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\begin{document}

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\begin{multicols}{3}

  {\huge Fields}\hfill Andrew Lorimer\hspace{2em}

\section*{Non-contact forces}
\begin{itemize}
\item electric fields (dipoles \& monopoles)
\item magnetic fields (dipoles only)
\item gravitational fields (monopoles only)
\end{itemize}

\begin{itemize}
\item monopoles: field lines radiate towards central object
\item dipoles - field lines go from + to -, or N to S
\item closer field lines means larger force
\item dot means out of page, cross means into page
\end{itemize}

\section*{Gravity}
\[
F_g=G{{m_1m_2}\over r^2}\tag{grav. force}
\]

\[
g={F_g \over m}=G{M_{\operatorname{planet}} \over r^2}\tag{grav. acceleration}
\]

\[
E_g = mg \Delta h\tag{grav. potential energy}
\]

\[
W = \Delta E_g = Fx\tag{work}
\]

Area under force-distance graph = $\Delta G.P.E$

Area under field-distance graph = $\Delta G.P.E / \operatorname{kg}$

% \columnbreak

\section*{Magnetic fields}
% \begin{itemize}
% \item field strength $B$ measured in tesla
% \item magnetic flux $\Phi$ measured in weber
% \item charge $q$ measured in coulombs
% \item emf $\mathcal{E}$ measured in volts
% \end{itemize}

% \[
% {E_1 \over E_2}={r_1 \over r_2}^2
% \]

\[
F=qvB\tag{force on moving charged particles}
\]

if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$


\includegraphics[height=3cm]{/mnt/andrew/graphics/field-lines.png}

\section*{Electric fields}

\begin{align*}
F=qE \tag{force on particle - $E$ is field strength} \\
W=q_{\operatorname{point}}\Delta V \tag{work in field or points} \\
F=k{{q_1q_2}\over r^2}\tag{Coulomb - force between particles} \\
E=k{Q \over r^2} \tag{field at distance from charge} \\
F=BInl \tag{force on a coil} \\
\Phi = B_{\perp}A\tag{magnetic flux} \\
\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{Faraday - induced emf} \\
{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \\
\end{align*}


\textbf{Lenz's law:}  ``$-n$'' in Faraday - emf opposes $\Delta \Phi$

\textbf{Eddy currents:} counter movement within a field

\textbf{Right hand grip:} thumb points to north or $I$

\textbf{Right hand slap:} field, current, force are $\perp$

\textbf{Flux-time graphs:} gradient $\times n = \operatorname{emf}$

\textbf{Transformers:} core strengthens \& focuses $\Phi$

% \columnbreak

\section*{Power transmission}

\begin{align*}
  V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}} \tag{rms to peak $\rightarrow$ peak} \\
  P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \tag{power loss}
\end{align*}

\includegraphics[height=4cm]{/mnt/andrew/graphics/ac-generator.png}

\section*{Motors}
% \begin{wrapfigure}{r}{-0.1\textwidth}

\includegraphics[height=4cm]{/mnt/andrew/graphics/dc-motor-2.png}
\includegraphics[height=3cm]{/mnt/andrew/graphics/ac-motor.png} \\
% \end{wrapfigure}
\textbf{DC:} split ring (one ring split into two halves)

% \begin{wrapfigure}{r}{0.3\textwidth}

% \end{wrapfigure}
\textbf{AC:} slip ring (separate rings with constant contact)





\end{multicols}
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