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 at an arbitrary location by an ideal point charge moving in a trajectory prescribed in advance (not calculated from any dynamical model). Because the trajectory is prescribed in advance, no account is taken of effects upon the charge motion due to its radiation of energy and momentum.

Point charges are an idealization, of course. The fields produced by charged bodies of non-zero extent, such as electrons, is more complicated. Among other matters, the dimensions of the moving charge are affected by its motion and potentially by its interaction with the fields it generates. Thus, dynamics intrudes once non-zero dimensions are introduced. For some specialized problems, quantum electrodynamics provides answers, but for a broader range of practical problems, non-relativistic electrodynamics is used.

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 for velocities much less than the speed of light the result is the Biot-Savart law. If the particle accelerates, the last term is called the radiation field, and is associated with the emission of energy and momentum by the moving charge. 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.