Electrodynamics

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Open Source Code: Github

Introduction to electrodynamics - by David J. Griffiths
Classical Electrodynamics - by John David Jackson

*(If you find any errors in the code, please let me know on Github)

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Electrodynamics

Lorenz Gauge

\begin{equation} \nabla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2}= -\frac{\rho}{\epsilon_0} \end{equation} and \begin{equation} \nabla^2 \vec{A} - \frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2}= -\frac{\vec{J}}{\epsilon_0 c^2} \end{equation} with \begin{equation} \vec{\nabla} \cdot \vec{A} - \frac{1}{c^2} \frac{\partial \phi}{\partial t}= 0 \end{equation}

The covariant basis

For Polar \begin{equation} \vec{e_r}= \frac{\partial R(r,\theta)}{\partial r}\end{equation} \begin{equation} \vec{e_\theta}= \frac{\partial R(r,\theta)}{\partial \theta}\end{equation} Similarly, for cartesian \begin{equation} \vec{e_x}= \frac{\partial R(x,y)}{\partial x}\end{equation} \begin{equation} \vec{e_y}= \frac{\partial R(x,y)}{\partial y}\end{equation}

Covariant Electrodynamics

Conservation of local charge \begin{equation} \partial_\alpha J^\alpha =0 \end{equation} Lorenz Gauge condition, where $A^\alpha=(\phi / c , \vec{A})$ \begin{equation} \partial_\alpha A^\alpha =0 \end{equation} Fields $\vec{E}$ and $\vec{B}$ expressed in therms of the potentials \begin{equation} \vec{E} = - \frac{1}{c} \frac{\partial \vec{A}}{\partial t} - \nabla \phi \end{equation} \begin{equation} \vec{B} = \vec{\nabla} \times \vec{A} \end{equation} The Field Strength Tensor is an Antisymentric Second Rank Contravariant Tensor ($F^{\alpha \beta}$) \begin{equation} F^{\alpha \beta}= \partial^\alpha A^\beta - \partial^\beta A^\alpha = \end{equation} \[ \left( \begin{array}{ccc} 0 & -E_x /c & -E_y /c& -E_z /c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{array} \right)\] The dual Field Strength Tensor $\mathscr{F}^{\alpha \beta} =\frac{1}{2}\epsilon^{\alpha \beta \gamma \delta} F_{\gamma \delta}$ \[ \left( \begin{array}{ccc} 0 & -B_x & -B_y & -B_z \\ B_x & 0 & E_z/c & -E_y/c \\ B_y & -E_z/c & 0 & E_x/c \\ B_z & E_y/c & -E_x/c & 0 \end{array} \right)\] Maxwell's inhomogeneous equation \begin{equation} \partial_\alpha F^{\alpha \beta} = \mu_0 J^\beta \end{equation} Maxwell's homogeneous equation \begin{equation} \partial_\alpha \mathscr{F}^{\alpha \beta} = 0 \end{equation} Lorentz force and rate of change of energy equations \begin{equation} \frac{d p^\mu}{d\tau}=m \frac{d U^\mu}{d\tau}= q U_\gamma F^{\mu \gamma} \end{equation} Identity ?? \begin{equation} \partial_\gamma F_{\alpha \beta} = 2\partial_\gamma \partial_\alpha A_\beta \end{equation}

Lagrangian

The Lagrangian $\mathcal{L}$ \begin{equation} \mathcal{L} =T-V \end{equation} where T is the kinetic energy and V is the potential energy. Also \begin{equation} \frac{d \mathcal{L}}{d \Phi}= \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \Phi )} \right) \end{equation} for example (Euler-Lagrange equation of motion): \begin{equation} \frac{\partial \mathcal{L}}{\partial x}= \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot x} \right) \end{equation} Equations of motion \begin{equation} \frac{d\vec{p}}{dt}= q \left( \vec{E} + \vec{v} \times \vec{B} \right) \end{equation} \begin{equation} \frac{dE}{dt}= q \vec{v} \cdot \vec{E} \end{equation}

Dynamics of relativistic particles and fields

The action integral is \begin{equation} S=\int_a^b \mathcal{L} dt \end{equation} Hamilton's principle (principle of least action) is \begin{equation} \delta S= \delta \int_a^b \mathcal{L} dt =0 \end{equation} Euler-Lagrange equation of motion: \begin{equation} \frac{\partial \mathcal{L}}{\partial x^\gamma}= \frac{d}{d\tau} \left( \frac{\partial \mathcal{L}}{\partial U^\gamma} \right) \end{equation} The canonical momentum , conjugate momentum 4-vector is \begin{equation} P^\alpha=\frac{\partial \mathcal{L}}{\partial U_\alpha} \end{equation} A Hamiltonian can be defined by \begin{equation} \tilde{H}=P_\alpha U^\alpha + \tilde{\mathcal{L}} \end{equation} Equation of motion of a vector field \begin{equation} \partial_\beta F^{\alpha \beta} + \mu_\gamma^2 A^\alpha = \frac{k_2}{2 k_1} J^\alpha \end{equation}

Other formulas

Green's theorem (Divergence theorem) \begin{equation} \oint_{\partial D} \vec{A} \cdot d\vec{\Sigma} = \int_D (\nabla \cdot \vec{A}) dV \end{equation} \begin{equation} \oint F^{\alpha \beta} \hat{n}_\beta d\sigma = \int \partial_\beta F^{\alpha \beta} d^4x \ \ \ \ \ ?check\end{equation}

Definitions

In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and a Lorentz scalar.