% vim: tw=50
% 19/01/2023 09AM

\section{Introduction}

\subsection{Charges and currents}

\emph{Electric charge} is a physical property of
elementary particles. It is
\begin{itemize}
    \item Positive, negative or zero.
    \item Quantized (an integer multiple of the
        \emph{elementary charge} $e$)
    \item Conserved (even if particles are created
        or destroyed).
\end{itemize}
By convention, the electron has charge $-e$,
proton has charge $+e$ and neutron has charge $0$.

\myskip
On macroscopic scales, the number of particles is
so large that charge can be considered to have a
continuous \emph{electric charge density}
$\rho(\bf{x}, t)$. The total charge in a volume
$V$ is
\[ Q = \int_V \rho \dd V \]
The \emph{electric current density}
$\bf{J}(\bf{x}, t)$ is the flux of electric charge
per unit area. The current flowing through a
surface $S$ is
\[ I = \int_S \bf{J} \cdot \dd \bf{S} .\]
Consider a time-independent volume $V$ with
boundary $S$. Since charge is conserved,
\[ \dfrac{Q}{t} = -I \]
\[ \dfrac{}{t} \int_V \rho \dd V + \int_S \bf{J}
\cdot \dd \bf{S} = 0 \]
\[ \int_V \left( \pfrac{\rho}{t} + \nabla \cdot
\bf{J} \right) \dd V = 0 \]
Since this is true for any $V$,
\[ \boxed{\pfrac{\rho}{t} + \nabla \cdot \bf{J} =
0} \tag{1} \]
This \emph{equation of charge conservation} has
the typical form of a conservation law. The
discrete charge distribution of a single particle
of charge $q_i$ and position vector $\bf{x}_i(t)$
is
\[ \rho = q_i \delta(\bf{x} - \bf{x}_i(t)) \]
\[ \bf{J} = q_i \dot{\bf{x}}_i \delta(\bf{x} -
\bf{x}_i(t)) \]
For $N$ particles it is
\[ \rho = \sum_{i = 1}^N q_i \delta(\bf{x} -
\bf{x}_i(t)) \]
\[ \bf{J} = \sum_{i = 1}^N q_i \dot{\bf{x}}_i
\delta(\bf{x} - \bf{x}_i(t)) \]
Exercise: Verify that these satisfy equation (1).

\subsection{Fields and Forces}

Electromagnetism is a \emph{field theory}. Charged
particles interact not directly but by generating
fields around them that are experienced by other
charged particles. In general we have two
time-dependent vector fields:
\begin{itemize}
    \item electric field $\bf{E}(\bf{x}, t)$.
    \item magnetic field $\bf{B}(\bf{x}, t)$.
\end{itemize}
The \emph{Lorentz force} on a particle of charge
$q$ and velocity $\bf{v}$ is
\[ \boxed{\bf{F} = q(\bf{E} + \bf{v} \times
\bf{B})} \]

\subsection{Maxwell's Equations}

In this course we will explore some consequences
of \emph{Maxwell's equations}:
\begin{equation*}
    \boxed{
    \begin{aligned}
        \nabla \cdot \bf{E}
        &= \frac{\rho}{\eps_0}
        &&\qquad\text{(M1)} \\
        \nabla \cdot \bf{B}
        &= 0 
        &&\qquad\text{(M2)} \\
        \nabla \times \bf{E}
        &= -\pfrac{\bf{B}}{t}
        &&\qquad\text{(M3)} \\
        \nabla \times \bf{B}
        &= \mu_0 \left( \bf{J} + \eps_0
        \pfrac{\bf{E}}{t} \right)
        &&\qquad\text{(M4)}
    \end{aligned}
    }
\end{equation*}
Some properties:
\begin{itemize}
    \item Coupled linear PDEs in space and time.
    \item Involve two positive constants:
        \begin{itemize}
            \item $\eps_0$ (vacuum permittivity)
            \item $\mu_0$ (vacuum permeability)
        \end{itemize}
    \item Charges ($\rho$) and currents ($\bf{J}$)
        are the sources of the electromagnetic
        fields.
    \item Each equation has an equivalent integral
        form (see later) related via the
        divergence theorem or Stoke's theorem.
    \item These are the vacuum equations that
        apply on microscopic scales (or in a
        vacuum). A related macroscopic version
        applies in media (for example air) (see
        Part II Electrodynamics).
    \item The equations are consistent with each
        other and with charge conservation.
        \begin{itemize}
            \item $\nabla \cdot \text{(M3)}$
                agrees with $\pfrac{}{t}
                \text{(M2)}$
            \item \begin{align*}
                \pfrac{\rho}{t} + \nabla \cdot \bf{J}
                &= \pfrac{}{t}\left( \eps_0 \nabla
                \cdot \bf{E} \right) + \nabla
                \cdot \left( -\eps_0
                \pfrac{\bf{E}}{t} +
                \frac{1}{\mu_0} \nabla \times
                \bf{B} \right) \\
                &= 0
            \end{align*}
        \end{itemize}
\end{itemize}

\subsection{Units}

The SI unit of electric charge is the coulomb
($\mathsf{C}$). The elementary charge is (exactly)
\[ e = 1.602176634 \times 10^{-19} \mathsf{C} .\]
The SI unit of electric current is the ampere or
amp ($\mathsf{A}$), equal to $1 \mathsf{C}
\mathsf{s^{-1}}$. The SI base units needed in
electromagnetism are
\begin{itemize}
    \item second ($\mathsf{s}$)
    \item metre ($\mathsf{m}$)
    \item kilogram ($\mathsf{k}\mathsf{g}$)
    \item ampere ($\mathsf{A}$)
\end{itemize}
From the Lorentz force law we see that the units
of $\bf{E}$ and $\bf{B}$ must be
$\mathsf{k}\mathsf{g}\mathsf{s^{-3}}
\mathsf{A^{-1}}$ and $\mathsf{k}\mathsf{g}
\mathsf{s^{-2}} \mathsf{A^{-1}}$ (also called
tesla ($\mathsf{T}$)). From Maxwell's equations we
can work out the units of $\eps_0$ and $\mu_0$.
Experimentally determined values are
\begin{align*}
    \eps_0
    &= 8.854\ldots \times 10^{-12}
    \mathsf{k}\mathsf{g^{-1}} \mathsf{m^{-3}}
    \mathsf{s^4} \mathsf{A^2} \\
    \mu_0
    &= 1.256\ldots \times 10^{-6}
    \mathsf{k}\mathsf{g} \mathsf{m}
    \mathsf{s^{-2}} \mathsf{A^{-2}} \\
    &\approx 4\pi \times 10^{-7}
\end{align*}
The speed of light is (exactly)
\[ c = \frac{1}{\sqrt{\mu_0\eps_0}} = 299792458
\mathsf{m}\mathsf{s^{-1}} \approx 3 \times 10^8
\mathsf{m}\mathsf{s^{-1}} \]