Classical Electrodynamics
Classical electrodynamics is the study of macroscopic electromagnetic behaviour. The classical formalism
generally employs vector calculus.
Basic Relations
The electric field is a non-rotational vector field that is postulated to permeate all space. The field uniquely determines the
force on a charge particle via the relation
$$
F=qE
$$
The energy of an arrangement of charges in an electric field is calculated via
$$
W=\frac{1}{2}\int_V \rho(\vec{x})\phi(\vec{x})d\vec{x}
$$
where \(\rho(\vec{x})\) is the charge density at point \(\vec{x}\).
Capacitance is generally defined as
$$
Q=CV.
$$
Jackson expands this definition to (with some paraphrasing): "the capacitance of a conductor is the total charge on a conducter when held
at unit potential, with all other surrounding conductors at zero potential (or with the conductor in free space)".
Electrostatics
Electrostatics is the description of static electric charge configurations. That is, scenarios in which
time dependence is eliminated and, thus, we only consider the form of electric fields over space.
Electrostatic Differential Equations
In electrostatic configurations, the electric field is entirely defined by it's two vector derivatives:
\begin{align}
\nabla \cdot \vec{E}&=\frac{\rho}{\epsilon_0}\\
\nabla \times\vec{E}&=0\\
\end{align}
Note that the electric field does have a curl (i.e. the electric field may be rotational) in dynamical scenarios, specifically
those with changing magnetic field. This curl-less property allows us to define
a (not neccesarily unique) scalar potential that may be used to completely specify the electric field.
Electrostatic Integral Equations
Equivalently, we may specify the field associated with an electrostatic configuration by the two vector integrals:
\begin{align}
\oint_{\partial V} \vec{E}\cdot d\vec{a}&=\int_V\frac{\rho(\vec{r})}{\epsilon_0}d\vec{r}\\
\oint_{\partial a} \vec{E}\cdot d\vec{\mathcal{l}}&=0\\
\end{align}
Once, again, the electric field does have a curl (i.e. the electric field may be rotational) in dynamical scenarios, specifically
those with changing magnetic field. The first integral above is often termed
Gauss' Law.
Scalar Potential
The electric field of electrostatic configurations (because of it's vanishing curl) may be specified as the gradient or exterior product
of some scalar potential.
$$
\vec{E}(\vec{r})=-\nabla \Phi(\vec{r})
$$
The potential is unique only up to the addition of some function with a gradient of zero divergence, which includes constant variations in the field
(except in the case of Dirichlet boundary conditions). This means that, at the least, the scalar potential's specfic value at any point in space is meaningless. Rather,
it's relative difference between points in space is what defines the behaviour of charges.
We may now define a volume integral that can be used to calculate a suitable scalar potential \(\Phi\) for a configuration of charge specified
by \(\rho(\vec{r}\)).
$$
\Phi(\vec{r})=\int_{V}\frac{\rho(\vec{r'})}{\vert \vec{r}-\vec{r}'\vert}d\vec{r}'
$$
Note that to calculate the potential at some point \(\vec{r}\), we are required to integrate over all of space; hence, the dummy variable
\(\vec{r}'\) appearing in the integral. As a result of this requirement, we can only really reasonably apply the above relation to configurations of
localized charge in otherwise free space (so we need not consider ALL space). Alternatively, we may specify boundary conditions on some surface
bounding a region of interest (addressed below).
Electrostatic Potential in Terms of Green's Functions
The most general procedure for calculating electrostatic potentials, given some boundary conditions, is achieved
through the use of Green's functions. Green's functions are effectively used as integral operators that invert the differential operator
through which boundary conditions are specified.
\begin{align}
\Phi(\vec{x})=\int_V \rho ( \vec{x}') &G(\vec{x},\vec{x}')d\vec{x}' \ \\
&+\frac{1}{4\pi} \oint_S\left[ \frac{\partial \Phi(\vec{x}')}{\partial \hat{n}}G(\vec{x},\vec{x}')
-\Phi(\vec{x}') \frac{\partial G(\vec{x},\vec{x}')}{\partial \hat{n}}\right]da'\\
\end{align}
The above relation is given in Gaussian units. To convert to SI units, simply multiply the first integral above by \(k=\frac{1}{4\pi \epsilon_0}.\)
Furthermore, Green's functions must generally satisfy the relation:
$$
\nabla^2G(\vec{x},\vec{x}')=-4\pi\delta(\vec{x}-\vec{x}')
$$
Types of Boundary Conditions
Boundary conditions are generally given as either one of two classes: Dirichlet or von Neumann. Both specify a certain equivalence
along some boundary, and have some corresponding behaviour neccesarily exhibited in their respective Green's functions.
Dirichlet
Dirichlet boundary condtions specify the potential along the boundary and hence specify \( \Phi.\) Under
these conditions, the Green's function along the boundary surface neccesarily vanishes, that is
$$
G_D\vert_S=0.
$$
Leaving one with the general solution for an electrostatic potential with a specified potential along a boundary:
$$
\Phi(\vec{x})=\int_V \rho ( \vec{x}') G_D(\vec{x},\vec{x}')d\vec{x}'-\frac{1}{4\pi} \oint_S\Phi(\vec{x}') \frac{\partial G_D(\vec{x},\vec{x}')}{\partial \hat{n}} da'.
$$
von Neumann
Neumann boundary condtions specify the electric field along the boundary and hence specify \(\frac{\partial \Phi}{\partial \hat{n}}.\) Under
these conditions, the Green's function derivative along the boundary surface neccesarily equals \(-\frac{4\pi}{S}\), where \(S\) is the surface area
of the boundary. That is,
$$
\frac{\partial G_N}{\partial \hat{n}}\vert_S=-\frac{4\pi}{S}.
$$
Leaving one with the general solution for an electrostatic potential with a specified electric field along a boundary:
$$
\Phi(\vec{x})=\langle \Phi\rangle_S +\int_V \rho ( \vec{x}') G_D(\vec{x},\vec{x}')d\vec{x}'+\frac{1}{4\pi} \oint_S \frac{\partial \Phi(\vec{x}')}{\partial \hat{n}}G(\vec{x},\vec{x}') da'
$$
where \(\langle \Phi\rangle_S \) is the average potential over the boundary surface(s) \(S\) (which results from the neccessary behaviour of the corresponding Green's function, discussed above).
Green's Functions for Common Coordinate Systems
Green's Function for Cylindrical Regions
Green's functions for unit point charges in cylindrical coordinates satisfy the relation
$$
\nabla^2_xG(\vec{x},\vec{x'})=-\frac{4\pi}{\rho}\delta (\rho-\rho')\delta (\phi-\phi')\delta (z-z')
$$
Expansions of Cylindrical Delta Functions
Generally termed the completeness relations, and traditionally normalized.
\begin{align}
\delta(z-z')&=\frac{2}{L}\sum_{n=1}^{\infty}\sin \left(\frac{n\pi z}{L}\right)\sin\left( \frac{n\pi z'}{L}\right)\\
\delta(\phi-\phi')&=\frac{1}{2\pi}\sum_{m=-\infty}^{\infty}e^{-im\phi'}e^{im\phi}\\
\frac{\delta(\rho-\rho')}{\rho'}&=\sum_{n=1}^{\infty}\frac{2}{a^2J_{m+1}^2(x_{mn})}J_m(x_{mn}\frac{\rho}{a})J_m(x_{mn}\frac{\rho'}{a}) \quad \forall m\\
\end{align}
Expansions of Cylindrical Inverse Vector Differences
Below are the point difference expansions in cylindrical coordinates.
\begin{align}
\frac{1}{\vert \vec{x}-\vec{x}'\vert}&=\frac{1}{\sqrt{\rho^2+\rho'^2-2\rho\rho' \cos (\phi-\phi')+(z-z')^2}}\\
&=\frac{2}{\pi}\sum_{m=-\infty}^{\infty}\int_0^{\infty}dk\ e^{im(\phi-\phi')}\cos [k(z-z')]
I_m(k\rho_{<})K_m(k\rho_{>})
\end{align}
Green's Function for Spherical Regions
Green's functions for unit point charges in spherical coordinates specified by spherical boundaries satisfy the relation
$$
\nabla^2_xG(\vec{x},\vec{x'})=\delta (r-r')\frac{\delta (\cos\phi-\cos\phi')}{r}\frac{\delta (\theta-\theta')}{r\sin\theta}
$$
Expansions of Spherical Delta Functions
\begin{align}
\delta(\cos\phi-\cos\phi')\delta(\theta -\theta')&=\sum_{l=0}^{l=\infty}\sum_{m=-l}^{l} Y^*_{lm}(\phi',\theta')Y_{lm}(\phi,\theta)\\
\end{align}
Expansions of Spherical Point Differences
Below are the point difference expansions in spherical coordinates.
\begin{align}
\frac{1}{\vert \vec{x}-\vec{x}'\vert}&=\frac{1}{\sqrt{r^2+r'^2-2rr'(\cos\theta\cos\theta'+\sin\theta\sin\theta'\cos(\phi-\phi'))}}\\
&=4\pi\sum_{l=-\infty}^{\infty}\sum_{m=-l}^{l}\frac{1}{2l+1}\frac{r_{<}^l}{r_{>}^{l+1}}Y^*_{lm}(\theta',\phi')Y_{lm}(\theta,\phi)
\end{align}
Dielectric Media
In insulating material, the particles composing the material react to the electric and magnetic fields such that
they decrease the overall applied external field. This reaction by dielectric material may be modeled by the relations below.
Dielectric Relations
Dielectric materials may be charachterized by their electric susceptibility \(\chi_e\), which measures the material's polarization
in response to an external electric field. As a special case, free space has a susceptibility of 0.
$$
\vec{P}=\chi_e \vec{E}
$$
We then may define the electric displacement \(\vec{D}\), which is suited for dealing
with the electric field in regions of non-zero susceptibility.
$$
\vec{D}=\vec{E}+\vec{P}
$$
Magnetostatics
Magnetoostatics is the description of static current configurations. That is, scenarios in which
time dependence is eliminated and, thus, we may only consider the form of magnetic fields over space.
Magnetostatic Differential Equations
In magnetostatic configurations, the magnetic field is entirely defined by it's two vector derivatives:
\begin{align}
\nabla \cdot \vec{B}&=0\\
\nabla \times\vec{B}&=\mu_0\vec{J}\\
\end{align}
The first derivative specified above, that of divergence being equal to zero, is equivalent to the statement that there are
no magnetic monopoles in all of reality. This has been experimentally tested to the limits of current technology, and allows us
to specify the magentic field as the curl of some vector potential.
Magnetostatic Integral Equations
Equivalently, we may specify the field associated with a magnetostatic configuration by the two vector integrals:
\begin{align}
\oint_{\partial V} \vec{B}\cdot d\vec{a}&=0\\
\oint_{\partial a} \vec{B}\cdot d\vec{\mathcal{l}}&=\mu_0 \int_{a} \vec{J}\cdot d\vec{a}\\
\end{align}
Vector Potential
Due to the curl-less nature of time independent magnetic fields, we may specify their behaviour entirely in terms of some
vector potential.
$$
\vec{B}(\vec{r})=-\nabla\times\vec{A}(\vec{r})
$$
The potential is unique only up to the addition of some function with a curl of zero. This additional function, added to the
vector potential, is termed the gauge. The gauge we choose may be used to satisfy certain appealing properties that may sinplify
or aid in calculations. Some common gauges include: the Coulomb gauge, which guarantees
\(\nabla\cdot\vec{A}=0\); the Lorentz gauge; and the Landau gauge.
We may now define a volume integral that can be used to calculate a suitable vector potential \(\vec{A}\) for a configuration of currents specified
by \(\vec{J}(\vec{r}\)).
$$
\vec{A}(\vec{r})=\int_{V}\frac{\vec{J}(\vec{r'})}{\vert \vec{r}-\vec{r}'\vert}d\vec{r}'
$$
Note that to calculate the potential at some point \(\vec{r}\), we are required to integrate over all of space; hence, the dummy variable
\(\vec{r}'\) appearing in the integral. As a result of this requirement, we can only really reasonably apply the above relation to configurations of
localized charge in otherwise free space (so we need not consider ALL space). Alternatively, we may specify boundary conditions on some surface
bounding a region of interest (addressed below).
Special Functions
Some common functions that arise in the above considerations (particularly in systems with certain symmetries) are discussed
below. These include the functions relevant to spherical cylindrical symmetries: namely bessel functions, spherical harmonics and legendre polynomials.
Bessel Functions
Bessel functions \(J_{\nu}(x)\) are one dimensional functions that satisfy their defining differential equation:
$$
x^2\frac{d^2J_{\nu}}{dx^2}+x\frac{dJ_{\nu}}{dx}+(x^2-\nu^2)J_{\nu}=0
$$
They often appear in the solution of spherical and cylindrical Laplace and Helmholtz equations.
Legendre Polynomials
Legendre polynomials \(P_{l}(x)\) are one dimensional functions that satisfy their defining differential equation:
$$
(1-x^2)\frac{d^2P_l}{dx^2}-2x\frac{dP_l}{dx}+l(l+1)P_l=0
$$
Legendre polynomials are often specified by Rodrigues' formula, given below:
$$
P_l(x)=\frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2-1)^l
$$
The above polynomials are actually just a special case of a more general set, known as the associated Legendre polynomials. The associated
Legendre polynomials are one dimensional functions that are solutions of the two parameter differential equation:
$$
(1-x^2)\frac{d^2P^m_l}{dx^2}-2x\frac{dP^m_l}{dx}+\left[l(l+1)-\frac{m^2}{1-x^2}\right]P^m_l=0
$$
Spherical Harmonics
Spherical harmonics are two dimensional functions specified as a product of two one-dimensional functions and, often, some normalization constant.
A general form with traditional normalization (integrated over all space equals one) is given as:
$$
Y_m^l(\theta,\phi)=\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}P_l^m(\cos\theta)e^{im\phi}
$$
They're defined as above due to their close relationship in the solution of certain differential equations (see Separation of Variables for Helmholtz Equation).
Resources
The classic reference text for the subject at hand is John David Jackson's Classical Electrodynamics. A good
introductory text is David J. Griffith's Introduction to Electrodynamics or Edward Purcell and David Morin's
Electricity and Magnetism. Another good reference text is Andrew Zangwill's Modern Electrodynamics.
Christopher Baird has a nice set of lecture notes that follow along with the Jackson text that can be found here:
Baird Notes. Other sets of lecture notes include
Gary Wysin's Notes and
Martin Houde's Notes.
