}\biggl{(}- \frac{1}{c}\biggr{)}^{n}\int d^{3}x_{1}\partial_{t}^{n}\vec{f}_{1}(\vec{x}_{1 },t)R^{n-1}\]  
As \(\frac{\mu_{0}}{4\pi}=\frac{\mu_{0}e_{0}}{4\pi e_{0}}=\frac{k}{c^{2}}\) it follows that  
\[\vec{A}_{1}(\vec{x}_{2}) = k\sum_{n=0}^{\infty}\frac{1}{n!}\biggl{(}-\frac{1}{c}\biggr{)}^{ n+2}\int d^{3}x_{1}\partial_{t}^{n}\vec{f}_{1}(\vec{x}_{1},t)R^{n-1} \tag{38}\] \[= k\sum_{n=2}^{\infty}\frac{1}{(n-2)!}\biggl{(}-\frac{1}{c} \biggr{)}^{n}\int d^{3}x_{1}\partial_{t}^{n-2}\vec{f}_{1}(\vec{x}_{1},t)R^{n-3}\]  
We are now ready to calculate the electric and magnetic fields using the expansions provided in Equations (36) and (38). Before proceeding, let us introduce the following notation: let \(G^{[n]}\) represent the contribution of order \(\frac{1}{c^{2}}\) to the quantity \(G\).