# Time-Dependent Retarded Microwave Electromagnetic Motors
## Appendix A Force and Momentum Formulae
By calculating the magnetic field using Equation (104), we obtain  
\[\vec{F}_{21}=\int d^{3}x_{2}\Big{(}\rho_{2}\vec{E}+\vec{J}_{2}\times\vec{B} \Big{)}. \tag{14}\]7 \[\vec{F}_{21}=\int d^{3}x_{2}\Big{(}\rho_{2}\vec{E}+\vec{J}_{2}\times\vec{B} \Big{)}. \tag{14}\]8 \[\vec{F}_{21}=\int d^{3}x_{2}\Big{(}\rho_{2}\vec{E}+\vec{J}_{2}\times\vec{B} \Big{)}. \tag{14}\]9  
However, we have  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]0and also  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]1  
Thus, we may write  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]2  
or more specifically  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]3  
We will now use the expressions for the electric and magnetic fields to derive the \(n^{\text{th}}\)-order contribution to the force by utilizing Equation (108):  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]4  
For the zeroth order, we conclude that  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]5  
By inserting Equation (107), we derive [PERSON]'s force:  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]6  
This (quasi)-static force adheres to [PERSON]'s third law:  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]7  
The force which is of the first order in \(\frac{1}{c}\) is zero, which implies that  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]8  
To calculate the second-order force term, we start by separating the force expressed in Equation (114) into its electric and magnetic components:  
\[\Phi(\vec{x}_{2})=k\int d^{3}x_{1}\frac{\rho_{1}(\vec{x}_{1},t_{ret})}{R}. \tag{15}\]9 \[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]0 \[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]1  
By inserting Equation (112) into Equation (119), we obtain the expression for the electric force:  
\[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]2  
The magnetic force is derived by substituting Equation (113) into Equation (119):  
\[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]3 \[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]4Notice that  
\[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]5 (A54)  
which results in  
\[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]6 (A55)  
We can inspect the integral  
\[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]7 (A56)  
By applying [PERSON]'s theorem and the continuity equation, we arrive at  
\[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]8 (A57)  
The surface surrounds the volume, and if the volume encompasses all of the space, then the surface will extend to infinity. If the currents at infinity are zero, then  
\[\vec{A}(\vec{x}_{2})=\frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\frac{\vec{J}_{1}(\vec {x}_{1},t_{ret})}{R}. \tag{16}\]9 (A58)  
By inserting Equation (A58) into Equation (A55), we derive  
\[\vec{B}(\vec{x}_{2})=\vec{\
abla}_{\vec{x}_{2}}\times\vec{A}(\vec{x}_{2})= \frac{\mu_{0}}{4\pi}\int d^{3}x_{1}\vec{\
abla}_{\vec{x}_{2}}\times\Bigg{(} \frac{\vec{J}_{1}(\vec{x}_{1},t_{ret})}{R}\Bigg{)}. \tag{17}\]0 (A59)  
We observe that the second order is the lowest order for the magnetic field. Consequently, the force consists of two components: one which complies with [PERSON]'s third law and another which does not.