Lienard-Wiechert Potentials

#Physics
The potentials for a point charge $\displaystyle q$ moving at velocity $\displaystyle \vec{v}$. All values like $\displaystyle \mathscr{r}$, $\displaystyle \vec{u}$, $\displaystyle v$, and $\displaystyle \vec{a}$ are evaluated at retarded time

Topics

Constant Velocity Lienard-Wiechert Potentials

$\displaystyle V(\vec{r},t)=k \frac{qc}{(\mathscr{r}c-\vec{\mathscr{r}}\cdot \vec{v})}$

$\displaystyle \mathscr{\vec{r}}$ is the separation vector for the charge at its retarded position to the field point $\displaystyle \vec{r}$

$\displaystyle \vec{A}(\vec{r},t)=\frac{\vec{v}}{c^{2}}V(\vec{r},t)$

$\displaystyle \mathscr{\vec{r}}=\vec{r}-\vec{w}(t_{r}),\lVert \mathscr{\vec{r}}\rVert=c(t-t_{r})$

The separation vector for moving charges, or the difference between the field point and where the charge once was
$\displaystyle \vec{r}$ is the position vector
$\displaystyle \vec{w}(t)$ is the position of the point charge at time $\displaystyle t$
$\displaystyle t_{r}$ is the retarded time
Use this equation to solve for $\displaystyle t_{r}$ and then plug it into the above equations

$\displaystyle \vec{E}(\vec{r},t)=k \frac{q \mathscr{r}}{(\mathscr{\vec{r}}\cdot \vec{u})^{3}}[(c^{2}-v^{2})\vec{u}+\mathscr{\vec{r}}\times (\vec{u}\times \vec{a})]$

The first term reduces to coloumb's law when there is minimal velocity and acceleration
The second term is the radiation field
$\displaystyle \vec{a}$ is the time rate of change of velocity $\displaystyle v$
$\displaystyle \vec{u}=c\mathscr{\hat{r}}-\vec{v}$