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z?x)>^w;w}aP@~G9zxx$rgINf5V=N+A!gJPch|61jmD+vFh0g^sHmmYA*{PW{Ev-Xn z=%(lK92m}j@YUZbnN%)~)p$s|=dLy|EBDpj$(j`WvZXbW=It8Zl-$=Cq3HG21VS0x zy3I%Vw)Osperyx61CkzhljQM1Yw^-QV^HNYlW8UTS!4%pZtu5( zsw-iK52_C5<)vn^gHDiXm}r~(2dg4Py+1@u{miLHzgXG26(R=P*D-g?ki6>Ff7`A3 zF~f|#{Q@z=aok0#c|cY>{A@SruR&|64y?o)x0clhnuPu&82dXYxq=wE?kLq8F)NXP zmzTf<0l>%;02~3Z^eT{kJdUKN;UXf?0YyvNpIO=Lhq9b$l94k5Q^x>P&yWBggGR!j z(vYJx;=Ge@Q{!Z|8>jh!T~)mQ0{)HYasU7T diff --git a/latex/main.tex b/latex/main.tex index 4bd63f2..f9c63f7 100644 --- a/latex/main.tex +++ b/latex/main.tex @@ -63,6 +63,9 @@ % Introduction \subfile{sections/introduction} +% Theory +\subfile{sections/theory} + % Methods \subfile{sections/methods} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 42348f6..125bd37 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -3,63 +3,6 @@ \begin{document} \section{Methods} -% problem 1 -When we study the Penning traps effect on a particle with a charge $q$, we need to consider the forces acting on the particle. The sum of all forces acting on the particle, is given by the Lorentz force \eqref{eq:lorentz_force}. -\begin{equation}\label{eq:lorentz_force} - \mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B}, -\end{equation} -We can use Newton's second law \eqref{eq:newton_second} to determine this sum by -\begin{align*} - \ddot{\mathbf{r}} &= \frac{1}{m} \sum_{i} \mathbf{F}_{i} \\ - &= \frac{1}{m} (q \mathbf{E} + q \mathbf{v} \times \mathbf{B}) \\ - &= \frac{q}{m} \big(\frac{V_{0}}{d^{2}} (x, y, -2z) + (B_{0}\dot{y}, -B_{0}\dot{x}, 0) \big), -\end{align*} -where the complete derivation can be found in \ref{sec:appendix_b}. We can now write the particle's position as -\begin{align} - \label{eq:motion_x} - \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x &= 0, \\ - \label{eq:motion_y} - \ddot{y} + \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y &= 0, \\ - \label{eq:motion_z} - \ddot{z} + \omega_{z}^{2} z &= 0, -\end{align} -% define w_0 og w_z i appendix? -In addition, we can find the general solution for eq. \eqref{eq:motion_z}, when we consider the characteristic equation of a second order differential equation \cite{lindstrom:2016:ch10:5}. -\begin{align*} - r^{2} + \omega_{z}^{2} &= 0 \\ - r &= \pm \sqrt{- \omega_{z}^{2}} = \mp i \omega_{z}, -\end{align*} -two complex roots which gives us solutions in the form of -\begin{equation}\label{eq:all_diff_sol} - z = c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t}, -\end{equation} -For a complex number $z = a + ib$, we can define $e^{z} \equiv e^{a} (\cos{b} + i \sin{b})$ \cite{lindstrom:2016:ch3}. We can rewrite \eqref{eq:all_diff_sol} as -\begin{align*} - c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t} &= c_{1} (\cos{\omega_{z} t} + i \sin{\omega_{z} t}) + c_{2} (\cos{\omega_{z} t} - i \sin{\omega_{z} t} \\ - &= E \cos{\omega_{z} t} + i F \sin{\omega_{z} t} -\end{align*} -% -Since \eqref{eq:motion_x} and \eqref{eq:motion_y} are coupled, we want to rewrite it as a single differential equation. We can obtain this by introducing $f(t) = x(t) + iy(t)$, -\begin{align*} - (\ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x) + i (\ddot{y} + \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y) &= 0 \\ - \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x + i\ddot{y} + i\omega_{0} \dot{x} - i \frac{1}{2} \omega_{z}^{2} y &= 0 \\ - \ddot{x} + i\ddot{y} + i\omega_{0} \dot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x - i \frac{1}{2} \omega_{z}^{2} y &= 0 \\ - \ddot{x} + i\ddot{y} + i\omega_{0} (\dot{x} + i \dot{y}) - \frac{1}{2} \omega_{z}^{2} x - i \frac{1}{2} \omega_{z}^{2} y &= 0 \text{where $i \omega_{0} \dot{x} + (-1) \omega_{0} \dot{y} = i \omega_{0} \dot{x} + i^{2} \omega_{0} \dot{y}$} \\ - \ddot{f} + i \omega_{0} \dot{f} - \frac{1}{2} \omega_{z}^{2} f &= 0 -\end{align*} - - -Physical properties given by newtons second law \eqref{eq:newton_second} -\begin{equation}\label{eq:general_solution} - f(t) = A_{+}e^{-i(\omega_{+} t + \phi_{+})} + A_{-}e^{-i(\omega_{-} t + \phi_{-})} -\end{equation} -The particle moves and its position can be determined using newton. where the electric field - -\subsection*{Algorithm} - - -\subsection*{Tools} -We used matplotlib \subsection{Units and constants} @@ -286,5 +229,8 @@ and would reduce the required amount of loops down to 4. \subsection{Relative error and error convergance rate} +\subsection{Tools} +We used matplotlib + %\biblio \end{document} diff --git a/latex/sections/theory.tex b/latex/sections/theory.tex new file mode 100644 index 0000000..e8d43c8 --- /dev/null +++ b/latex/sections/theory.tex @@ -0,0 +1,61 @@ +\documentclass[../main.tex]{subfiles} +\graphicspath{{\subfix{../images/}}} + +\begin{document} +\section{Theory} +% problem 1 +When we study the Penning traps effect on a particle with a charge $q$, we need to consider the forces acting on the particle. The sum of all forces acting on the particle, is given by the Lorentz force \eqref{eq:lorentz_force}. +\begin{equation}\label{eq:lorentz_force} + \mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B}, +\end{equation} +We can use Newton's second law \eqref{eq:newton_second} to determine this sum by +\begin{align*} + \ddot{\mathbf{r}} &= \frac{1}{m} \sum_{i} \mathbf{F}_{i} \\ + &= \frac{1}{m} (q \mathbf{E} + q \mathbf{v} \times \mathbf{B}) \\ + &= \frac{q}{m} \big(\frac{V_{0}}{d^{2}} (x, y, -2z) + (B_{0}\dot{y}, -B_{0}\dot{x}, 0) \big), +\end{align*} +where the complete derivation can be found in \ref{sec:appendix_b}. We can now write the particle's position as +\begin{align} + \label{eq:motion_x} + \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x &= 0, \\ + \label{eq:motion_y} + \ddot{y} + \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y &= 0, \\ + \label{eq:motion_z} + \ddot{z} + \omega_{z}^{2} z &= 0, +\end{align} +% define w_0 og w_z i appendix? +In addition, we can find the general solution for eq. \eqref{eq:motion_z}, when we consider the characteristic equation of a second order differential equation \cite{lindstrom:2016:ch10:5}. +\begin{align*} + r^{2} + \omega_{z}^{2} &= 0 \\ + r &= \pm \sqrt{- \omega_{z}^{2}} = \mp i \omega_{z}, +\end{align*} +two complex roots which gives us solutions in the form of +\begin{equation}\label{eq:all_diff_sol} + z = c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t}, +\end{equation} +For a complex number $z = a + ib$, we can define $e^{z} \equiv e^{a} (\cos{b} + i \sin{b})$ \cite{lindstrom:2016:ch3}. We can rewrite \eqref{eq:all_diff_sol} as +\begin{align*} + c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t} &= c_{1} (\cos{\omega_{z} t} + i \sin{\omega_{z} t}) + c_{2} (\cos{\omega_{z} t} - i \sin{\omega_{z} t} \\ + &= E \cos{\omega_{z} t} + i F \sin{\omega_{z} t} +\end{align*} +% +Since \eqref{eq:motion_x} and \eqref{eq:motion_y} are coupled, we want to rewrite it as a single differential equation. We can obtain this by introducing $f(t) = x(t) + iy(t)$, +\begin{align*} + (\ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x) + i (\ddot{y} + \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y) &= 0 \\ + \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x + i\ddot{y} + i\omega_{0} \dot{x} - i \frac{1}{2} \omega_{z}^{2} y &= 0 \\ + \ddot{x} + i\ddot{y} + i\omega_{0} \dot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x - i \frac{1}{2} \omega_{z}^{2} y &= 0 \\ + \ddot{x} + i\ddot{y} + i\omega_{0} (\dot{x} + i \dot{y}) - \frac{1}{2} \omega_{z}^{2} x - i \frac{1}{2} \omega_{z}^{2} y &= 0 \text{where $i \omega_{0} \dot{x} + (-1) \omega_{0} \dot{y} = i \omega_{0} \dot{x} + i^{2} \omega_{0} \dot{y}$} \\ + \ddot{f} + i \omega_{0} \dot{f} - \frac{1}{2} \omega_{z}^{2} f &= 0 +\end{align*} + + +Physical properties given by newtons second law \eqref{eq:newton_second} +\begin{equation}\label{eq:general_solution} + f(t) = A_{+}e^{-i(\omega_{+} t + \phi_{+})} + A_{-}e^{-i(\omega_{-} t + \phi_{-})} +\end{equation} +The particle moves and its position can be determined using newton. where the electric field + + + +%\biblio +\end{document}