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Liénard–Wiechert potentials

The Liénard–Wiechert potentials are scalar and vector potentials that allow determination of exact solutions of the Maxwell equations for the electric field and magnetic flux density generated by a moving ideal point charge.

Mathematical results

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

${\boldsymbol {\beta }}={\boldsymbol {v}}/c_{0}\ ,$

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, c₀ the speed of light in classical vacuum. Then the Liénard–Wiechert potentials consist of a scalar potential Φ and a vector potential A. The scalar potential is:

$\Phi ({\boldsymbol {r}},\ t)=\left.{\frac {q}{4\pi \varepsilon _{0}}}{\frac {1}{(1-\mathbf {{\hat {u}}\cdot } {\boldsymbol {v}})R}}\right|_{\tilde {t}}\ ,$

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

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

c₀ being the speed of light in classical vacuum, r the location of the observation point, and r_O being the location of the particle on its trajectory. The symbol ε₀ is the electric constant of the SI units.

The vector potential is:

${\boldsymbol {A}}({\boldsymbol {r}},\ t)=\left.{\frac {q\mu _{0}}{4\pi }}{\frac {\boldsymbol {v}}{(1-\mathbf {{\hat {u}}\cdot } {\boldsymbol {\beta }})R}}\right|_{\tilde {t}}\ .$

The symbol μ₀ is the magnetic constant of the SI units.With these potentials the electric field and the magnetic flux density are found to be (dots over symbols are time derivatives):

${\boldsymbol {E}}({\boldsymbol {r}},\ t)=-{\boldsymbol {\nabla }}\Phi -{\frac {\partial }{\partial t}}{\boldsymbol {A}}={\frac {q}{4\pi \varepsilon _{0}}}\left[{\frac {(\mathbf {\hat {u}} -{\boldsymbol {\beta }})(1-\beta ^{2})}{(1-\mathbf {\hat {u}} \mathbf {\cdot } {\boldsymbol {\beta }})^{3}R^{2}}}+{\frac {\mathbf {{\hat {u}}\ \mathbf {\times } \ } [({\hat {\mathbf {u} }}-{\boldsymbol {\beta }})\ \mathbf {\times } \ {\boldsymbol {\dot {\beta }}}]}{c_{0}(1-\mathbf {{\hat {u}}\cdot } {\boldsymbol {\beta }})^{3}R}}\right]_{\tilde {t}}$

${\boldsymbol {B}}({\boldsymbol {r}},\ t)={\boldsymbol {\nabla }}\ \mathbf {\times } \ {\boldsymbol {A}}={\boldsymbol {{\hat {u}}\ \times }}\ {\boldsymbol {E}}/{c_{0}}\ .$

If the particle does not accelerate, the first term alone survives and the result is the Biot-Savart law. If the particle accelerates, the last term is called the radiation field. The Biot-Savart term drops off more quickly with distance, and is called the near field term. The radiation field drops off more slowly with distance, so it dominates the result at large distances and is called the far field term.

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