Lienard-Wiechert Potentials

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

Topics

• Constant Velocity Lienard-Wiechert Potentials

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

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

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

$\vec{r}=\vec{r}-\vec{w}(t_r),\Vert \vec{r}\Vert =c(t-t_r)$

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

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

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

$\vec{B}(\vec{r},t)=\frac{1}{c}\hat{r}\times \vec{E}(\vec{r},t)$