Maxwell's equations
Maxwell's equations are:
In vacuum:
\begin{align}
\nabla \cdot \mathbf{E} & = \frac{\rho}{\epsilon_0} \\
\nabla \times \mathbf{E} & = -\frac{\partial\mathbf{B}}{\partial t}\\
\nabla \cdot \mathbf{B} & = 0\\
\nabla \times \mathbf{B} & = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial\mathbf{E}}{\partial t}
\end{align}
In matter:
\begin{align}
\nabla \cdot \mathbf{D} & = \rho_f \\
\nabla \times \mathbf{E} & = -\frac{\partial\mathbf{B}}{\partial t}\\
\nabla \cdot \mathbf{B} & = 0\\
\nabla \times \mathbf{H} & = \mathbf{J}_f + \frac{\partial\mathbf{D}}{\partial t}
\end{align}
Auxillary fields:
$$\mathbf{D}=\epsilon_0\mathbf{E}+\mathbf{P},\quad \rho_b=-\nabla\cdot\mathbf{P},\quad \mathbf{J}_p=\frac{\partial\mathbf{P}}{\partial t}$$
$$\mathbf{H}=\frac{1}{\mu_0}\mathbf{B}-\mathbf{M},\quad J_b = \nabla\times \mathbf{M}$$
In linear media:
$$\mathbf{P}=\epsilon_0\chi_e\mathbf{E}, \quad \mathbf{D}=\epsilon\mathbf{E},\quad \epsilon=\epsilon_r\epsilon_0=\epsilon_0(1+\chi_e)$$
$$\mathbf{M}=\chi_m\mathbf{H}, \quad \mathbf{H}=\frac{1}{\mu}\mathbf{B},\quad \mu=\mu_r\mu_0=\mu_0(1+\chi_m)$$
Both sets of Maxwell's equations are equally valid in all situations. However, the set on the left, in terms of $\mathbf{E}$ and $\mathbf{B}$ is more useful
in vacuum where the total charge density $\rho=\rho_b+\rho_f$ and the total current density $\mathbf{J}=\mathbf{J}_f+\mathbf{J}_b+\mathbf{J}_p$ are known.
The ones in the center, in terms of $\mathbf{D}$ and $\mathbf{H}$ are more useful in media, where only the free charge and current are known.
The force that a particle of charge $q$ and velocity $\mathbf{v}$ experiences due to the electromagnetic fields
is given by the Lorentz force law $$\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}),$$
which together with Maxwell's equations, fully describe the classical theory of electrodynamics.
Statics
In the static case, one can define a scalar potential $V$ and a vector potential $\mathbf{A}$ as
$$\mathbf{E}=-\nabla V$$ $$\mathbf{B}=\nabla \times \mathbf{A}$$
Electrostatics
Electrostatics can be described by a Poisson equation:
$$\nabla^2 V = -\frac{\rho}{\epsilon_0}$$
The solution can be written as
\begin{align}
V(\mathbf{r}) &= \frac{1}{4\pi \epsilon_0}\int_\mathcal{V}\frac{\rho(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}d^3r'\\
\mathbf{E}(\mathbf{r}) &= \frac{1}{4\pi \epsilon_0}\int_\mathcal{V}\frac{\rho(\mathbf{r}')(\mathbf{r}-\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|^3}d^3r'
\end{align}
Magnetostatics
Magnetostatics can be described by 3 Poisson equations(for each Cartesian component):
$$\nabla^2 \mathbf{A} = -\mu_0\mathbf{J}$$
The solution can be written as
\begin{align}
\mathbf{A}(\mathbf{r}) &= \frac{\mu_0}{4\pi}\int_\mathcal{V}\frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}d^3r'\\
\mathbf{B}(\mathbf{r}) &= \frac{\mu_0}{4\pi}\int_\mathcal{V}\frac{\mathbf{J}(\mathbf{r}')\times(\mathbf{r}-\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|^3}d^3r'
\end{align}
Multipole expansion
We can obtain approximate solutions to the scalar and vector potentials using the following expansion
$$\frac{1}{|\mathbf{r}-\mathbf{r}'|} = \frac{1}{r}\sum_{n=0}^\infty\left(\frac{r'}{r}\right)^nP_n(\cos\alpha)$$
where $P_n$ are
Legendre polynomials and $\alpha$ is the angle between $\mathbf{r}$ and $\mathbf{r}'$.
The monopole and dipole terms are
Electric
The monopole term is
$$V_{mon}(\mathbf{r}) = \frac{1}{4\pi \epsilon_0}\frac{Q}{r}$$
The dipole term is
$$V_{dip}(\mathbf{r}) = \frac{1}{4\pi \epsilon_0}\frac{\mathbf{p}\cdot\mathbf{\hat{r}}}{r^2}$$
$$\mathbf{p} =\int_\mathcal{V}\mathbf{r}'\rho(\mathbf{r}')d^3r'$$
Magnetic
The magnetic monopole term is always zero.
The dipole term is
$$\mathbf{A}_{dip}(\mathbf{r}) = \frac{\mu_0}{4\pi}\frac{\mathbf{m}\times\mathbf{\hat{r}}}{r^2}$$
$$\mathbf{m} = \frac{1}{2}\int_\mathcal{V}\mathbf{r}\times\mathbf{J}d^3r$$
Energy & Momentum
The Poynting's theorem relates the rate at which work $W$ is done on the charges by the electromagnetic force to the decrease in the internal energy stored in the fields $u$, and
the energy flowing through the surface $\mathcal{S}$:
$$\frac{dW}{dt}=-\frac{d}{dt}\int_\mathcal{V} u\, d^3r' -\oint_\mathcal{S} \mathbf{S}\cdot d\mathbf{a},$$
where $$u = \frac{1}{2}\left(\epsilon_0 E^2 +\frac{1}{\mu_0}B^2\right)$$ is the energy density and
$$\mathbf{S}=\frac{1}{\mu_0}(\mathbf{E}\times\mathbf{B})$$ the Poynting vector.
The momentum stored inside the fields is $$\mathbf{P}=\mu_0\epsilon_0\int_\mathcal{V}\mathbf{S}\,d^3r'.$$
Waves
In a linear medium without free charges, Maxwell's equations reduce to two wave equations
$$ \nabla^2 \mathbf{E}=\mu\epsilon \frac{\partial^2\mathbf{E}}{\partial t^2} +\mu\sigma \frac{\partial\mathbf{E}}{\partial t}, \quad
\nabla^2 \mathbf{B}=\mu\epsilon \frac{\partial^2\mathbf{B}}{\partial t^2} +\mu\sigma \frac{\partial\mathbf{B}}{\partial t}$$
where the conductivity $\sigma$ is defined in terms of Ohm's law $\mathbf{J}_f=\sigma\mathbf{E}$.
These equations admit a plane wave solution of the form
$$ \mathbf{E}=\mathbf{E_0}e^{i(\mathbf{\tilde{k}}\cdot \mathbf{r}-\omega t)}, \quad \mathbf{B}=\mathbf{B_0}e^{i(\mathbf{\tilde{k}}\cdot \mathbf{r}-\omega t)}$$
where $\mathbf{B_0} = \frac{1}{\omega}\left(\mathbf{\tilde{k}}\times \mathbf{E_0}\right)$ and ${\tilde{k}}^2=\mu\epsilon\omega^2 +i\mu\sigma\omega$ is the complex wavenumber ($\tilde{k}=k+ik_2$).
We can also define a complex refractive index $\tilde{n}=n+in_2$ and a complex relative permittivity $\tilde{\epsilon}_r$ by $\tilde{n}^2= \frac{c^2}{\omega^2}\tilde{k}^2=\frac{\mu}{\mu_0}\tilde{\epsilon}_r$.
The speed of propagation of waves can be characterized with two quantities: The phase velocity $v_{phase}=\frac{c}{n} = \frac{\omega}{k}$
is the velocity at which the phase of any one frequency component of the wave travels. The group velocity $v_{group}=\frac{d\omega}{dk}$ is the speed of propagation of
the envelope of the wave, often, but not always, corresponding to the speed of energy or information propagation.
Fresnel's equations
A plane wave propagating through an interface will be partly reflected and partly transmitted. For a wave propagating from medium 1 with refractive index $\tilde{n}_1$
into medium 2 with refractive index $\tilde{n}_2$, the reflection coefficient, $r$, is the ratio of the reflected electric field amplitude to the incident amplitude. The transmission coefficient, $t$,
is the ratio of the transmitted amplitude to the incident amplitude.
S-polarization
\begin{align}
r_\mathrm{s} &= \frac{ \tilde{n}_1 \cos \theta_1 - \tilde{n}_2 \cos \theta_2}{\tilde{n}_1 \cos \theta_1 + \tilde{n}_2 \cos \theta_2} = t_\text{s} - 1, \\
t_\mathrm{s} &= \frac{2 \tilde{n}_1 \cos \theta_1} {\tilde{n}_1 \cos \theta_1 + \tilde{n}_2 \cos \theta_2} = r_\text{s} + 1.
\end{align}
P-polarization
\begin{align}
r_\mathrm{p} &= \frac{ \tilde{n}_2 \cos \theta_1 - \tilde{n}_1 \cos \theta_2}{\tilde{n}_1 \cos \theta_2 + \tilde{n}_2 \cos \theta_1}, \\
t_\mathrm{p} &= \frac{2 \tilde{n}_1 \cos \theta_1} {\tilde{n}_1 \cos \theta_2 + \tilde{n}_2 \cos \theta_1}.
\end{align}
Convention: for s-polarization, a positive $r$ means that the electric fields of the incoming and reflected waves are parallel,
while negative means anti-parallel. For p-polarization, a positive $r$ means that the magnetic fields of the waves are parallel.
It is also assumed that the magnetic permeability, $\mu$, of both media are approximately equal.
The power reflection and transmission coefficients are
$$R=|r|^2,\quad T = \frac{\operatorname{Re}(\tilde{n}_2 \cos \theta_2)}{\operatorname{Re}(\tilde{n}_1 \cos \theta_1)} |t|^2.$$
The incidence and transmittance angles are related by Snell's law:
$$n_1\sin\theta_1=n_2\sin\theta_2,$$
while the angle of incidence and the angle of reflection are equal: $\theta_i=\theta_r$.
Potentials
In general, the scalar and vector potentials are defined so that
$$\mathbf{E}=-\nabla V -\frac{\partial\mathbf{A}}{\partial t},$$ $$\mathbf{B}=\nabla \times \mathbf{A}.$$
Gauge transformation
The potentials can be transformed by an arbitrary scalar function $\lambda$ without affecting the physical quantities $\mathbf{E}$ and $\mathbf{B}$
\begin{align}
\mathbf{A}'&=\mathbf{A}+\nabla \lambda,\\
V'&=V-\frac{\partial\lambda}{\partial t}.
\end{align}
Coulomb Gauge
Definition:
$$\nabla\cdot \mathbf{A}=0$$
Maxwell's equations reduce to
$$\nabla^2 V = -\frac{1}{\epsilon_0}\rho$$
$$\nabla^2 \mathbf{A} -\mu_0\epsilon_0\frac{\partial^2\mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J}+\mu_0\epsilon_0\nabla\left(\frac{\partial V}{\partial t}\right)$$
Lorenz Gauge
Definition:
$$\nabla\cdot \mathbf{A}=-\mu_0\epsilon_0\frac{\partial V}{\partial t} $$
Maxwell's equations reduce to
$$\nabla^2 \mathbf{A} -\mu_0\epsilon_0\frac{\partial^2\mathbf{A}}{\partial t^2} = -\mu_0 \mathbf{J}$$
$$\nabla^2 V -\mu_0\epsilon_0\frac{\partial^2 V}{\partial t^2} = -\frac{1}{\epsilon_0}\rho$$
In the Lorenz gauge, the scalar and vector potentials can be solved as
$$V(\mathbf{r},t) = \frac{1}{4\pi \epsilon_0}\int_\mathcal{V}\frac{\rho(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|}d^3r',\qquad
\mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi}\int_\mathcal{V}\frac{\mathbf{J}(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|}d^3r'$$
where $t_r=t-\frac{|\mathbf{r}-\mathbf{r}'|}{c}$ is the retarded time.
Radiation
Radiation can be defined as the process in which energy is transported out to infinity. The power passing through the surface of sphere whose radius $r\to \infty$ is
$$P_{rad} = \lim_{r\to \infty} \oint \mathbf{S}\cdot d\mathbf{a}.$$
Electric dipole radiation
For a dipole moment $p(t)=p_0\cos(\omega t)$
$$P=\frac{\mu_0p_0^2\omega^4}{12 \pi c}$$
Arbitrary source
$$P=\frac{\mu_0\langle|\ddot{\mathbf{p}}|^2\rangle}{6 \pi c}$$
Magnetic dipole radiation
For a dipole moment $m(t)=m_0\cos(\omega t)$
$$P=\frac{\mu_0m_0^2\omega^4}{12 \pi c^3}$$
Arbitrary source
$$P=\frac{\mu_0\langle|\ddot{\mathbf{m}}|^2\rangle}{6 \pi c^3}$$
where the brackets denote time average. Here, the power $P$ is the power averaged over a period.
The power radiated by a point charge $q$ with acceleration $a$ is given by the Larmor formula
$$P=\frac{\mu_0q^2a^2}{6 \pi c}$$
which is valid for non-relativistic speeds. Liénard's generalization of the Larmor formula is valid for relativistic speeds:
$$P=\frac{\mu_0 q^2 \gamma^6}{6\pi c}\left(a^2-\left|\frac{\mathbf{v}\times\mathbf{a}}{c}\right|^2\right),$$
where $\mathbf{a}$ is the acceleration of the particle, $\mathbf{v}$ its velocity, $\gamma = \frac{1}{\sqrt{1-\beta^2}}$ and $\beta = \frac{v}{c}$.
Relativity
We can define a current density 4 vector as $J^\mu=(c\rho,J_x,J_y,J_z)$ and a 4 vector potential $A^{\mu}=(V/c,A_x,A_y,A_z)$.
With these, Maxwell equations can be combined into a single 4-vector equation (in the Lorenz gauge):
$$\Box^2 A^\mu =-\mu_0 J^\mu, \quad \Box^2 =\nabla^2 -\frac{1}{c^2}\frac{\partial^2}{\partial t^2}.$$
In terms of the field tensor, $F^{\mu\nu}=\frac{\partial A^\nu}{\partial x_\mu} - \frac{\partial A^\mu}{\partial x_\nu}$, and its dual
$G^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$, Maxwell equations can also be written as
$$\frac{\partial F^{\mu\nu}}{\partial x^\nu} = \mu_0 J^\mu, \quad \frac{\partial G^{\mu\nu}}{\partial x^\nu}=0.$$
The transformation of the electric and magnetic fields between inertial frames $S$ and $S'$ is
$$F'^{\mu\nu}=\Lambda_\lambda^\mu \Lambda_\sigma^\nu F^{\lambda \sigma}$$
where $\Lambda$ is the Lorentz transformation matrix. If frame $S'$ is moving in the $+x$ direction with speed $v$ relative to $S$,
$\Lambda$ has the form
\begin{bmatrix} \gamma & -\gamma \beta &0 &0 \\ -\gamma\beta & \gamma &0 &0 \\0 & 0 &1 &0 \\0 & 0&0 &1 \end{bmatrix}