LaTeX and Electromagnetics +1
…and maybe you finished studying for this exam, or you think you did, but really you can’t be sure, and you want to eat batteries, and this kind of looks like shit but you don’t care, and this is every equation you need to know for emag but it still won’t help goddamnit. Vita meretrix est.
\documentclass[11pt]{amsart}
\usepackage[margin=.75in]{geometry} % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper} % ... or a4paper or a5paper or ...
%\geometry{landscape} % Activate for for rotated page geometry
%\usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\usepackage{amsmath}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}
%%%%%%%%%%% Bold style Vectors%%%%%%%%%%%%%
\newcommand{\vect}[1]{\ensuremath{\boldsymbol{#1}}}
\newcommand{\unit}[1]{\ensuremath{\boldsymbol{\widehat{#1}}}}
\newcommand{\del}{\ensuremath{\boldsymbol{\nabla}}}
%%%%%%%%%%% arrow style vectors %%%%%%%%%%%%%%
%\newcommand{\vect}[1]{\ensuremath{\overrightarrow{#1}}}
%\newcommand{\unit}[1]{\ensuremath{\widehat{#1}}}
%\newcommand{\del}{\ensuremath{\nabla}}
\newcommand{\dotp}{\ensuremath{\!\cdot\!}}
\newcommand{\cross}{\ensuremath{\!\times\!}}
\newcommand{\entry}[2]{\item[#1] \hfill $#2$}
\begin{document}
\section{Electric}
\subsection{Charge}
\begin{description}
\entry{Electric Field Examples}{~}
\begin{description}
\entry{due to point charge}{\vect{E}=\frac{Q\unit{R}}{4\pi\epsilon R^2} = \frac{Q\vect{R}}{4\pi\epsilon R^3}}
\entry{due to an infinitely long wire}{E=\frac{\rho_l}{2\pi\epsilon_0R}}
\end{description}
\entry{Electric Flux Density}{\vect{D} = \epsilon_0\vect{E} + \vect P= \epsilon\vect{E}=\frac{Q\unit{R}}{4\pi R^2} }
\entry{Gauss's Law}{Q=\oint\vect{D}\dotp\vect{ds} = \oint\vect{D}\dotp\unit{n}ds = \int\del \dotp\vect{D}dv = \int\rho_vdv} \\ \hfill (``$v$'' as in volume.)
\entry{Voltage}{V_{\left(\textit{1 2}\right)}=-\int_2^1\vect{E}\dotp\vect{dl}}
\entry{Capacitance}{C=\frac{Q}{V}}
\item[Charge Density] ~
\begin{description}
\entry{Surface}{\rho_s=\frac{Q}{S}}
\entry{Volume}{\rho_v=\del\dotp\vect{D}}
\entry{Polarization Surface}{\rho_{\textit{ps}}=\vect{P}\dotp\unit{n}}
\entry{Polarization Volume}{\rho_{\textit{ps}}=-\del\dotp\vect{P}}
\end{description}
\entry{Polarization}{\oint\rho_{\textit{ps}}\vect{ds}+\int\rho_{\textit{pv}}\vect{dv} = 0}
\item[Boundary Conditions] ~
\begin{description}
\entry{Tangential}{E_{t1} = E_{t2}}
\entry{Normal}{D_{n1} - D_{n2}=\rho_s}
\end{description}
\entry{Energy Density}{W=\frac{1}{2}\vect{D}\dotp\vect{E}=\frac{1}{2}\epsilon\vect{E}\dotp\vect{E}}
\entry{Total Energy}{W=\frac{1}{2}\int_{v'}\rho_vvdv = \frac{1}{2}\int_{v'} \epsilon \vect{E}\dotp\vect{E} dv = \frac{1}{2}CV^2= \frac{ 1}{2}QV=\frac{1}{2}\frac{Q^2}{C}}
\entry{Force}{\vect{F}=-\del W=QE}
\entry{Poisson's Equation}{\del^2V=\frac{-\rho_v}{\epsilon}}
\entry{Laplace Equation}{\del^2V=0}
\end{description}
\subsection{Current}
\begin{description}
\entry{Current Density}{\vect J = \sigma \vect E }
\entry{Resistance}{ R = \frac{l}{\sigma s}}
\entry{Calculus with $\vect J$}{ \del \dotp \vect J = 0} ; $\oint\vect J \dotp \vect{ds} = 0$
\item[Current Boundary Conditions] ~
\begin{description}
\entry{Tangential}{\frac{J_{t1}}{\sigma_1} = \frac{J_{t2}}{\sigma_2} }
\entry{Normal}{J_{n1} = J_{n2}}
\end{description}
\end{description}
\section{Magnetic}
\begin{description}
\entry{Ampere's Law}{\oint \vect B \dotp \vect{dl} = \mu I} ; $\del \cross \vect B = \mu_0 \vect J \therefore \int \del \cross\vect B \dotp \vect{ds}=\mu_0\int\vect J \dotp \vect ds \therefore \del \cross \vect H = \vect J$
\entry{Current Density}{J=\frac I A} where $A= \textrm{cross-sectional area}$
\entry{Biot--Savart's Law}{\vect B = \frac {\mu_0I}{4\pi}\int\frac{dl\cross\unit R}{R^2}=\del\cross\vect A} where $\vect A$ is Magnetic Vector Potential.
\entry{Magnetic Field Intensity}{ H=\frac 1 {\mu_0}\vect B - \vect M} ; $\vect B=\mu\vect H$
\entry{Vector Poisson's Equation}{\del^2\vect A = -\mu_0 \vect J}
\entry{Magnetization Current}{\vect{J_\textit{ms}}=\vect M \cross \unit{n}}
\entry{Magnetic Flux}{\Phi = \int_s\vect B \dotp \vect{ds} = \oint \vect A \cdot \vect{dl}}
\item[Boundary Conditions] ~
\begin{description}
\entry{Normal}{\vect{B_{1n}}=\vect{B_{2n}}}
\entry{Tangential}{\vect{H_{1t}}=\vect{H_{2t}}}
\end{description}
\entry{Flux Linkage}{\Lambda_{\left(\textit{1 2}\right)}=L_{\left(\textit{1 2}\right)}I_1=N_2\Phi_{\left(\textit{1 2}\right)}}
\entry{\textit{M\'as} Flux Linkage}{L_{\left(\textit{1 2}\right)} = \frac{\Lambda_{\left(\textit{1 2}\right)}}{I_1}=\frac{N_2}{I_1}\int_{S_2}\vect{B_1} \dotp \vect{ds_2}}
\entry{Energy Density}{W = \frac 1 2 \vect H \dotp \vect B = \frac 1 2 \frac{ B^2}{\mu}}
\entry{Energy}{W = \frac 1 2 LI^2=\frac 1 2 \int_{v'}\frac{B^2}{\mu}dv}
\entry{Force}{F=I\oint_c\vect{dl}\cross\vect B}
\entry{Lorentz's Force Equation}{\vect F = \vect{F_e}+\vect{F_m}=q\left(\vect E + \vect u \cross \vect B\right)} where $\vect u$ is velocity
\entry{Torque for Dipole or Loop}{\vect T = \vect m \cross \vect B $ where $ \vect m= IS \unit n$ and $ F_\Phi = -\unit y \frac{I^2}{\mu_0I}}
\entry{Vector Magnetic Potential}{\vect A = \frac{\mu_0}{4\pi}\int_{v'}\frac {\vect J} R dv'} in Webers per Meter $\left(\frac {\mathrm{Wb}}{\mathrm{m}}\right)$
\entry{Magnetic field examples}{~}
\begin{description}
\entry{Axis of a Loop on $z=0$ plane with axis at origin}{\vect B = \unit z \frac{\mu I}{2}\left(\frac{a^2}{\left(a^2+z^2\right)^{3/2}}\right)}
\entry{Distance from a wire}{\vect B = \unit \Phi \frac {\mu_0I}{4\pi}\left[\frac {z}{r\sqrt{z^2+r^2}} \right]_{z=-L}^{L} = \unit \Phi \frac {\mu_0I}{4\pi r}\left(\frac {2L}{\sqrt{L^2+r^2}} \right) }
\end{description}
\entry{Modeling magnetic circuits as their Electrical Equivalents}{~}
\begin{description}
\entry{Resistance}{R = \frac{l}{\mu S}}
\entry{Equivalency}{\Phi R = NI}
\end{description}
\end{description}
\end{document}
