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\section{Equations of electrodynamics and classical mechanics}\label{sec:appendix_a}
Newton's second law
\begin{equation}\label{eq:newton_second}
    m \ddot{\mathbf{r}} = \sum_{i} \mathbf{F}_{i}
\end{equation}%
%
Lorentz force
\begin{equation}\label{eq:lorentz_force}
    \mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B},
\end{equation}%
%
Electric field
\begin{equation}\label{eq:e_field_potential}
    \mathbf{E} = - \nabla V
\end{equation} %
%
Electric field at a point $\mathbf{r}$
\begin{equation}\label{eq:e_field_point}
    \mathbf{E} = k_{e} \sum_{j=1}^{n} q_j \frac{\mathbf{r} - \mathbf{r}_{j}}{|\mathbf{r} - \mathbf{r}_{j}|^{3}}
\end{equation}



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