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\title{EMT Quick Reference}
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\section*{Maxwell's Equations}
In Vacuum:
\begin{eqnarray*}
\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \\
\nabla \times \mathbf{B} = \frac{\partial \mathbf{E}}{c^2 \partial \mathbf{t}} + \mu_0 \mathbf{J} \\
\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} \\
\nabla \cdot \mathbf{B} = 0 \\
\end{eqnarray*}
 
In Material:
\begin{eqnarray*}
\nabla \cdot \mathbf{D} = \rho \\
\nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial \mathbf{t}} + \mathbf{J} \\
\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} \\
\nabla \cdot \mathbf{B} = 0 \\
\end{eqnarray*}

\section*{Useful Stuff}
\[\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \]
\[\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0 \]
\[\mathbf{S} = \mathbf{E} \times \mathbf{H}\]
\[\langle \mathbf{S} \rangle = 
\frac{1}{2} \mbox{Re}[\mathbf{E} \times \mathbf{H}^\ast]\]
\[Z_0=\sqrt{\frac{\mu_0}{\epsilon_0}}\] 
\section*{Materials}
\[\mathbf{D} = \epsilon_0 \mathbf{E}+\mathbf{P}\]
\[\mathbf{P} = \epsilon_0 \chi_0 \mathbf{E}\]
\[\mathbf{H} = \frac{\mathbf{B}}{\mu_0}-\mathbf{M}\]
The index of refraction is
\[n=\sqrt{\frac{\mu \epsilon}{\mu_0 \epsilon_0}}\]

\subsection*{Linear Response}
\[\mathbf{D} = \epsilon \mathbf{E}\]
\[\mathbf{B} = \mu \mathbf{H}\]
 
\subsection*{Boundary Conditions}

\[\mathbf{n} \cdot (\mathbf{D}_2 -\mathbf{D}_1) = \sigma\]
\[\mathbf{n} \cdot (\mathbf{B}_2 -\mathbf{B}_1) = 0\]
\[\mathbf{n} \times (\mathbf{E}_2 -\mathbf{E}_1) = 0\]
\[\mathbf{n} \times (\mathbf{H}_2 -\mathbf{H}_1) = \mathbf{K}\]

\section*{Potentials}

\begin{eqnarray*}
\mathbf{B} = \nabla \times \mathbf{A} \\
\mathbf{E} = -\nabla \Phi - \frac{\partial \mathbf{A}}{\partial t} \\
\nabla^2 \Phi = \frac{\rho}{\epsilon_0}
\end{eqnarray*}

\subsection*{Guage Transformation}
Coulomb Guage:
\[\nabla \cdot \mathbf{A} = 0\]

Lorentz Guage:
\[\mathbf{A} \rightarrow \mathbf{A} + \nabla \Lambda  \]
\[\Psi \rightarrow \Psi - \frac{\partial \Lambda}{\partial t}  \]
where:
\[\nabla^2 \Lambda - 
\frac{1}{c^2} \frac{\partial^2 \Lambda}{\partial t^2} = 0\]

\subsection*{Wave Equations}
%J. p. 240
\[\nabla^2 \Phi - \frac{1}{c^2} \frac{\partial^2 \Phi}{\partial t^2} 
= -\rho/\epsilon_0\]
\[\nabla^2 \mathbf{A} - \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}
{\partial t^2} = -\mu_0 \mathbf{J}\]

\section*{Green's Functions}

\[\nabla^2 \Psi - \frac{1}{c^2} \frac{\partial^2 \Psi}
{\partial t^2} = -4 \pi f(\mathbf{x},t)  \Rightarrow\]
\[\Psi(\mathbf{x},t) = 
\int \frac{[f(\mathbf{x}',t')]_{ret}}{|\mathbf{x}-\mathbf{x}'|} d^3 x'\]


\section*{Field Energy and Momentum}
\[u = \frac{1}{2} (\mathbf{E} \cdot \mathbf{D} + 
\mathbf{B} \cdot \mathbf{H})\]
\[\mathbf{S} = \mathbf{E} \times \mathbf{H}\]
\[\mathbf{g} = \mathbf{P}_{field} = 
\frac{1}{c^2}(\mathbf{E} \times \mathbf{H})  \]
For harmonic time dependance:
\[\langle \mathbf{S} \rangle =
\frac{1}{2} Re\left(\mathbf{E} \times \mathbf{H}^\star\right)\]
\[u = \frac{1}{2} Re\left(\frac{\epsilon}{2}\mathbf{E} \cdot \mathbf{E}^\star + 
\frac{\mu}{2}\mathbf{H} \cdot \mathbf{H}^\star\right)\]
 
\section*{Plane Waves}
\[\mathbf{E}=\mathbf{E}_0 e^{i\mathbf{k} \cdot \mathbf{r} -i\omega t}\]
\[\mathbf{B}=\sqrt{\mu \epsilon}\frac{\mathbf{k} \times \mathbf{E}_0}{k} 
e^{i\mathbf{k} \cdot \mathbf{r} -i\omega t}\]
In this case:
\[\frac{\partial}{\partial t} \rightarrow -i \omega\]
\[\nabla \rightarrow \mathbf{k}\]

\section*{Special Relativity}
\[\vec{\beta}= \frac{\vec{v}}{c},\, \gamma = \sqrt{1-\beta^2}\]
4-vector:
\[x^\nu = (ct,x^i) = (ct, x^1,x^2,x^3) \]
Lorentz Transformation:
\[x_\parallel' =\gamma(x_\parallel-v t)\]
\[ct' = \gamma (ct-\beta x_\parallel)\]
Addition of 4-Velocity:
\[u_\parallel = \frac{u_\parallel'+v}
{1+\frac{\mathbf{v} \cdot \mathbf{u}'}{c^2}}\]
\[\mathbf{u}_\perp = \frac{\mathbf{u}_\perp/\gamma_v}
{1+\frac{\mathbf{v} \cdot \mathbf{u}'}{c^2}}\]
\subsection*{Formalism}
A Lorentz transformation, $\Lambda_\rho^\nu$, obeys this equation
with the spacetime metric, $\eta_{\mu \nu}$:
\[\eta_{\mu \nu}\Lambda_\rho^\mu \Lambda_\sigma^\nu 
= \eta_{\rho \sigma}\]
For Scalar and Vector fields respectively:
\[\phi'(x')= \phi(x)=\phi(\Lambda^{-1}x')\]
\[J'^\mu (x') = \Lambda^\mu_\nu J^\nu (x) = 
\Lambda^\mu_\nu J^\nu (\Lambda^{-1} x')\]
For higher rank tensors, you just need another $\Lambda$ out front for
each index.
\subsection*{Maxwell's Equations}
\[A^\nu = (\Phi/c,\mathbf{A}^i)\]
\[E_k = c\partial_k A_0 - c\partial_0 A_k\]
\[\epsilon_{ijk} B^k = \partial_i A_j - \partial_j A_i\]
\[F_{\mu \nu}= \partial_\mu A_\nu - \partial_\nu A_\mu\]
\[F_{\mu \nu} = \left(
\begin{array}{cccc}
0 & -E_1 & -E_2 & -E_3 \\
E^1 & 0 & cB^3 & -cB^2 \\
E_2 & -cB^3 & 0 & cB^1 \\
E_3 & cB^2 & -B^1 & 0 
\end{array} \right)\]
\[E_\parallel' = E_\parallel, \, \, B_\parallel' = B_\parallel\]
\[E_\perp' = \gamma(E_\perp + v \times B)\]
\[B_\perp' = \gamma(B_\perp - \frac{v}{c^2} \times E)\]
Maxwell Dynamic Equation:
\[c\epsilon_0 \partial_\nu F^{\mu \nu}=J^\mu\]
Charge/current conservation:
\[\partial_\mu J^\mu = 0\]

\section*{Field Theory}
Scalar Field:
\[(-\partial^2 + \mu^2)\phi = 0\]
\[\mathcal{L} = -\partial_\mu \phi \partial^\mu \phi - \mu^2 \phi^2\]
General relations for field lagrangian densities:
\[\partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu \psi_i)}
= \frac{\partial \mathcal{L}}{\partial \psi_i}\]
\[T^{\mu \nu} = - \sum_i \partial^\mu \psi_i \frac{\partial \mathcal{L}}{\partial(\partial_\nu \psi_i)} + \eta^{\mu \nu} \mathcal{L}\]

\section*{Wave Guides}
This is an eigenvalue problem:
\[\nabla^2_\perp \psi = \gamma \psi\]
\[\gamma^2 = \omega^2 \mu \epsilon - k^2\]
Transverse Electric (TE) Mode:
\[\mathbf{B}_\perp = \frac{i k}{\gamma^2} \nabla_\perp B_z\]
\[\mathbf{\hat{z}} \times \mathbf{E}_\perp = 
\frac{i \omega}{\gamma^2} \nabla_\perp B_z\]
\[(-\nabla^2_\perp -\gamma^2)B_z=0\]
\[\hat{n} \cdot \nabla_\perp B_z \mbox{, at boundary}\]
Transverse Magnetic (TM) Mode:
\[\mathbf{B}_\perp = \frac{i \omega \mu \epsilon}{\gamma^2} 
\mathbf{\hat{z}} \times \nabla_\perp E_z\]
\[\mathbf{\hat{z}} \times \mathbf{E}_\perp
=\frac{i k}{k^2} \mathbf{\hat{z}} \times \nabla_\perp E_z\]
\[E_z=0 \mbox{, at boundary}\]
Transverse Electric and Magnetic (TEM) Mode:
\[\nabla^2 \phi = 0\]
\[\phi = \mbox{constant, at boundary}\]

\subsection*{Cavity Quality Factor}
\[Q = \omega \frac{\mbox{Eneregy Stored}}{\mbox{Power Loss}}\]

\subsection*{Driven Waveguides}
\[\mathbf{E} = \sum_\lambda (A^+_\lambda \mathbf{E}^+_\lambda +
A^-_\lambda \mathbf{E}^-_\lambda)\]
\[\mathbf{B} = \sum_\lambda (A^+_\lambda \mathbf{B}^+_\lambda +
A^-_\lambda \mathbf{B}^-_\lambda)\]

\[A^\pm_{\lambda R/L} = -\frac{Z_\lambda}{2} 
\int d^3x \mathbf{J} \cdot \mathbf{E}^{\mp}_\lambda\]
\[ = -\frac{Z_\lambda}{2} \int d^3x 
(\mathbf{J}_\perp \cdot \mathbf{E}_{\perp \lambda} \pm J_z E_{z \lambda})
e^{\mp ik_\lambda z}\]

\section*{Radiation}
The defining thing for radiation is fields falling off as $\frac{1}{r}$ as 
$r\rightarrow \infty$. The 4-potential is given in time 
and frequency space respectively, for large $r$, by:
\[A_\mu(\mathbf{r},t) \simeq \frac{\mu_0}{4 \pi r} \int d^3 r' 
J_\mu (\mathbf{r}',t-R/c)\]
\[A_\mu(\mathbf{r},\omega) \simeq \frac{\mu_0 e^{ikr}}{4 \pi r} \int d^3 r' 
J_\mu (\mathbf{r}',\omega) e^{-ik\mathbf{r} \cdot \mathbf{r}'/r}\]
\subsection*{Long Wavelength Approximation}
We can expand the exponential in $A_\mu(\mathbf{r},\omega)$:
\[e^{-i \mathbf{k} \cdot \mathbf{r}} = 1 - i\mathbf{k} \cdot \mathbf{r}
 - \frac{1}{2}(\mathbf{k} \cdot \mathbf{r})^2\]
These turn into electric dipole, magnetic dipole, and electric quodropole contributions:
\[A_j(\mathbf{r},\omega) \simeq \frac{\mu_0 e^{ikr}}{4 \pi r}(
-i\omega p_j + 
i(\mathbf{k} \times \mathbf{m})_j
-\frac{\omega}{6} Q^l_j k^l
)\]
For the electric dipole:
\[\frac{dP}{d\Omega}=\frac{Z_0 \omega^2}{32 \pi^2} (k^2 |\mathbf{p}|^2-
|\mathbf{k} \cdot \mathbf{p}|^2)\]
\[P = \frac{Z_0 k^4 c^2}{12 \pi} |\mathbf{p}|^2\]
\subsection*{Multipole Expansion}
Vector Spherical Harmonics:
\[\mathbf{X}_{lm} = \frac{\mathbf{L} Y_{lm}}{\sqrt{l(l+1)}}\]
We use the appropriate spherical bessel functions,
 generally denoted by $f_l$ and $g_l$,
combined with sperical bessel functions to expand solutions.
\[
\mathbf{H} = \sum_{lm} \left(a_{lm}^E f_l(kr) \mathbf{X}_{lm} - 
\frac{i}{k} a_{lm}^M \nabla \times (g_l(kr) \mathbf{X}_{lm})
\right)\]
\[
\mathbf{E} = Z_0 \sum_{lm} \left(\frac{i}{k}a_{lm}^E \nabla \times 
(f_l(kr) \mathbf{X}_{lm}) + 
a_{lm}^M g_l(kr) \mathbf{X}_{lm}
\right)\]

\section*{Scattering}
We look at the general scattering problem as an incident plane wave 
iteracting with something producing a spherical wave:
\[\mathbf{E} = \hat{\epsilon}_0 e^{ikz} + \mathbf{f} \frac{e^{ikr}}{r}\]
To find $\mathbf{f}$, the methods of the ``Radiation'' section are used, either 
approximation or expansion.

\subsection*{Optical Theorem}
\[\sigma_{\mbox{total}} = \frac{4 \pi}{k} Im(\hat{\epsilon}_0^\star \cdot 
\mathbf{f})|_{\mathbf{k}=\mathbf{k}_0}\]

\section*{References}
Almost all of this has been taken from Professor Charles Thorn's 
class lectures and notes during fall 2009-spring 2010.  Also, 
J. D. Jackson, \emph{Classical Electrodynamics}, 3rd Ed., 
John Wiley \& Sons, 1998.
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