Liénard-Wiechert potential

The Liénard-Wiechert potentials ( Emil Wiechert introduced the Liénard-Wiechert potentials of a moving charge, named after both of them , independently of Alfred-Marie Liénard (1898) in an article in 1900) describe the electric and magnetic fields generated by a moving electrical point charge be generated. They generalize the Coulomb potential , which is generated by a point charge at rest and has no magnetic component, and represent an approximation of the potential that would result from the Doppler effect at high energies.

The scalar Liénard-Wiechert potential is a modified Coulomb potential. The vector potential , which contains the information about the magnetic field, is essentially the scalar potential multiplied by the particle speed.

There are the following differences compared to the Coulomb potential:

• The fields that are observed at the point in time are generated by the particle at a previous (retarded) point in time . The difference is equal to the transit time from the particle to the observer with the speed of light . ${\displaystyle t}$ ${\displaystyle t_{\mathrm {ret} }}$ ${\displaystyle t-t_{\mathrm {ret} }}$
• There is an amplification factor when the particle moves towards the observer (attenuation factor when it moves away). The gain factor approaches infinity when the particle speed goes against the speed of light.

The electrical and magnetic field strengths can be obtained from the potentials by deriving them according to space and time coordinates (see also potentials and wave equations in electrodynamics ). The field strengths are divided into a speed part and an acceleration part. The part that only contains the particle velocity is strong in the vicinity of the particle, but weak at a greater distance (no far field ). The proportion proportional to the acceleration leads to the emission of energy into infinity.

The formulas

The location of the particle is considered to be a given function . How the trajectory comes about (e.g. through electromagnetic fields that exert forces on the particle) is not taken into account. The speed of the particle is calculated using the time derivative of the function . One of the following practical quantities is this speed divided by the speed of light: ${\displaystyle {\vec {R}}(t)}$ ${\displaystyle {\vec {R}}(t)}$

${\displaystyle {\vec {\beta }}:={\frac {\dot {\vec {R}}}{c}}}$

In the international system of units, the Liénard-Wiechert potentials are (according to Nolting, but formulated for fields in material-free space)

${\displaystyle \Phi ({\vec {r}},t)={\frac {1}{4\pi \varepsilon _{0}}}\left[{\frac {q}{(1-{\vec {\beta }}\cdot {\vec {n}})\;|{\vec {r}}-{\vec {R}}|}}\right]_{\mathrm {ret} }\qquad \qquad {\vec {A}}({\vec {r}},t)={\frac {{\vec {\beta }}_{\mathrm {ret} }}{c}}\,\Phi ({\vec {r}},t)}$

The index “ret” means that the particle position and velocity are to be taken at the retarded point in time. The implicit equation applies to the retarded point in time

${\displaystyle t_{\mathrm {ret} }=t-{\frac {1}{c}}|{\vec {r}}-{\vec {R}}(t_{\mathrm {ret} })|}$

Apart from the special case of uniform movement, the resolution according to is often only possible approximately. ${\displaystyle t_{\mathrm {ret} }}$

The vector is the unit vector that points from the position of the particle to the position vector . The following applies: ${\displaystyle {\vec {n}}}$ ${\displaystyle \,{\vec {r}}\,}$

${\displaystyle {\vec {n}}={\frac {{\vec {r}}-{\vec {R}}}{|{\vec {r}}-{\vec {R}}|}}}$

Applications

Synchrotron radiation

The particle moves on a circular path at a speed close to the speed of light . The speed-dependent factor then assumes a high peak value with each revolution. Because if the tangential direction of the velocity coincides with the direction to the observer, i.e. H. if is parallel to , then applies with ${\displaystyle v}$ ${\displaystyle c}$ ${\displaystyle {\dot {\vec {R}}}}$ ${\displaystyle {\vec {n}}}$ ${\displaystyle v\approx c}$

${\displaystyle {\frac {1}{1-{\textstyle {\frac {1}{c}}}{\dot {\vec {R}}}\cdot {\vec {n}}}}={\frac {1}{1-{\frac {v}{c}}}}\approx {\frac {1}{1-{\frac {v}{c}}}}\,{\frac {2}{1+{\frac {v}{c}}}}={\frac {2}{1-{\frac {v^{2}}{c^{2}}}}}=2\gamma ^{2}}$

where denotes the Lorentz factor . The potentials and field strengths are thus proportional to . Because the field strengths enter the radiant energy as a square (see Poynting vector ), the energy of the synchrotron radiation is proportional to . ${\displaystyle \gamma }$ ${\displaystyle \gamma ^{2}}$ ${\displaystyle \gamma ^{4}}$

Accelerated particle at low speed at a great distance

For example, you have a low speed at the beginning of an acceleration process. Long distances are the area that is relevant for electromagnetic radiation. With this specialization, the expressions for the electric and magnetic field strength are simplified (see Nolting in Limes ). The following applies to the magnetic field ${\displaystyle \beta =0}$

${\displaystyle {\vec {B}}={\frac {\mu _{0}q}{4\pi c|{\vec {r}}-{\vec {R}}_{\mathrm {ret} }|}}\left({\ddot {\vec {R}}}\times {\vec {n}}\right)_{\mathrm {ret} }}$

The electric field strength follows from this with a general relation for fields in the far zone

${\displaystyle {\vec {E}}=c{\vec {B}}\times {\vec {n}}}$

The energy flux density ( Poynting vector ) , which goes into infinity, is the same in terms of amount ${\displaystyle r}$

${\displaystyle S={\frac {1}{\mu _{0}}}|{\vec {E}}\times {\vec {B}}|={\frac {\mu _{0}q^{2}}{16\pi ^{2}cr^{2}}}\,\left|{\ddot {\vec {R}}}\right|^{2}\,\sin ^{2}\!\vartheta }$

where is the angle between the acceleration vector and the viewing direction. The energy flow per solid angle is obtained by omitting the in the denominator. ${\displaystyle \vartheta \!\,}$ ${\displaystyle r^{2}}$

Derivation of the B-field: The particle speed should be small compared to the speed of light, so that all terms that contain a factor in the result can be neglected . If derivatives act on the retarded time, the particle location does not need to be differentiated. This applies approximately ${\displaystyle {\dot {R}}/c}$ ${\displaystyle x,y,z,t}$

${\displaystyle {\frac {\partial }{\partial t}}t_{\mathrm {ret} }=1\qquad \qquad \operatorname {grad} \,t_{\mathrm {ret} }=-{\frac {1}{c}}\operatorname {grad} \,|{\vec {r}}-{\vec {R}}_{\mathrm {ret} }|=-{\frac {\vec {n}}{c}}}$

The factor must not stop when the vector potential is derived ; so only this factor has to be differentiated. For the magnetic field you get this ${\displaystyle {\dot {R}}/c}$

${\displaystyle {\vec {B}}=\operatorname {rot} \,{\vec {A}}={\frac {\Phi }{c^{2}}}\,\operatorname {rot} \,{\dot {\vec {R}}}(t_{\mathrm {ret} })={\frac {\Phi }{c^{2}}}\,\operatorname {grad} \,t_{\mathrm {ret} }\times {\ddot {\vec {R}}}=-{\frac {\mu _{0}q}{4\pi c|{\vec {r}}-{\vec {R}}_{\mathrm {ret} }|}}\,{\vec {n}}\times {\ddot {\vec {R}}}}$

using a chain rule for the rotation . In addition, was used. ${\displaystyle c^{2}=1/\varepsilon _{0}\mu _{0}}$

Individual evidence

1. ↑ ^{a b c} W. Nolting , Basic Course Theoretical Physics , Volume 3 Electrodynamics , 8th edition, Springer 2007, Section 4.5.5
2. ↑ JD Jackson , Classical Electrodynamics , 4th Edition, de Gruyter 2006, Section 9.8. There it is shown that the relation holds for all multipole orders. Caution: The printout for on page 472 contains two typographical errors; is correct for the vacuum, see section "field wave resistance" in the main article wave impedance ${\displaystyle Z_{0}}$ ${\displaystyle Z_{0}={\sqrt {\mu _{0}/\varepsilon _{0}}}}$