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, r the location of the observation point, and r_O being the location of the particle on its trajectory.

The vector potential is:

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

With these potentials the electric field and the magnetic flux density are found to be (dots over symbols are time derivatives):

${\boldsymbol {E}}({\boldsymbol {r}},\ t)=q\left[{\frac {(\mathbf {\hat {u}} -{\boldsymbol {\beta }})(1-\beta ^{2})}{(1-\mathbf {\hat {u}} \mathbf {\cdot } {\boldsymbol {\beta }})^{3}R^{2}}}+{\frac {\mathbf {{\hat {u}}\ \mathbf {\times } \ } [({\hat {\mathbf {u} }}-{\boldsymbol {\beta }})\ \mathbf {\times } \ {\boldsymbol {\dot {\beta }}}]}{c(1-\mathbf {{\hat {u}}\cdot } {\boldsymbol {\beta }})^{3}R}}\right]_{\tilde {t}}$

${\boldsymbol {B}}({\boldsymbol {r}},\ t)={\boldsymbol {{\hat {u}}\ \times }}\ {\boldsymbol {E}}\ .$

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