User:John R. Brews/Sample

Liénard–Wiechert potentials

Define β in terms of the velocity of a point charge at time t as:

${\boldsymbol {\beta }}={\boldsymbol {v}}/c\ ,$

and unit vector û as

$\mathbf {\hat {u}} ={\frac {\boldsymbol {R}}{R}}\ ,$

where R is the vector joining the observation point P to the moving charge q at the time of observation. Then the Liénard–Wiechert potentials consist of a scalar potential Φ and a vector potential A. The scalar potential is:

$\Phi ({\boldsymbol {r}},\ t)=\left.{\frac {q}{(1-\mathbf {{\hat {u}}\cdot } {\boldsymbol {\beta }})|{\boldsymbol {r}}-{\boldsymbol {\tilde {r}}}|}}\right|_{\tilde {t}}=\left.{\frac {q}{(1-\mathbf {{\hat {u}}\cdot } {\boldsymbol {\beta }})R}}\right|_{\tilde {t}}\ ,$

where the tilde ‘ ^~ ’ denotes evaluation at the retarded time ,

${\tilde {t}}=t-{\frac {{\boldsymbol {r}}-{\boldsymbol {r}}_{0}({\tilde {t}})|}{c}}\ ,$

c being the speed of light and r_O being the location of the particle on its trajectory.

The vector potential is:

${\boldsymbol {A}}({\boldsymbol {r}},\ t)=\left.{\frac {q{\boldsymbol {\beta }}}{(1-\mathbf {{\hat {u}}\cdot } {\boldsymbol {\beta }})|{\boldsymbol {r}}-{\boldsymbol {\tilde {r}}}|}}\right|_{\tilde {t}}=\left.{\frac {q{\boldsymbol {\beta }}}{(1-\mathbf {{\hat {u}}\cdot } {\boldsymbol {\beta }})R}}\right|_{\tilde {t}}\ .$

Notes

Feynman Belušević Gould Schwartz Schwartz Oughstun Eichler Müller-Kirsten Panat Palit Camara Smith classical distributed charge Florian Scheck Radiation reaction Fulvio Melia Radiative reaction Fulvio Melia Barut Radiative reaction Distributed charges: history Lorentz-Dirac equation Gould Fourier space description