Electromagnetism

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%Kiyoshi Shiraishi Oct. 05, 1997
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%%%%%%%%%%%%%%%
%\hfill {ver. 1.0}
%%%%%%%%%%%%%%%
\title{
Electromagnetism
}
\author{Based on Stevens' book,
Kiyoshi Shiraishi
}
\date{ver. 1.0}
\begin{document}
\maketitle
\begin{abstract}
Yamaguchi University
\end{abstract}

\section{mathematical identities} %%%%%%%%%%%%%%%%%%%%

\begin{itemize}

\item rot grad vanishes

\be
\vnabla\times\vnabla\phi=0
\ee

\item div rot vanishes

\be
\vnabla\cdot\vnabla\times\vA=0
\ee

\item Gauss' theorem

\be 
\int_V \vnabla\cdot\vE ~d^3\vx=\int_S \vE\cdot d\vsigma
\ee 

\item Stokes' theorem

\be 
\int_S \vnabla\times\vE\cdot d\vsigma=\oint_C \vE\cdot d\vs
\ee 


\end{itemize}

\section{The electrostatic field} %%%%%%%%%%%%%%%%%%%%

\begin{itemize}

\item the force $\vF$ exerted on a test particle with charge $q$ in the electric field $\vE$

\be
\vF=q\vE
\ee

\item Coulomb's law

\be
F=\frac{1}{4\pi\epsilon_{0}}\frac{qq'}{r^2}
\ee

\be
\vE (\vx)=\frac{1}{4\pi\epsilon_{0}}\frac{q~\vx}{|\vx|^3}
\ee

\be
\vE (\vx)=\frac{1}{4\pi\epsilon_{0}}\sum_{i} q_{i} \frac{\vx-\vx_{i}}{|\vx-\vx_{i}|^3}
\ee

\item generalized Coulomb's law

\be
\vE (\vx)=\frac{1}{4\pi\epsilon_{0}}\int \rho(\vx')\frac{\vx-\vx'}{|\vx-\vx'|^3} d^3\vx'
\ee

\be 
\vnabla\frac{1}{|\vx-\vx'|}=-\frac{\vx-\vx'}{|\vx-\vx'|^3}
\ee 

\be
\vE (\vx)=-\frac{1}{4\pi\epsilon_{0}}\vnabla\int \frac{\rho(\vx')}{|\vx-\vx'|} d^3\vx'
\ee

\item the scalar potential

\be
\phi (\vx)=\frac{1}{4\pi\epsilon_{0}}\int \frac{\rho(\vx')}{|\vx-\vx'|} d^3\vx'
\ee

\be 
\vE (\vx)=-\vnabla\phi (\vx)
\ee 

\be 
\vnabla\times\vE=-\vnabla\times\vnabla\phi=0
\ee 

\item Gauss'law

\be
\int_S \vE\cdot d\vsigma=\frac{1}{\epsilon_{0}}\int \rho (\vx) ~d^3\vx
\ee

\be
\int \vnabla\cdot\vE ~d^3\vx=\frac{1}{\epsilon_{0}}\int \rho (\vx) ~d^3\vx
\ee

\item differential Coulomb's law

\be
\vnabla\cdot\vE=\frac{1}{\epsilon_{0}}\rho
\ee

\item equations for the electrostatic field

\bea
\vnabla\cdot\vE&=&\frac{1}{\epsilon_{0}}\rho \\
\vnabla\times\vE&=&0
\eea

\item Poisson's equation

\be
\nabla^2 \phi=-\frac{1}{\epsilon_{0}}~\rho
\ee

\end{itemize}

\section{The magnetostatic field}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{itemize}

\item magnetic monopoles do not exist

\be
\vnabla\cdot\vB=0
\ee

\item Amp\`ere's law

\be
\oint_C \vB\cdot d\vs=\mu_{0} \vI
\ee

\be 
\oint \vB\cdot d\vs=\mu_{0} \vI=\mu_{0} \int \vj\cdot d\vsigma
\ee

\be
\int_S \vnabla\times\vB\cdot d\vsigma=\mu_{0}\int_S \vj\cdot d\vsigma
\ee

\be
\vnabla\times\vB=\mu_{0} \vj
\ee

\item equations for the magnetostatic field

\bea
\vnabla\times\vB&=&\mu_{0} \vj \\
\vnabla\cdot\vB&=&0
\eea



\end{itemize}

\section{The electromagnetic field}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{itemize}

\item Faraday's law

\be
\oint_C \vE\cdot d\vs=-\frac{\partial}{\partial t} \int_S \vB\cdot d\vsigma
\ee

\be
\oint_C \vE\cdot d\vs=\int_S \vnabla\times\vE\cdot d\vsigma=
-\frac{\partial}{\partial t} \int_S \vB\cdot d\vsigma
\ee

\be
\vnabla\times\vE=-\frac{\partial}{\partial t}\vB
\ee


\item continuity equation

\be
\frac{\partial\rho}{\partial t}+\vnabla\cdot\vj=0
\ee

\be
\int_V \vnabla\cdot\vj ~d^3\vx=\int_S \vj\cdot d\vsigma=
-\frac{\partial}{\partial t}\int_V \rho(\vx) ~d^3\vx
\ee

\item to maintain the charge conservation

\be
\vnabla\times\vB=\mu_{0} \vJ
\ee

\be
\vnabla\cdot(\vnabla\times\vB)=0=\mu_{0} \vnabla\cdot\vJ
\ee

\be
\vnabla\cdot\vJ=\vnabla\cdot\vj+\frac{\partial\rho}{\partial t}=0
\ee

\be
\frac{\partial\rho}{\partial t}=
\frac{\partial}{\partial t}\epsilon_{0}\vnabla\cdot\vE
\ee

\be
\vJ=\vj+\epsilon_{0}\frac{\partial}{\partial t}\vE
\ee

\be
\vnabla\times\vB=\mu_{0} \vj+\epsilon_{0}\mu_{0}\frac{\partial}{\partial t}\vE
\ee

\item Maxwell's equations

\begin{tabular}{|ll|}
\hline
$\vnabla\times\vE=-\frac{\partial}{\partial t}\vB$ & {\small Faraday's law}\\
$\vnabla\times\vB=\mu_{0} \vj+\frac{1}{c^2}\frac{\partial}{\partial t}\vE$ & 
{\small Amp\`ere's law and conservation of charges}\\
$\vnabla\cdot\vE=\frac{\rho}{\epsilon_{0}}$ & {\small Coulomb's law}\\
$\vnabla\cdot\vB=0$ & {\small No magnetic monopoles}\\
\hline
\end{tabular}

where $c^2=\frac{1}{\epsilon_{0}\mu_{0}}$.

\end{itemize}



\section{Gauge transformations}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{itemize}

\item vector potential

\be
\vnabla\cdot\vB=0 \Rightarrow \vB=\vnabla\times\vA
\ee

\item scalar potential

\be
\vnabla\times\vE+\frac{\partial}{\partial t}\vB=0
\ee

\be
\vnabla\times(\vE+\frac{\partial}{\partial t}\vA)=0 \Rightarrow 
-\vnabla\phi=\vE+\frac{\partial}{\partial t}\vA
\ee

\item gauge transformations

\be
\vA \rightarrow \vA'=\vA+\vnabla\lambda
\ee

\be
\vB \rightarrow \vB'=\vnabla\times(\vA+\vnabla\lambda)=\vB
\ee

\be
\phi \rightarrow \phi'=\phi-\frac{\partial}{\partial t}\lambda
\ee

\be
\vE \rightarrow \vE'=-\vnabla\left(\phi-\frac{\partial}{\partial t}\lambda\right)
-\frac{\partial}{\partial t}(\vA+\vnabla\lambda)=\vE
\ee

\item summary of definition of potentials

\begin{tabular}{|c|}
\hline
$\vB=\vnabla\times\vA$ \\
$\vE=-\vnabla\phi-\frac{\partial}{\partial t}\vA$ \\
\hline
\end{tabular}

\item summary of gauge transformations

\begin{tabular}{|c|}
\hline
$\vA \rightarrow \vA+\vnabla\lambda$ \\
$\phi \rightarrow \phi-\frac{\partial}{\partial t}\lambda$ \\
\hline
\end{tabular}


\end{itemize}


\section{Maxwell's equations for the Lorentz gauge}%%%%%%%%%%%

\begin{itemize}

\item Amp\`ere's law and conservation of charges

\bea
\vnabla\times\vB&=&\vnabla\times(\vnabla\times\vA)=
\vnabla(\vnabla\cdot\vA)-\nabla^2\vA \nn
&=&\mu_{0} \vj+\frac{1}{c^2}\frac{\partial\vE}{\partial t}=
\mu_{0} \vj+\frac{1}{c^2}\frac{\partial}{\partial t}
\left(-\vnabla\phi-\frac{\partial\vA}{\partial t}\right)
\eea

\be
-\nabla^2\vA+\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\vA+
\vnabla\cdot\left(\vnabla\cdot\vA+\frac{1}{c^2}\frac{\partial\phi}{\partial t}\right)=
\mu_{0}\vj
\ee

\item Gauss' law

\be
\vnabla\cdot\vE=-\vnabla\cdot\left(\vnabla\phi+\frac{\partial\vA}{\partial t}\right)=
\frac{\rho}{\epsilon_{0}}
\ee

\be
\nabla^2\phi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\phi+
\frac{\partial}{\partial t}
\left(\vnabla\cdot\vA+\frac{1}{c^2}\frac{\partial\phi}{\partial t}\right)=
-\frac{\rho}{\epsilon_{0}}
\ee

\item the Lorentz gauge

\be
\vnabla\cdot\vA+\frac{1}{c^2}\frac{\partial\phi}{\partial t}=0
\ee

\item Maxwell's equations for the Lorentz gauge

\bea
\nabla^2\vA-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\vA&=&-\mu_{0}\vj \\
\nabla^2\phi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\phi&=&-\frac{\rho}{\epsilon_{0}}
\eea

\item wave equation for the magnetic field in vacua ($\rho=0$, $\vj=0$)
for the Lorentz gauge

\be
\nabla^2\vB-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\vB=0 
\ee

\item wave equation for the electric field in vacua ($\rho=0$, $\vj=0$)
for the Lorentz gauge

\be
\nabla^2\vE-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\vE=0 
\ee


\end{itemize}

\section{The macroscopic Maxwell's equations}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{itemize}

\item the electrostatic field in material media

\be 
\epsilon_{0}\vnabla\cdot\vE=\rho \stackrel{material~media}{\Rightarrow} 
\epsilon_{0}\vnabla\cdot\vE=\rho+\rho_{i}(\vE)
\ee

\item polarization

\be
\vnabla\cdot\vP=-\rho_{i}
\ee

\be 
\vnabla\cdot(\epsilon_{0}\vE+\vP)=\vnabla\cdot\vD=\rho
\ee

\item electric displacement

\be
\vD=\epsilon_{0}\vE+\vP
\ee

\item macroscopic differential Coulomb's law

\be
\vnabla\cdot\vD=\rho
\ee

\item the electric susceptibility $\chi$

\be
\vP(\vE)=\chi\vE
\ee

\be
\vD=\epsilon_{0}\vE+\vP=\epsilon_{0}\vE+\chi\vE=
(\epsilon_{0}+\chi)\vE\equiv \epsilon\vE
\ee

\item the magnetostatic field in material media

\be 
\vnabla\times\vB=\mu_{0} \vj \stackrel{material~media}{\Rightarrow} 
\vnabla\times\vB=\mu_{0} (\vj+\vj_{i}(\vB))
\ee

\item magnetization

\be
\vnabla\times\vM= \vj_{i}
\ee

\be 
\vnabla\times(\vB-\mu_{0}\vM)=\vnabla\times\mu_{0}\vH=\mu_{0} \vj
\ee

\item $H$ field

\be
\vH\equiv\frac{1}{\mu_{0}}\vB-\vM
\ee

\item macroscopic differential Amp\`ere's law

\be
\vnabla\times\vH=\vj
\ee

\be
\vB=\mu\vH
\ee


\item the macroscopic Maxwell's equations

\begin{tabular}{|ll|}
\hline
$\vnabla\times\vE=-\frac{\partial}{\partial t}\vB$ & {\small Faraday's law}\\
$\vnabla\times\vH=\vj+\frac{\partial}{\partial t}\vD$ & 
{\small Amp\`ere's law and conservation of charges}\\
$\vnabla\cdot\vD=\rho$ & {\small Coulomb's law}\\
$\vnabla\cdot\vB=0$ & {\small No magnetic monopoles}\\
\hline
\end{tabular}

\end{itemize}

\section{Lorentz force}%%%%%%%%%%%

\begin{itemize}

\item Lorentz force

\be
\vF=q (\vE+\vv\times\vB)
\ee

\end{itemize}

\section{Conservation of energy and momentum}%%%%%%%%%%%

\begin{itemize}

\item the rate of doing work for a charge $q$

\be
W=q\vv\cdot\vE
\ee

\item for a continuous distribution of charge and current in $V$

\be
W=\int_V \vj\cdot\vE ~d^3\vx
\ee

\bea
W&=&\int_V \left(\vnabla\times\vH-\frac{\partial}{\partial t}\vD\right)\cdot\vE ~d^3\vx \nn
&=&\int_V \left[\left(\vnabla\times\vE\right)\cdot\vH-
\left(\frac{\partial}{\partial t}\vD\right)\cdot\vE\right] ~d^3\vx-
\int_S (\vE\times\vH)\cdot d\vsigma \nn
&=&-\int_V \left[\left(\frac{\partial}{\partial t}\vB\right)\cdot\vH+
\left(\frac{\partial}{\partial t}\vD\right)\cdot\vE\right] ~d^3\vx-
\int_S (\vE\times\vH)\cdot d\vsigma \nn
&=&-\frac{1}{2}\frac{\partial}{\partial t}\int_V \left[\vE\cdot\vD+\vB\cdot\vH\right] ~d^3\vx-
\int_S (\vE\times\vH)\cdot d\vsigma 
\eea

here we have used

\be
\vnabla\cdot(\vE\times\vH)=\vH\cdot(\vnabla\times\vE)-\vE\cdot(\vnabla\times\vH)
\ee

\item the energy density of the electromagenetic field

\be
u=-\frac{1}{2}\left(\vE\cdot\vD+\vB\cdot\vH\right)
\ee

\item the Poynting vector

\be
\vS=\vE\times\vH
\ee

\item the electromagnetic momentum density

\be
\vg=\vS/c^2
\ee



\item conservation of energy and momentum

\be
\frac{\partial}{\partial t}\int_V u ~d^3\vx+\int_S \vS\cdot d\vsigma+W=0
\ee

\item differential conservation law

\be
\frac{\partial u}{\partial t}+\vnabla\cdot\vS+\vj\cdot\vE=0
\ee


\end{itemize}



\end{document}
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