# Time-Dependent Retarded Microwave Electromagnetic Motors
## 2 Engine Optimization  
We investigate what the conditions are for a retarded electromagnetic engine in terms of composition and structure to generate a maximal amount of momentum. To this end, we write the charge density and current density in the form  
\[\rho(\vec{x},t)=\bar{\rho}\sum_{n}\rho^{n}(\vec{x})f^{n}(t) \tag{11}\]  
in which \(\bar{\rho}\) is a constant which has the units of charge density, \(\rho^{n}(\vec{x})\) represents spatial functions, and \(f^{n}(t)\) represents temporal functions, where both functions are dimensionless. Similarly, we have  
\[\vec{f}(\vec{x},t)=\int\sum_{n}\bar{f}^{n}(\vec{x})g^{n}(t) \tag{12}\]  
in which \(\bar{J}\) is a constant which has the units of current density, \(\bar{J}^{n}(\vec{x})\) represents spatial vector functions, and \(g^{n}(t)\) represents temporal functions, where both functions are dimensionless. We shall write the dimensional constants in terms of a generic charge \(Q\), typical length scale \(h\), and typical time scale \(\tau\):  
\[\bar{\rho}\equiv\frac{Q}{h^{3}},\qquad\bar{J}\equiv\bar{\rho}\frac{h}{\tau}= \frac{Q}{h^{2}\tau}. \tag{13}\]  
Next, we shall attribute an expansion of the type given in Equations (11) and (12) to each of the subsystems introduced previously. This is accomplished by adding a subscript with the relevant subsystem number:  
\[\rho_{1}(\vec{x},t)=\bar{\rho}_{1}\sum_{n}\rho_{1}^{n}(\vec{x})f^{n}(t), \qquad\rho_{2}(\vec{x},t)=\bar{\rho}_{2}\sum_{n}\rho_{2}^{n}(\vec{x})f^{n}(t). \tag{14}\]  
\[\vec{f}_{1}(\vec{x},t)=\bar{f}_{1}\sum_{n}\vec{f}_{1}^{n}(\vec{x})g^{n}(t), \qquad\vec{f}_{2}(\vec{x},t)=\bar{f}_{2}\sum_{n}\vec{f}_{2}^{n}(\vec{x})g^{n}( t). \tag{15}\]  
Next, we define the following dimensionless vector constants:  
\[\vec{\Gamma}^{mn}\equiv\vec{\Gamma}^{mn}_{21}\equiv\frac{1}{2h^{6}}\int d^{3 }x_{1}d^{3}x_{2}\rho_{1}^{n}\rho_{2}^{m}\frac{\vec{x}_{1}-\vec{x}_{2}}{|\vec{ x}_{1}-\vec{x}_{2}|}=-\vec{\Gamma}^{nm}_{12} \tag{16}\]  
which depend on the spatial structure of the two charged systems. Similarly, we define  
\[\vec{\Lambda}^{mn}_{21}\equiv\frac{1}{h^{5}}\int d^{3}x_{1}d^{3}x_{2}\frac{ \rho_{2}^{m}\vec{f}_{1}^{n}}{|\vec{x}_{1}-\vec{x}_{2}|},\qquad\vec{\Lambda}^{ mn}_{12}\equiv\frac{1}{h^{5}}\int d^{3}x_{1}d^{3}x_{2}\frac{\rho_{1}^{m} \vec{f}_{2}^{n}}{|\vec{x}_{1}-\vec{x}_{2}|}, \tag{17}\]  
We notice that generically, we have  
\[\vec{\Lambda}^{mn}_{21}\
eq\vec{\Lambda}^{mn}_{12}. \tag{18}\]  
Using the typical time scales of each of the systems \(\tau_{1}\) and \(\tau_{2}\), we define  
\[\vec{\Lambda}^{mn}\equiv\frac{1}{\tau_{1}+\tau_{2}}\Big{(}\tau_{2}\vec{\Lambda }^{mn}_{21}+\tau_{1}\vec{\Lambda}^{mn}_{12}\Big{)},\qquad\bar{\tau}=\frac{\tau_ {1}\tau_{2}}{\tau_{1}+\tau_{2}}. \tag{19}\]  
By plugging Equations (14) and (15) into Equation (10) (Equation (83) in [11]) and using the definitions above, we arrive at the expression  
\[\vec{P}(t)=\frac{\mu_{0}}{4\pi}Q_{1}Q_{2}\sum_{mn}\biggl{[}\vec{\Gamma}^{mn}( f^{m}\partial_{t}f^{n}-f^{n}\partial_{t}f^{m})-\frac{1}{\bar{\tau}}\vec{ \Lambda}^{mn}f^{m}g^{n}\biggr{]}. \tag{20}\]We remind the reader that \(\rho\) and \(\vec{J}\) are not independent, as they are connected through the continuity in Equation (16) from [11]:  
\[\vec{\
abla}\cdot\vec{J}+\partial_{t}\rho=0\Rightarrow\sum_{n}\Bigl{[}\bar{\rho} \rho^{n}(\vec{x})\partial_{t}f^{n}(t)+\vec{J}\vec{\
abla}\cdot\vec{J}^{n}g^{n}( t)\Bigr{]}=0. \tag{21}\]  
By taking into account Equation (13), it follows that  
\[\sum_{n}\Bigl{[}\rho^{n}(\vec{x})\tau\partial_{t}f^{n}(t)+h\vec{\
abla}\cdot \vec{J}^{n}(\vec{x})g^{n}(t)\Bigr{]}=0. \tag{22}\]  
As we can always choose  
\[g^{n}(t)=\tau\partial_{t}f^{n}(t), \tag{23}\]  
then the following Equation must be satisfied:  
\[\sum_{n}\Bigl{[}(\rho^{n}(\vec{x})+h\vec{\
abla}\cdot\vec{J}^{n}(\vec{x}))g^{ n}(t)\Bigr{]}=0. \tag{24}\]  
One possible solution to this is choosing  
\[\rho^{n}(\vec{x})=-h\vec{\
abla}\cdot\vec{J}^{n}(\vec{x}), \tag{25}\]  
for all \(n\). In this case, we can choose \(\vec{J}^{n}\), but \(\rho^{n}\) is dictated.