Liénard–Wiechert potential






























Part of a series of articles about
Electromagnetism
Solenoid


  • Electricity

  • Magnetism






Electrodynamics



  • Lorentz force law

  • Electromagnetic induction

  • Faraday's law

  • Lenz's law

  • Displacement current

  • Magnetic potential

  • Maxwell's equations

  • Electromagnetic field

  • Electromagnetic pulse

  • Electromagnetic radiation

  • Maxwell tensor

  • Poynting vector

  • Liénard–Wiechert potential

  • Jefimenko's equations

  • Eddy current

  • London equations


  • Mathematical descriptions of the electromagnetic field







  • v

  • t

  • e


The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Built directly from Maxwell's equations, these potentials describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum-mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials. These expressions were developed in part by Alfred-Marie Liénard in 1898[1] and independently by Emil Wiechert in 1900.[2][3]




Contents






  • 1 Equations


    • 1.1 Definition of Liénard–Wiechert potentials


    • 1.2 Corresponding values of electric and magnetic fields




  • 2 Derivation


    • 2.1 Lorenz gauge, electric and magnetic fields




  • 3 Implications


  • 4 Universal speed limit


    • 4.1 Existence and uniqueness of the retarded time


      • 4.1.1 Existence


      • 4.1.2 Uniqueness






  • 5 See also


  • 6 References





Equations



Definition of Liénard–Wiechert potentials


The Liénard–Wiechert potentials φ{displaystyle varphi }varphi (scalar potential field) and A{displaystyle mathbf {A} }mathbf {A} (vector potential field) are for a source point charge q{displaystyle q}q at position rs{displaystyle mathbf {r} _{s}}{mathbf {r}}_{s} traveling with velocity vs{displaystyle mathbf {v} _{s}}{mathbf {v}}_{s}:


φ(r,t)=14πϵ0(q(1−n⋅βs)|r−rs|)tr{displaystyle varphi (mathbf {r} ,t)={frac {1}{4pi epsilon _{0}}}left({frac {q}{(1-mathbf {n} cdot {boldsymbol {beta }}_{s})|mathbf {r} -mathbf {r} _{s}|}}right)_{t_{r}}}varphi ({mathbf {r}},t)={frac {1}{4pi epsilon _{0}}}left({frac {q}{(1-{mathbf {n}}cdot {boldsymbol {beta }}_{s})|{mathbf {r}}-{mathbf {r}}_{s}|}}right)_{{t_{r}}}

and


A(r,t)=μ0c4π(qβs(1−n⋅βs)|r−rs|)tr=βs(tr)cφ(r,t){displaystyle mathbf {A} (mathbf {r} ,t)={frac {mu _{0}c}{4pi }}left({frac {q{boldsymbol {beta }}_{s}}{(1-mathbf {n} cdot {boldsymbol {beta }}_{s})|mathbf {r} -mathbf {r} _{s}|}}right)_{t_{r}}={frac {{boldsymbol {beta }}_{s}(t_{r})}{c}}varphi (mathbf {r} ,t)}{mathbf {A}}({mathbf {r}},t)={frac {mu _{0}c}{4pi }}left({frac {q{boldsymbol {beta }}_{s}}{(1-{mathbf {n}}cdot {boldsymbol {beta }}_{s})|{mathbf {r}}-{mathbf {r}}_{s}|}}right)_{{t_{r}}}={frac {{boldsymbol {beta }}_{s}(t_{r})}{c}}varphi ({mathbf {r}},t)

where:


βs(t)=vs(t)c{displaystyle {boldsymbol {beta }}_{s}(t)={frac {mathbf {v} _{s}(t)}{c}}}{boldsymbol {beta }}_{s}(t)={frac {{mathbf {v}}_{s}(t)}{c}} is the velocity of the source expressed as a fraction of the speed of light


|r−rs|{displaystyle {|mathbf {r} -mathbf {r} _{s}|}}{displaystyle {|mathbf {r} -mathbf {r} _{s}|}} is the distance from the source


and
n=r−rs|r−rs|{displaystyle mathbf {n} ={frac {mathbf {r} -mathbf {r} _{s}}{|mathbf {r} -mathbf {r} _{s}|}}}{mathbf {n}}={frac {{mathbf {r}}-{mathbf {r}}_{s}}{|{mathbf {r}}-{mathbf {r}}_{s}|}} is the unit vector pointing in the direction from the source.


(see also retarded potential.)



Corresponding values of electric and magnetic fields


We can calculate the electric and magnetic fields directly from the potentials using the definitions:



E=−∇φ−∂A∂t{displaystyle mathbf {E} =-nabla varphi -{dfrac {partial mathbf {A} }{partial t}}}{mathbf {E}}=-nabla varphi -{dfrac {partial {mathbf {A}}}{partial t}} and B=∇×A{displaystyle mathbf {B} =nabla times mathbf {A} }mathbf {B} =nabla times mathbf {A}

The calculation is nontrivial and requires a number of steps. The electric and magnetic fields are (in non-covariant form):


E(r,t)=14πε0(q(n−β)γ2(1−n⋅β)3|r−rs|2+qn×((n−β)×β˙)c(1−n⋅β)3|r−rs|)tr{displaystyle mathbf {E} (mathbf {r} ,t)={frac {1}{4pi varepsilon _{0}}}left({frac {q(mathbf {n} -{boldsymbol {beta }})}{gamma ^{2}(1-mathbf {n} cdot {boldsymbol {beta }})^{3}|mathbf {r} -mathbf {r} _{s}|^{2}}}+{frac {qmathbf {n} times {big (}(mathbf {n} -{boldsymbol {beta }})times {dot {boldsymbol {beta }}}{big )}}{c(1-mathbf {n} cdot {boldsymbol {beta }})^{3}|mathbf {r} -mathbf {r} _{s}|}}right)_{t_{r}}}{displaystyle mathbf {E} (mathbf {r} ,t)={frac {1}{4pi varepsilon _{0}}}left({frac {q(mathbf {n} -{boldsymbol {beta }})}{gamma ^{2}(1-mathbf {n} cdot {boldsymbol {beta }})^{3}|mathbf {r} -mathbf {r} _{s}|^{2}}}+{frac {qmathbf {n} times {big (}(mathbf {n} -{boldsymbol {beta }})times {dot {boldsymbol {beta }}}{big )}}{c(1-mathbf {n} cdot {boldsymbol {beta }})^{3}|mathbf {r} -mathbf {r} _{s}|}}right)_{t_{r}}}

and


B(r,t)=μ04π(qc(β×n)γ2(1−n⋅β)3|r−rs|2+qn×(n×((n−β)×β˙))(1−n⋅β)3|r−rs|)tr=n(tr)c×E(r,t){displaystyle mathbf {B} (mathbf {r} ,t)={frac {mu _{0}}{4pi }}left({frac {qc({boldsymbol {beta }}times mathbf {n} )}{gamma ^{2}(1-mathbf {n} cdot {boldsymbol {beta }})^{3}|mathbf {r} -mathbf {r} _{s}|^{2}}}+{frac {qmathbf {n} times {Big (}mathbf {n} times {big (}(mathbf {n} -{boldsymbol {beta }})times {dot {boldsymbol {beta }}}{big )}{Big )}}{(1-mathbf {n} cdot {boldsymbol {beta }})^{3}|mathbf {r} -mathbf {r} _{s}|}}right)_{t_{r}}={frac {mathbf {n} (t_{r})}{c}}times mathbf {E} (mathbf {r} ,t)}{mathbf {B}}({mathbf {r}},t)={frac {mu _{0}}{4pi }}left({frac {qc({boldsymbol {beta }}times {mathbf {n}})}{gamma ^{2}(1-{mathbf {n}}cdot {boldsymbol {beta }})^{3}|{mathbf {r}}-{mathbf {r}}_{s}|^{2}}}+{frac {q{mathbf {n}}times {Big (}{mathbf {n}}times {big (}({mathbf {n}}-{boldsymbol {beta }})times {dot {{boldsymbol {beta }}}}{big )}{Big )}}{(1-{mathbf {n}}cdot {boldsymbol {beta }})^{3}|{mathbf {r}}-{mathbf {r}}_{s}|}}right)_{{t_{r}}}={frac {{mathbf {n}}(t_{r})}{c}}times {mathbf {E}}({mathbf {r}},t)

where β(t)=vs(t)c{displaystyle {boldsymbol {beta }}(t)={frac {mathbf {v} _{s}(t)}{c}}}{boldsymbol {beta }}(t)={frac {{mathbf {v}}_{s}(t)}{c}}, n(t)=r−rs(t)|r−rs(t)|{displaystyle mathbf {n} (t)={frac {mathbf {r} -mathbf {r} _{s}(t)}{|mathbf {r} -mathbf {r} _{s}(t)|}}}{mathbf {n}}(t)={frac {{mathbf {r}}-{mathbf {r}}_{s}(t)}{|{mathbf {r}}-{mathbf {r}}_{s}(t)|}} and γ(t)=11−|β(t)|2{displaystyle gamma (t)={frac {1}{sqrt {1-|{boldsymbol {beta }}(t)|^{2}}}}}gamma (t)={frac {1}{{sqrt {1-|{boldsymbol {beta }}(t)|^{2}}}}} (the Lorentz factor).


Note that the n−β{displaystyle mathbf {n} -{boldsymbol {beta }}}{mathbf {n}}-{boldsymbol {beta }} part of the first term updates the direction of the field toward the instantaneous position of the charge, if it continues to move with constant velocity cβ{displaystyle c{boldsymbol {beta }}}c{boldsymbol {beta }}. This term is connected with the "static" part of the electromagnetic field of the charge.


The second term, which is connected with electromagnetic radiation by the moving charge, requires charge acceleration β˙{displaystyle {dot {boldsymbol {beta }}}}{dot {boldsymbol {beta }}} and if this is zero, the value of this term is zero, and the charge does not radiate (emit electromagnetic radiation). This term requires additionally that a component of the charge acceleration be in a direction transverse to the line which connects the charge q{displaystyle q}q and the observer of the field E(r,t){displaystyle mathbf {E} (mathbf {r} ,t)}{mathbf {E}}({mathbf {r}},t). The direction of the field associated with this radiative term is toward the fully time-retarded position of the charge (i.e. where the charge was when it was accelerated).



Derivation


In the case that there are no boundaries surrounding the sources, the retarded solutions for the scalar and vector potentials (SI units) of the nonhomogeneous wave equations with sources given by the charge and current densities ρ(r,t){displaystyle rho (mathbf {r} ,t)}rho ({mathbf {r}},t) and J(r,t){displaystyle mathbf {J} (mathbf {r} ,t)}{mathbf {J}}({mathbf {r}},t) are in the Lorenz gauge (see Nonhomogeneous electromagnetic wave equation)


φ(r,t)=14πϵ0∫ρ(r′,tr′)|r−r′|d3r′{displaystyle varphi (mathbf {r} ,t)={frac {1}{4pi epsilon _{0}}}int {frac {rho (mathbf {r} ',t_{r}')}{|mathbf {r} -mathbf {r} '|}}d^{3}mathbf {r} '}varphi ({mathbf {r}},t)={frac {1}{4pi epsilon _{0}}}int {frac {rho ({mathbf {r}}',t_{r}')}{|{mathbf {r}}-{mathbf {r}}'|}}d^{3}{mathbf {r}}'

and


A(r,t)=μ04π∫J(r′,tr′)|r−r′|d3r′{displaystyle mathbf {A} (mathbf {r} ,t)={frac {mu _{0}}{4pi }}int {frac {mathbf {J} (mathbf {r} ',t_{r}')}{|mathbf {r} -mathbf {r} '|}}d^{3}mathbf {r} '}{mathbf {A}}({mathbf {r}},t)={frac {mu _{0}}{4pi }}int {frac {{mathbf {J}}({mathbf {r}}',t_{r}')}{|{mathbf {r}}-{mathbf {r}}'|}}d^{3}{mathbf {r}}'

where tr′=t−1c|r−r′|{displaystyle t_{r}'=t-{frac {1}{c}}|mathbf {r} -mathbf {r} '|}t_{r}'=t-{frac {1}{c}}|{mathbf {r}}-{mathbf {r}}'| is the retarded time.


For a moving point charge whose trajectory is given as a function of time by rs(t′){displaystyle mathbf {r} _{s}(t')}{mathbf {r}}_{s}(t'), the charge and current densities are as follows:



ρ(r′,t′)=qδ3(r′−rs(t′)){displaystyle rho (mathbf {r} ',t')=qdelta ^{3}(mathbf {r'} -mathbf {r} _{s}(t'))}<br /> rho(mathbf{r}', t') = q delta^3(mathbf{r'} - mathbf{r}_s(t'))<br />

J(r′,t′)=qvs(t′)δ3(r′−rs(t′)){displaystyle mathbf {J} (mathbf {r} ',t')=qmathbf {v} _{s}(t')delta ^{3}(mathbf {r'} -mathbf {r} _{s}(t'))}{mathbf {J}}({mathbf {r}}',t')=q{mathbf {v}}_{s}(t')delta ^{3}({mathbf {r'}}-{mathbf {r}}_{s}(t'))


where δ3{displaystyle delta ^{3}}delta ^{3} is the three-dimensional Dirac delta function and vs(t′){displaystyle mathbf {v} _{s}(t')}{mathbf {v}}_{s}(t') is the velocity of the point charge.


Substituting into the expressions for the potential gives



φ(r,t)=14πϵ0∫qδ3(r′−rs(tr′))|r−r′|d3r′{displaystyle varphi (mathbf {r} ,t)={frac {1}{4pi epsilon _{0}}}int {frac {qdelta ^{3}(mathbf {r'} -mathbf {r} _{s}(t_{r}'))}{|mathbf {r} -mathbf {r} '|}}d^{3}mathbf {r} '}varphi ({mathbf {r}},t)={frac {1}{4pi epsilon _{0}}}int {frac {qdelta ^{3}({mathbf {r'}}-{mathbf {r}}_{s}(t_{r}'))}{|{mathbf {r}}-{mathbf {r}}'|}}d^{3}{mathbf {r}}'

A(r,t)=μ04π∫qvs(tr′)δ3(r′−rs(tr′))|r−r′|d3r′{displaystyle mathbf {A} (mathbf {r} ,t)={frac {mu _{0}}{4pi }}int {frac {qmathbf {v} _{s}(t_{r}')delta ^{3}(mathbf {r'} -mathbf {r} _{s}(t_{r}'))}{|mathbf {r} -mathbf {r} '|}}d^{3}mathbf {r} '}{mathbf {A}}({mathbf {r}},t)={frac {mu _{0}}{4pi }}int {frac {q{mathbf {v}}_{s}(t_{r}')delta ^{3}({mathbf {r'}}-{mathbf {r}}_{s}(t_{r}'))}{|{mathbf {r}}-{mathbf {r}}'|}}d^{3}{mathbf {r}}'


These integrals are difficult to evaluate in their present form, so we will rewrite them by replacing tr′{displaystyle t_{r}'}t_{r}' with t′{displaystyle t'}t' and integrating over the delta distribution δ(t′−tr′){displaystyle delta (t'-t_{r}')}delta (t'-t_{r}'):



φ(r,t)=14πϵ0∬qδ3(r′−rs(t′))|r−r′|δ(t′−tr′)dt′d3r′{displaystyle varphi (mathbf {r} ,t)={frac {1}{4pi epsilon _{0}}}iint {frac {qdelta ^{3}(mathbf {r'} -mathbf {r} _{s}(t'))}{|mathbf {r} -mathbf {r} '|}}delta (t'-t_{r}'),dt',d^{3}mathbf {r} '}varphi ({mathbf {r}},t)={frac {1}{4pi epsilon _{0}}}iint {frac {qdelta ^{3}({mathbf {r'}}-{mathbf {r}}_{s}(t'))}{|{mathbf {r}}-{mathbf {r}}'|}}delta (t'-t_{r}'),dt',d^{3}{mathbf {r}}'

A(r,t)=μ04π∬qvs(t′)δ3(r′−rs(t′))|r−r′|δ(t′−tr′)dt′d3r′{displaystyle mathbf {A} (mathbf {r} ,t)={frac {mu _{0}}{4pi }}iint {frac {qmathbf {v} _{s}(t')delta ^{3}(mathbf {r'} -mathbf {r} _{s}(t'))}{|mathbf {r} -mathbf {r} '|}}delta (t'-t_{r}'),dt',d^{3}mathbf {r} '}{mathbf {A}}({mathbf {r}},t)={frac {mu _{0}}{4pi }}iint {frac {q{mathbf {v}}_{s}(t')delta ^{3}({mathbf {r'}}-{mathbf {r}}_{s}(t'))}{|{mathbf {r}}-{mathbf {r}}'|}}delta (t'-t_{r}'),dt',d^{3}{mathbf {r}}'


We exchange the order of integration:



φ(r,t)=14πϵ0∬δ(t′−tr′)|r−r′|qδ3(r′−rs(t′))d3r′dt′{displaystyle varphi (mathbf {r} ,t)={frac {1}{4pi epsilon _{0}}}iint {frac {delta (t'-t_{r}')}{|mathbf {r} -mathbf {r} '|}}qdelta ^{3}(mathbf {r'} -mathbf {r} _{s}(t')),d^{3}mathbf {r} 'dt'}varphi ({mathbf {r}},t)={frac {1}{4pi epsilon _{0}}}iint {frac {delta (t'-t_{r}')}{|{mathbf {r}}-{mathbf {r}}'|}}qdelta ^{3}({mathbf {r'}}-{mathbf {r}}_{s}(t')),d^{3}{mathbf {r}}'dt'

A(r,t)=μ04π∬δ(t′−tr′)|r−r′|qvs(t′)δ3(r′−rs(t′))d3r′dt′{displaystyle mathbf {A} (mathbf {r} ,t)={frac {mu _{0}}{4pi }}iint {frac {delta (t'-t_{r}')}{|mathbf {r} -mathbf {r} '|}}qmathbf {v} _{s}(t')delta ^{3}(mathbf {r'} -mathbf {r} _{s}(t')),d^{3}mathbf {r} 'dt'}{mathbf {A}}({mathbf {r}},t)={frac {mu _{0}}{4pi }}iint {frac {delta (t'-t_{r}')}{|{mathbf {r}}-{mathbf {r}}'|}}q{mathbf {v}}_{s}(t')delta ^{3}({mathbf {r'}}-{mathbf {r}}_{s}(t')),d^{3}{mathbf {r}}'dt'


The delta function picks out r′=rs(t′){displaystyle mathbf {r} '=mathbf {r} _{s}(t')}{mathbf {r}}'={mathbf {r}}_{s}(t') which allows us to perform the inner integration with ease. Note that tr′{displaystyle t_{r}'}t_{r}' is a function of r′{displaystyle mathbf {r} '}mathbf {r} ', so this integration also fixes tr=tr(rs(t′),t′){displaystyle t_{r}=t_{r}(mathbf {r} _{s}(t'),t')}t_{r}=t_{r}({mathbf {r}}_{s}(t'),t').



φ(r,t)=14πϵ0∫qδ(t′−tr′)|r−rs(t′)|dt′{displaystyle varphi (mathbf {r} ,t)={frac {1}{4pi epsilon _{0}}}int q{frac {delta (t'-t_{r}')}{|mathbf {r} -mathbf {r} _{s}(t')|}}dt'}varphi ({mathbf {r}},t)={frac {1}{4pi epsilon _{0}}}int q{frac {delta (t'-t_{r}')}{|{mathbf {r}}-{mathbf {r}}_{s}(t')|}}dt'

A(r,t)=μ04π∫qvs(t′)δ(t′−tr′)|r−rs(t′)|dt′{displaystyle mathbf {A} (mathbf {r} ,t)={frac {mu _{0}}{4pi }}int qmathbf {v} _{s}(t'){frac {delta (t'-t_{r}')}{|mathbf {r} -mathbf {r} _{s}(t')|}},dt'}{mathbf {A}}({mathbf {r}},t)={frac {mu _{0}}{4pi }}int q{mathbf {v}}_{s}(t'){frac {delta (t'-t_{r}')}{|{mathbf {r}}-{mathbf {r}}_{s}(t')|}},dt'


The retarded time tr′{displaystyle t_{r}'}t_{r}' is a function of the field point (r,t){displaystyle (mathbf {r} ,t)}({mathbf {r}},t) and the source trajectory rs(t′){displaystyle mathbf {r} _{s}(t')}{mathbf {r}}_{s}(t'), and hence depends on t′{displaystyle t'}t'. To evaluate this integral, therefore, we need the identity


δ(f(t′))=∑iδ(t′−ti)|f′(ti)|{displaystyle delta (f(t'))=sum _{i}{frac {delta (t'-t_{i})}{|f'(t_{i})|}}}delta (f(t'))=sum _{i}{frac {delta (t'-t_{i})}{|f'(t_{i})|}}

where each ti{displaystyle t_{i}}t_{i} is a zero of f{displaystyle f}f. Because there is only one retarded time tr{displaystyle t_{r}}t_{r} for any given space-time coordinates (r,t){displaystyle (mathbf {r} ,t)}({mathbf {r}},t) and source trajectory rs(t′){displaystyle mathbf {r} _{s}(t')}{mathbf {r}}_{s}(t'), this reduces to:


δ(t′−tr′)=δ(t′−tr)∂∂t′(t′−tr′)|t′=tr=δ(t′−tr)∂∂t′(t′−(t−1c|r−rs(t′)|))|t′=tr=δ(t′−tr)1+1c(r−rs(t′))/|r−rs(t′)|⋅(−vs(t′))|t′=tr==δ(t′−tr)1−βs⋅(r−rs)/|r−rs|{displaystyle {begin{aligned}delta (t'-t_{r}')={frac {delta (t'-t_{r})}{{frac {partial }{partial t'}}(t'-t_{r}')|_{t'=t_{r}}}}=&{frac {delta (t'-t_{r})}{{frac {partial }{partial t'}}(t'-(t-{frac {1}{c}}|mathbf {r} -mathbf {r} _{s}(t')|))|_{t'=t_{r}}}}={frac {delta (t'-t_{r})}{1+{frac {1}{c}}(mathbf {r} -mathbf {r} _{s}(t'))/|mathbf {r} -mathbf {r} _{s}(t')|cdot (-mathbf {v} _{s}(t'))|_{t'=t_{r}}}}=\&={frac {delta (t'-t_{r})}{1-{boldsymbol {beta }}_{s}cdot (mathbf {r} -mathbf {r} _{s})/|mathbf {r} -mathbf {r} _{s}|}}end{aligned}}}{begin{aligned}delta (t'-t_{r}')={frac {delta (t'-t_{r})}{{frac {partial }{partial t'}}(t'-t_{r}')|_{{t'=t_{r}}}}}=&{frac {delta (t'-t_{r})}{{frac {partial }{partial t'}}(t'-(t-{frac {1}{c}}|{mathbf {r}}-{mathbf {r}}_{s}(t')|))|_{{t'=t_{r}}}}}={frac {delta (t'-t_{r})}{1+{frac {1}{c}}({mathbf {r}}-{mathbf {r}}_{s}(t'))/|{mathbf {r}}-{mathbf {r}}_{s}(t')|cdot (-{mathbf {v}}_{s}(t'))|_{{t'=t_{r}}}}}=\&={frac {delta (t'-t_{r})}{1-{boldsymbol {beta }}_{s}cdot ({mathbf {r}}-{mathbf {r}}_{s})/|{mathbf {r}}-{mathbf {r}}_{s}|}}end{aligned}}

where βs=vs/c{displaystyle {boldsymbol {beta }}_{s}=mathbf {v} _{s}/c}{boldsymbol {beta }}_{s}={mathbf {v}}_{s}/c and rs{displaystyle mathbf {r} _{s}}{mathbf {r}}_{s} are evaluated at the retarded time, and we have used the identity |x|′=x^⋅v{displaystyle |mathbf {x} |'={hat {mathbf {x} }}cdot mathbf {v} }|{mathbf {x}}|'={hat {{mathbf {x}}}}cdot {mathbf {v}}. Finally, the delta function picks out t′=tr{displaystyle t'=t_{r}}t'=t_{r}, and



φ(r,t)=14πϵ0(q|r−rs|(1−βs⋅(r−rs)/|r−rs|))tr=14πϵ0(q(1−n⋅βs)|r−rs|)tr{displaystyle varphi (mathbf {r} ,t)={frac {1}{4pi epsilon _{0}}}left({frac {q}{|mathbf {r} -mathbf {r} _{s}|(1-{boldsymbol {beta }}_{s}cdot (mathbf {r} -mathbf {r} _{s})/|mathbf {r} -mathbf {r} _{s}|)}}right)_{t_{r}}={frac {1}{4pi epsilon _{0}}}left({frac {q}{(1-mathbf {n} cdot {boldsymbol {beta }}_{s})|mathbf {r} -mathbf {r} _{s}|}}right)_{t_{r}}}varphi ({mathbf {r}},t)={frac {1}{4pi epsilon _{0}}}left({frac {q}{|{mathbf {r}}-{mathbf {r}}_{s}|(1-{boldsymbol {beta }}_{s}cdot ({mathbf {r}}-{mathbf {r}}_{s})/|{mathbf {r}}-{mathbf {r}}_{s}|)}}right)_{{t_{r}}}={frac {1}{4pi epsilon _{0}}}left({frac {q}{(1-{mathbf {n}}cdot {boldsymbol {beta }}_{s})|{mathbf {r}}-{mathbf {r}}_{s}|}}right)_{{t_{r}}}

A(r,t)=μ04π(qv|r−rs|(1−βs⋅(r−rs)/|r−rs|))tr=μ0c4π(qβs(1−n⋅βs)|r−rs|)tr{displaystyle mathbf {A} (mathbf {r} ,t)={frac {mu _{0}}{4pi }}left({frac {qmathbf {v} }{|mathbf {r} -mathbf {r} _{s}|(1-{boldsymbol {beta }}_{s}cdot (mathbf {r} -mathbf {r} _{s})/|mathbf {r} -mathbf {r} _{s}|)}}right)_{t_{r}}={frac {mu _{0}c}{4pi }}left({frac {q{boldsymbol {beta }}_{s}}{(1-mathbf {n} cdot {boldsymbol {beta }}_{s})|mathbf {r} -mathbf {r} _{s}|}}right)_{t_{r}}}{mathbf {A}}({mathbf {r}},t)={frac {mu _{0}}{4pi }}left({frac {q{mathbf {v}}}{|{mathbf {r}}-{mathbf {r}}_{s}|(1-{boldsymbol {beta }}_{s}cdot ({mathbf {r}}-{mathbf {r}}_{s})/|{mathbf {r}}-{mathbf {r}}_{s}|)}}right)_{{t_{r}}}={frac {mu _{0}c}{4pi }}left({frac {q{boldsymbol {beta }}_{s}}{(1-{mathbf {n}}cdot {boldsymbol {beta }}_{s})|{mathbf {r}}-{mathbf {r}}_{s}|}}right)_{{t_{r}}}


which are the Liénard–Wiechert potentials.



Lorenz gauge, electric and magnetic fields


In order to calculate the derivatives of φ{displaystyle varphi }varphi and A{displaystyle mathbf {A} }mathbf {A} it is convenient to first compute the derivatives of the retarded time. Taking the derivatives of both sides of its defining equation (remembering that rs=rs(tr){displaystyle mathbf {r_{s}} =mathbf {r_{s}} (t_{r})}{mathbf {r_{s}}}={mathbf {r_{s}}}(t_{r})):


tr+1c|r−rs|=t{displaystyle t_{r}+{frac {1}{c}}|mathbf {r} -mathbf {r_{s}} |=t}t_{r}+{frac {1}{c}}|{mathbf {r}}-{mathbf {r_{s}}}|=t

Differentiating with respect to t,


dtrdt+1cdtrdtd|r−rs|dtr=1{displaystyle {frac {dt_{r}}{dt}}+{frac {1}{c}}{frac {dt_{r}}{dt}}{frac {d|mathbf {r} -mathbf {r_{s}} |}{dt_{r}}}=1}{frac {dt_{r}}{dt}}+{frac {1}{c}}{frac {dt_{r}}{dt}}{frac {d|{mathbf {r}}-{mathbf {r_{s}}}|}{dt_{r}}}=1

dtrdt(1−n⋅βs)=1{displaystyle {frac {dt_{r}}{dt}}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)=1}{frac {dt_{r}}{dt}}left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)=1

dtrdt=1(1−n⋅βs){displaystyle {frac {dt_{r}}{dt}}={frac {1}{left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)}}}{frac {dt_{r}}{dt}}={frac {1}{left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)}}

Similarly, Taking the gradient with respect to r{displaystyle mathbf {r} }mathbf {r} gives


∇tr+1c∇|r−rs|=0{displaystyle {boldsymbol {nabla }}t_{r}+{frac {1}{c}}{boldsymbol {nabla }}|mathbf {r} -mathbf {r_{s}} |=0}{{boldsymbol nabla }}t_{r}+{frac {1}{c}}{{boldsymbol nabla }}|{mathbf {r}}-{mathbf {r_{s}}}|=0

∇tr+1c(∇trd|r−rs|dtr+n)=0{displaystyle {boldsymbol {nabla }}t_{r}+{frac {1}{c}}left({boldsymbol {nabla }}t_{r}{frac {d|mathbf {r} -mathbf {r_{s}} |}{dt_{r}}}+mathbf {n} right)=0}{{boldsymbol nabla }}t_{r}+{frac {1}{c}}left({{boldsymbol nabla }}t_{r}{frac {d|{mathbf {r}}-{mathbf {r_{s}}}|}{dt_{r}}}+{mathbf {n}}right)=0

∇tr(1−n⋅βs)=−n/c{displaystyle {boldsymbol {nabla }}t_{r}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)=-mathbf {n} /c}{{boldsymbol nabla }}t_{r}left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)=-{mathbf {n}}/c

∇tr=−n/c(1−n⋅βs){displaystyle {boldsymbol {nabla }}t_{r}=-{frac {mathbf {n} /c}{left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)}}}{{boldsymbol nabla }}t_{r}=-{frac {{mathbf {n}}/c}{left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)}}

It follows that


d|r−rs|dt=dtrdtd|r−rs|dtr=−n⋅βsc(1−n⋅βs){displaystyle {frac {d|mathbf {r} -mathbf {r_{s}} |}{dt}}={frac {dt_{r}}{dt}}{frac {d|mathbf {r} -mathbf {r_{s}} |}{dt_{r}}}={frac {-mathbf {n} cdot {boldsymbol {beta }}_{s}c}{left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)}}}{frac {d|{mathbf {r}}-{mathbf {r_{s}}}|}{dt}}={frac {dt_{r}}{dt}}{frac {d|{mathbf {r}}-{mathbf {r_{s}}}|}{dt_{r}}}={frac {-{mathbf {n}}cdot {{boldsymbol beta }}_{s}c}{left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)}}

∇|r−rs|=∇trd|r−rs|dtr+n=n(1−n⋅βs){displaystyle {boldsymbol {nabla }}|mathbf {r} -mathbf {r_{s}} |={boldsymbol {nabla }}t_{r}{frac {d|mathbf {r} -mathbf {r_{s}} |}{dt_{r}}}+mathbf {n} ={frac {mathbf {n} }{left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)}}}{{boldsymbol nabla }}|{mathbf {r}}-{mathbf {r_{s}}}|={{boldsymbol nabla }}t_{r}{frac {d|{mathbf {r}}-{mathbf {r_{s}}}|}{dt_{r}}}+{mathbf {n}}={frac {{mathbf {n}}}{left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)}}

These can be used in calculating the derivatives of the vector potential and the resulting expressions are


dφdt=−q4πϵ01|r−rs|2(1−n⋅βs)2ddt[(|r−rs|(1−n⋅βs)]=−q4πϵ01|r−rs|2(1−n⋅βs)2ddt[|r−rs|−(r−rs)⋅βs]=−qc4πϵ01|r−rs|2(1−n⋅βs)3[−n⋅βs+βs2−(r−rs)⋅β˙s/c]{displaystyle {begin{aligned}{frac {dvarphi }{dt}}=&-{frac {q}{4pi epsilon _{0}}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{2}}}{frac {d}{dt}}left[(|mathbf {r} -mathbf {r_{s}} |(1-mathbf {n} cdot {boldsymbol {beta }}_{s})right]\=&-{frac {q}{4pi epsilon _{0}}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{2}}}{frac {d}{dt}}left[|mathbf {r} -mathbf {r_{s}} |-(mathbf {r} -mathbf {r_{s}} )cdot {boldsymbol {beta }}_{s}right]\=&-{frac {qc}{4pi epsilon _{0}}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{3}}}left[-mathbf {n} cdot {boldsymbol {beta }}_{s}+{beta _{s}}^{2}-(mathbf {r} -mathbf {r_{s}} )cdot {dot {boldsymbol {beta }}}_{s}/cright]end{aligned}}}begin{align}frac{d varphi}{d t} =&<br /> -frac{q}{4piepsilon_0}frac{1}{|mathbf{r}-mathbf{r_s}|^2left(1-mathbf{n}cdot{boldsymbol beta}_sright)^2}frac{d}{d t}left[(|mathbf{r}-mathbf{r_s}|(1-mathbf{n}cdot{boldsymbol beta}_s)right]\<br /> =& -frac{q}{4piepsilon_0}frac{1}{|mathbf{r}-mathbf{r_s}|^2left(1-mathbf{n}cdot{boldsymbol beta}_sright)^2}frac{d}{d t}left[|mathbf{r}-mathbf{r_s}|-(mathbf{r}-mathbf{r_s})cdot{boldsymbol beta}_sright]\<br /> =& -frac{q c}{4piepsilon_0}frac{1}{|mathbf{r}-mathbf{r_s}|^2left(1-mathbf{n}cdot{boldsymbol beta}_sright)^3}left[- mathbf{n}cdot{boldsymbol beta}_s + {beta_s}^2 - (mathbf{r}-mathbf{r_s})cdot dot {boldsymbol beta}_s /c right]end{align}

∇⋅A=−q4πϵ0c1|r−rs|2(1−n⋅βs)2(∇[(|r−rs|−(r−rs)⋅βs)]⋅βs−[(|r−rs|−(r−rs)⋅βs)]∇⋅βs)=−q4πϵ0c1|r−rs|2(1−n⋅βs)3⋅[(n⋅βs)−βs2(1−n⋅βs)−βs2n⋅βs+((r−rs)⋅β˙s/c)(n⋅βs)+(|r−rs|−(r−rs)⋅βs)(n⋅β˙s/c)]=q4πϵ0c1|r−rs|2(1−n⋅βs)3[βs2−n⋅βs−(r−rs)⋅β˙s/c]{displaystyle {begin{aligned}{boldsymbol {nabla }}cdot mathbf {A} =&-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{2}}}{big (}{boldsymbol {nabla }}left[left(|mathbf {r} -mathbf {r_{s}} |-(mathbf {r} -mathbf {r_{s}} )cdot {boldsymbol {beta }}_{s}right)right]cdot {boldsymbol {beta }}_{s}-left[left(|mathbf {r} -mathbf {r_{s}} |-(mathbf {r} -mathbf {r_{s}} )cdot {boldsymbol {beta }}_{s}right)right]{boldsymbol {nabla }}cdot {boldsymbol {beta }}_{s}{big )}\=&-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{3}}}cdot \&left[(mathbf {n} cdot {boldsymbol {beta }}_{s})-{beta }_{s}^{2}(1-mathbf {n} cdot {boldsymbol {beta }}_{s})-{beta }_{s}^{2}mathbf {n} cdot {boldsymbol {beta }}_{s}+left((mathbf {r} -mathbf {r_{s}} )cdot {dot {boldsymbol {beta }}}_{s}/cright)(mathbf {n} cdot {boldsymbol {beta }}_{s})+{big (}|mathbf {r} -mathbf {r_{s}} |-(mathbf {r} -mathbf {r_{s}} )cdot {boldsymbol {beta }}_{s}{big )}(mathbf {n} cdot {dot {boldsymbol {beta }}}_{s}/c)right]\=&{frac {q}{4pi epsilon _{0}c}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{3}}}left[beta _{s}^{2}-mathbf {n} cdot {boldsymbol {beta }}_{s}-(mathbf {r} -mathbf {r_{s}} )cdot {dot {boldsymbol {beta }}}_{s}/cright]end{aligned}}}{begin{aligned}{{boldsymbol nabla }}cdot {mathbf {A}}=&-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|{mathbf {r}}-{mathbf {r_{s}}}|^{2}left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)^{2}}}{big (}{{boldsymbol nabla }}left[left(|{mathbf {r}}-{mathbf {r_{s}}}|-({mathbf {r}}-{mathbf {r_{s}}})cdot {{boldsymbol beta }}_{s}right)right]cdot {{boldsymbol beta }}_{s}-left[left(|{mathbf {r}}-{mathbf {r_{s}}}|-({mathbf {r}}-{mathbf {r_{s}}})cdot {{boldsymbol beta }}_{s}right)right]{{boldsymbol nabla }}cdot {{boldsymbol beta }}_{s}{big )}\=&-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|{mathbf {r}}-{mathbf {r_{s}}}|^{2}left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)^{3}}}cdot \&left[({mathbf {n}}cdot {{boldsymbol beta }}_{s})-{beta }_{s}^{2}(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s})-{beta }_{s}^{2}{mathbf {n}}cdot {{boldsymbol beta }}_{s}+left(({mathbf {r}}-{mathbf {r_{s}}})cdot {dot {{boldsymbol beta }}}_{s}/cright)({mathbf {n}}cdot {{boldsymbol beta }}_{s})+{big (}|{mathbf {r}}-{mathbf {r_{s}}}|-({mathbf {r}}-{mathbf {r_{s}}})cdot {{boldsymbol beta }}_{s}{big )}({mathbf {n}}cdot {dot {{boldsymbol beta }}}_{s}/c)right]\=&{frac {q}{4pi epsilon _{0}c}}{frac {1}{|{mathbf {r}}-{mathbf {r_{s}}}|^{2}left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)^{3}}}left[beta _{s}^{2}-{mathbf {n}}cdot {{boldsymbol beta }}_{s}-({mathbf {r}}-{mathbf {r_{s}}})cdot {dot {{boldsymbol beta }}}_{s}/cright]end{aligned}}

These show that the Lorenz gauge is satisfied, namely that dφdt+c2∇⋅A=0{displaystyle {frac {dvarphi }{dt}}+c^{2}{boldsymbol {nabla }}cdot mathbf {A} =0}{frac {dvarphi }{dt}}+c^{2}{{boldsymbol nabla }}cdot {mathbf {A}}=0.


Similarly one calculates:


∇φ=−q4πϵ01|r−rs|2(1−n⋅βs)3[n(1−βs2+(r−rs)⋅β˙s/c)−βs(1−n⋅βs)]{displaystyle {boldsymbol {nabla }}varphi =-{frac {q}{4pi epsilon _{0}}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{3}}}left[mathbf {n} left(1-{beta _{s}}^{2}+(mathbf {r} -mathbf {r_{s}} )cdot {dot {boldsymbol {beta }}}_{s}/cright)-{boldsymbol {beta }}_{s}(1-mathbf {n} cdot {boldsymbol {beta }}_{s})right]}{{boldsymbol nabla }}varphi =-{frac {q}{4pi epsilon _{0}}}{frac {1}{|{mathbf {r}}-{mathbf {r_{s}}}|^{2}left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)^{3}}}left[{mathbf {n}}left(1-{beta _{s}}^{2}+({mathbf {r}}-{mathbf {r_{s}}})cdot {dot {{boldsymbol beta }}}_{s}/cright)-{{boldsymbol beta }}_{s}(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s})right]

dAdt=q4πϵ01|r−rs|2(1−n⋅βs)3[βs(n⋅βs−βs2+(r−rs)⋅β˙s/c)+|r−rs|β˙s(1−n⋅βs)/c]{displaystyle {frac {dmathbf {A} }{dt}}={frac {q}{4pi epsilon _{0}}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{3}}}left[{boldsymbol {beta }}_{s}left(mathbf {n} cdot {boldsymbol {beta }}_{s}-{beta _{s}}^{2}+(mathbf {r} -mathbf {r_{s}} )cdot {dot {boldsymbol {beta }}}_{s}/cright)+|mathbf {r} -mathbf {r_{s}} |{dot {boldsymbol {beta }}}_{s}(1-mathbf {n} cdot {boldsymbol {beta }}_{s})/cright]}{frac {d{mathbf {A}}}{dt}}={frac {q}{4pi epsilon _{0}}}{frac {1}{|{mathbf {r}}-{mathbf {r_{s}}}|^{2}left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)^{3}}}left[{{boldsymbol beta }}_{s}left({mathbf {n}}cdot {{boldsymbol beta }}_{s}-{beta _{s}}^{2}+({mathbf {r}}-{mathbf {r_{s}}})cdot {dot {{boldsymbol beta }}}_{s}/cright)+|{mathbf {r}}-{mathbf {r_{s}}}|{dot {{boldsymbol beta }}}_{s}(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s})/cright]

By noting that for any vectors u{displaystyle mathbf {u} }mathbf {u} , v{displaystyle mathbf {v} }mathbf {v} , w{displaystyle mathbf {w} }mathbf {w} :


u×(v×w)=(u⋅w)v−(u⋅v)w{displaystyle mathbf {u} times (mathbf {v} times mathbf {w} )=(mathbf {u} cdot mathbf {w} )mathbf {v} -(mathbf {u} cdot mathbf {v} )mathbf {w} }{mathbf {u}}times ({mathbf {v}}times {mathbf {w}})=({mathbf {u}}cdot {mathbf {w}}){mathbf {v}}-({mathbf {u}}cdot {mathbf {v}}){mathbf {w}}

The expression for the electric field mentioned above becomes


E(r,t)=q4πϵ01|r−rs|2(1−n⋅βs)3⋅[(n−βs)(1−βs2)+|r−rs|(n⋅β˙s/c)(n−βs)−|r−rs|(n⋅(n−βs))β˙s/c]{displaystyle {begin{aligned}mathbf {E} (mathbf {r} ,t)=&{frac {q}{4pi epsilon _{0}}}{frac {1}{|mathbf {r} -mathbf {r} _{s}|^{2}(1-mathbf {n} cdot {boldsymbol {beta }}_{s})^{3}}}cdot \&left[left(mathbf {n} -{boldsymbol {beta }}_{s}right)(1-{beta _{s}}^{2})+|mathbf {r} -mathbf {r} _{s}|(mathbf {n} cdot {dot {boldsymbol {beta }}}_{s}/c)(mathbf {n} -{boldsymbol {beta }}_{s})-|mathbf {r} -mathbf {r} _{s}|{big (}mathbf {n} cdot (mathbf {n} -{boldsymbol {beta }}_{s}){big )}{dot {boldsymbol {beta }}}_{s}/cright]end{aligned}}}{begin{aligned}{mathbf {E}}({mathbf {r}},t)=&{frac {q}{4pi epsilon _{0}}}{frac {1}{|{mathbf {r}}-{mathbf {r}}_{s}|^{2}(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s})^{3}}}cdot \&left[left({mathbf {n}}-{{boldsymbol beta }}_{s}right)(1-{beta _{s}}^{2})+|{mathbf {r}}-{mathbf {r}}_{s}|({mathbf {n}}cdot {dot {{boldsymbol beta }}}_{s}/c)({mathbf {n}}-{{boldsymbol beta }}_{s})-|{mathbf {r}}-{mathbf {r}}_{s}|{big (}{mathbf {n}}cdot ({mathbf {n}}-{{boldsymbol beta }}_{s}){big )}{dot {{boldsymbol beta }}}_{s}/cright]end{aligned}}

which is easily seen to be equal to −∇φ−dAdt{displaystyle -{boldsymbol {nabla }}varphi -{frac {dmathbf {A} }{dt}}}-{{boldsymbol nabla }}varphi -{frac {d{mathbf {A}}}{dt}}


Similarly ∇×A{displaystyle {boldsymbol {nabla }}times mathbf {A} }{{boldsymbol nabla }}times {mathbf {A}} gives the expression of the magnetic field mentioned above:


B=∇×A=−q4πϵ0c1|r−rs|2(1−n⋅βs)2(∇[(|r−rs|−(r−rs)⋅βs)]×βs−[(|r−rs|−(r−rs)⋅βs)]∇×βs)=−q4πϵ0c1|r−rs|2(1−n⋅βs)3⋅[(n×βs)−(βs×βs)(1−n⋅βs)−βs2n×βs+((r−rs)⋅β˙s/c)(n×βs)+(|r−rs|−(r−rs)⋅βs)(n×β˙s/c)]=−q4πϵ0c1|r−rs|2(1−n⋅βs)3⋅[(n×βs)(1−βs2)+|r−rs|(n⋅β˙s/c)(n×βs)+|r−rs|(n⋅(n−βs))n×β˙s/c]=nc×E{displaystyle {begin{aligned}{mathbf {B} }=&{boldsymbol {nabla }}times mathbf {A} =-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{2}}}{big (}{boldsymbol {nabla }}left[left(|mathbf {r} -mathbf {r_{s}} |-(mathbf {r} -mathbf {r_{s}} )cdot {boldsymbol {beta }}_{s}right)right]times {boldsymbol {beta }}_{s}-left[left(|mathbf {r} -mathbf {r_{s}} |-(mathbf {r} -mathbf {r_{s}} )cdot {boldsymbol {beta }}_{s}right)right]{boldsymbol {nabla }}times {boldsymbol {beta }}_{s}{big )}\=&-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|mathbf {r} -mathbf {r_{s}} |^{2}left(1-mathbf {n} cdot {boldsymbol {beta }}_{s}right)^{3}}}cdot \&left[(mathbf {n} times {boldsymbol {beta }}_{s})-({boldsymbol {beta }}_{s}times {boldsymbol {beta }}_{s})(1-mathbf {n} cdot {boldsymbol {beta }}_{s})-{beta }_{s}^{2}mathbf {n} times {boldsymbol {beta }}_{s}+left((mathbf {r} -mathbf {r_{s}} )cdot {dot {boldsymbol {beta }}}_{s}/cright)(mathbf {n} times {boldsymbol {beta }}_{s})+{big (}|mathbf {r} -mathbf {r_{s}} |-(mathbf {r} -mathbf {r_{s}} )cdot {boldsymbol {beta }}_{s}{big )}(mathbf {n} times {dot {boldsymbol {beta }}}_{s}/c)right]\=&-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|mathbf {r} -mathbf {r} _{s}|^{2}(1-mathbf {n} cdot {boldsymbol {beta }}_{s})^{3}}}cdot \&left[left(mathbf {n} times {boldsymbol {beta }}_{s}right)(1-{beta _{s}}^{2})+|mathbf {r} -mathbf {r} _{s}|(mathbf {n} cdot {dot {boldsymbol {beta }}}_{s}/c)(mathbf {n} times {boldsymbol {beta }}_{s})+|mathbf {r} -mathbf {r} _{s}|{big (}mathbf {n} cdot (mathbf {n} -{boldsymbol {beta }}_{s}){big )}mathbf {n} times {dot {boldsymbol {beta }}}_{s}/cright]={frac {mathbf {n} }{c}}times mathbf {E} end{aligned}}}{begin{aligned}{{mathbf {B}}}=&{{boldsymbol nabla }}times {mathbf {A}}=-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|{mathbf {r}}-{mathbf {r_{s}}}|^{2}left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)^{2}}}{big (}{{boldsymbol nabla }}left[left(|{mathbf {r}}-{mathbf {r_{s}}}|-({mathbf {r}}-{mathbf {r_{s}}})cdot {{boldsymbol beta }}_{s}right)right]times {{boldsymbol beta }}_{s}-left[left(|{mathbf {r}}-{mathbf {r_{s}}}|-({mathbf {r}}-{mathbf {r_{s}}})cdot {{boldsymbol beta }}_{s}right)right]{{boldsymbol nabla }}times {{boldsymbol beta }}_{s}{big )}\=&-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|{mathbf {r}}-{mathbf {r_{s}}}|^{2}left(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s}right)^{3}}}cdot \&left[({mathbf {n}}times {{boldsymbol beta }}_{s})-({{boldsymbol beta }}_{s}times {{boldsymbol beta }}_{s})(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s})-{beta }_{s}^{2}{mathbf {n}}times {{boldsymbol beta }}_{s}+left(({mathbf {r}}-{mathbf {r_{s}}})cdot {dot {{boldsymbol beta }}}_{s}/cright)({mathbf {n}}times {{boldsymbol beta }}_{s})+{big (}|{mathbf {r}}-{mathbf {r_{s}}}|-({mathbf {r}}-{mathbf {r_{s}}})cdot {{boldsymbol beta }}_{s}{big )}({mathbf {n}}times {dot {{boldsymbol beta }}}_{s}/c)right]\=&-{frac {q}{4pi epsilon _{0}c}}{frac {1}{|{mathbf {r}}-{mathbf {r}}_{s}|^{2}(1-{mathbf {n}}cdot {{boldsymbol beta }}_{s})^{3}}}cdot \&left[left({mathbf {n}}times {{boldsymbol beta }}_{s}right)(1-{beta _{s}}^{2})+|{mathbf {r}}-{mathbf {r}}_{s}|({mathbf {n}}cdot {dot {{boldsymbol beta }}}_{s}/c)({mathbf {n}}times {{boldsymbol beta }}_{s})+|{mathbf {r}}-{mathbf {r}}_{s}|{big (}{mathbf {n}}cdot ({mathbf {n}}-{{boldsymbol beta }}_{s}){big )}{mathbf {n}}times {dot {{boldsymbol beta }}}_{s}/cright]={frac {{mathbf {n}}}{c}}times {mathbf {E}}end{aligned}}


Implications


The study of classical electrodynamics was instrumental in Einstein's development of the theory of relativity. Analysis of the motion and propagation of electromagnetic waves led to the special relativity description of space and time. The Liénard–Wiechert formulation is an important launchpad into a deeper analysis of relativistic moving particles.


The Liénard–Wiechert description is accurate for a large (i.e., not quantum mechanical)[why?], independent (i.e., free of external influence)[clarification needed] moving particle. The Liénard–Wiechert formulation always provides two sets of solutions: Advanced fields are absorbed by the charges and retarded fields are emitted. Schwarzschild and Fokker considered the advanced field of a system of moving charges, and the retarded field of a system of charges having the same geometry and opposite charges. Linearity of Maxwell's equations in vacuum allows one to add both systems, so that the charges disappear: This trick allows Maxwell's equations to become linear in matter.
Multiplying electric parameters of both problems by arbitrary real constants produces a coherent interaction of light with matter which generalizes Einstein's theory (A. Einstein, “Zur Quantentheorie der Strahlung.” Phys. Z. 18 121-128, 1917) which is now considered as founding theory of lasers: it is not necessary to study a large set of identical molecules to get coherent amplification in the mode obtained by arbitrary multiplications of advanced and retarded fields.
To compute energy, it is necessary to use the absolute fields which includes the zero point field; otherwise, an error appears, for instance in photon counting.


It is important to take into account the zero point field discovered by Planck (M. Planck, Deutsche Physikalische Gesellschaft, Vol. 13, 1911, pp. 138–175.). It replaces Einstein's "A" coefficient and explains that the classical electron is stable on Rydberg's classical orbits. Moreover, introducing the fluctuations of the zero point field produces Willis E. Lamb's correction of levels of H atom.


Quantum electrodynamics helped bring together the radiative behavior with the quantum constraints. It introduces quantization of normal modes of the electromagnetic field in assumed perfect optical resonators.



Universal speed limit


The force on a particle at a given location r and time t depends in a complicated way on the position of the source particles at an earlier time tr due to the finite speed, c, at which electromagnetic information travels. A particle on Earth 'sees' a charged particle accelerate on the Moon as this acceleration happened 1.5 seconds ago, and a charged particle's acceleration on the Sun as happened 500 seconds ago. This earlier time in which an event happens such that a particle at location r 'sees' this event at a later time t is called the retarded time, tr. The retarded time varies with position; for example the retarded time at the Moon is 1.5 seconds before the current time and the retarded time on the Sun is 500 s before the current time on the Earth. The retarded time tr=tr(r,t) is defined implicitly by


tr=t−R(tr)c{displaystyle t_{r}=t-{frac {R(t_{r})}{c}}}t_{r}=t-{frac {R(t_{r})}{c}}

where R(tr){displaystyle R(t_{r})}R(t_{r}) is the distance of the particle from the source at the retarded time. Only electromagnetic wave effects depend fully on the retarded time.


A novel feature in the Liénard–Wiechert potential is seen in the breakup of its terms into two types of field terms (see below), only one of which depends fully on the retarded time. The first of these is the static electric (or magnetic) field term that depends only on the distance to the moving charge, and does not depend on the retarded time at all, if the velocity of the source is constant. The other term is dynamic, in that it requires that the moving charge be accelerating with a component perpendicular to the line connecting the charge and the observer and does not appear unless the source changes velocity. This second term is connected with electromagnetic radiation.


The first term describes near field effects from the charge, and its direction in space is updated with a term that corrects for any constant-velocity motion of the charge on its distant static field, so that the distant static field appears at distance from the charge, with no aberration of light or light-time correction. This term, which corrects for time-retardation delays in the direction of the static field, is required by Lorentz invariance. A charge moving with a constant velocity must appear to a distant observer in exactly the same way as a static charge appears to a moving observer, and in the latter case, the direction of the static field must change instantaneously, with no time-delay. Thus, static fields (the first term) point exactly at the true instantaneous (non-retarded) position of the charged object if its velocity has not changed over the retarded time delay. This is true over any distance separating objects.


The second term, however, which contains information about the acceleration and other unique behavior of the charge that cannot be removed by changing the Lorentz frame (inertial reference frame of the observer), is fully dependent for direction on the time-retarded position of the source. Thus, electromagnetic radiation (described by the second term) always appears to come from the direction of the position of the emitting charge at the retarded time. Only this second term describes information transfer about the behavior of the charge, which transfer occurs (radiates from the charge) at the speed of light. At "far" distances (longer than several wavelengths of radiation), the 1/R dependence of this term makes electromagnetic field effects (the value of this field term) more powerful than "static" field effects, which are described by the 1/R2 field of the first (static) term and thus decay more rapidly with distance from the charge.



Existence and uniqueness of the retarded time



Existence


The retarded time is not guaranteed to exist in general. For example, if, in a given frame of reference, an electron has just been created, then at this very moment another electron does not yet feel its electromagnetic force at all. However, under certain conditions, there always exists a retarded time. For example, if the source charge has existed for an unlimited amount of time, during which it has always travelled at a speed not exceeding vM<c{displaystyle v_{M}<c}v_{M}<c, then there exists a valid retarded time tr{displaystyle t_{r}}t_{r}. This can be seen by considering the function f(t′)=|r−rs(t′)|−c(t−t′){displaystyle f(t')=|mathbf {r} -mathbf {r} _{s}(t')|-c(t-t')}f(t')=|{mathbf {r}}-{mathbf {r}}_{s}(t')|-c(t-t'). At the present time t′=t{displaystyle t'=t}t'=t; f(t′)=|r−rs(t′)|−c(t−t′)=|r−rs(t′)|≥0{displaystyle f(t')=|mathbf {r} -mathbf {r} _{s}(t')|-c(t-t')=|mathbf {r} -mathbf {r} _{s}(t')|geq 0}f(t')=|{mathbf {r}}-{mathbf {r}}_{s}(t')|-c(t-t')=|{mathbf {r}}-{mathbf {r}}_{s}(t')|geq 0. The derivative f′(t′){displaystyle f'(t')}f'(t') is given by


f′(t′)=r−rs(tr)|r−rs(tr)|⋅(−vs(t′))+c≥c−|r−rs(tr)|r−rs(tr)|||vs(t′)|=c−|vs(t′)|≥c−vM>0{displaystyle f'(t')={frac {mathbf {r} -mathbf {r} _{s}(t_{r})}{|mathbf {r} -mathbf {r} _{s}(t_{r})|}}cdot (-mathbf {v} _{s}(t'))+cgeq c-left|{frac {mathbf {r} -mathbf {r} _{s}(t_{r})}{|mathbf {r} -mathbf {r} _{s}(t_{r})|}}right|,|mathbf {v} _{s}(t')|=c-|mathbf {v} _{s}(t')|geq c-v_{M}>0}f'(t')={frac {{mathbf {r}}-{mathbf {r}}_{s}(t_{r})}{|{mathbf {r}}-{mathbf {r}}_{s}(t_{r})|}}cdot (-{mathbf {v}}_{s}(t'))+cgeq c-left|{frac {{mathbf {r}}-{mathbf {r}}_{s}(t_{r})}{|{mathbf {r}}-{mathbf {r}}_{s}(t_{r})|}}right|,|{mathbf {v}}_{s}(t')|=c-|{mathbf {v}}_{s}(t')|geq c-v_{M}>0

By the mean value theorem, f(t−Δt)≤f(t)−f′(t)Δt≤f(t)−(c−vM)Δt{displaystyle f(t-Delta t)leq f(t)-f'(t)Delta tleq f(t)-(c-v_{M})Delta t}f(t-Delta t)leq f(t)-f'(t)Delta tleq f(t)-(c-v_{M})Delta t. By making Δt{displaystyle Delta t}Delta t sufficiently large, this can become negative, i.e., at some point in the past, f(t′)<0{displaystyle f(t')<0}f(t')<0. By the intermediate value theorem, there exists an intermediate tr{displaystyle t_{r}}t_{r} with f(tr)=0{displaystyle f(t_{r})=0}f(t_{r})=0, the defining equation of the retarded time. Intuitively, as the source charge moves back in time, the cross section of its light cone at present time expands faster than it can recede, so eventually it must reach the point r{displaystyle mathbf {r} }mathbf {r} . This is not necessarily true if the source charge's speed is allowed to be arbitrarily close to c{displaystyle c}c, i.e., if for any given speed v<c{displaystyle v<c}v<c there was some time in the past when the charge was moving at this speed. In this case the cross section of the light cone at present time approaches the point r{displaystyle mathbf {r} }mathbf {r} as the observer travels back in time but does not necessarily ever reach it.



Uniqueness


For a given point (r,t){displaystyle (mathbf {r} ,t)}({mathbf {r}},t) and trajectory of the point source rs(t′){displaystyle mathbf {r} _{s}(t')}{mathbf {r}}_{s}(t'), there is at most one value of the retarded time tr{displaystyle t_{r}}t_{r}, i.e., one value tr{displaystyle t_{r}}t_{r} such that |r−rs(tr)|=c(t−tr){displaystyle |mathbf {r} -mathbf {r} _{s}(t_{r})|=c(t-t_{r})}|{mathbf {r}}-{mathbf {r}}_{s}(t_{r})|=c(t-t_{r}). This can be realized by assuming that there are two retarded times t1{displaystyle t_{1}}t_{1} and t2{displaystyle t_{2}}t_{2}, with t1≤t2{displaystyle t_{1}leq t_{2}}t_{1}leq t_{2}. Then, |r−rs(t1)|=c(t−t1){displaystyle |mathbf {r} -mathbf {r} _{s}(t_{1})|=c(t-t_{1})}|{mathbf {r}}-{mathbf {r}}_{s}(t_{1})|=c(t-t_{1}) and |r−rs(t2)|=c(t−t2){displaystyle |mathbf {r} -mathbf {r} _{s}(t_{2})|=c(t-t_{2})}|{mathbf {r}}-{mathbf {r}}_{s}(t_{2})|=c(t-t_{2}). Subtracting gives c(t2−t1)=|r−rs(t1)|−|r−rs(t2)|≤|rs(t2)−rs(t1)|{displaystyle c(t_{2}-t_{1})=|mathbf {r} -mathbf {r} _{s}(t_{1})|-|mathbf {r} -mathbf {r} _{s}(t_{2})|leq |mathbf {r} _{s}(t_{2})-mathbf {r} _{s}(t_{1})|}c(t_{2}-t_{1})=|{mathbf {r}}-{mathbf {r}}_{s}(t_{1})|-|{mathbf {r}}-{mathbf {r}}_{s}(t_{2})|leq |{mathbf {r}}_{s}(t_{2})-{mathbf {r}}_{s}(t_{1})| by the triangle inequality. Unless t2=t1{displaystyle t_{2}=t_{1}}t_{2}=t_{1}, this then implies that the average velocity of the charge between t1{displaystyle t_{1}}t_{1} and t2{displaystyle t_{2}}t_{2} is |rs(t2)−rs(t1)|/(t2−t1)≥c{displaystyle |mathbf {r} _{s}(t_{2})-mathbf {r} _{s}(t_{1})|/(t_{2}-t_{1})geq c}|{mathbf {r}}_{s}(t_{2})-{mathbf {r}}_{s}(t_{1})|/(t_{2}-t_{1})geq c, which is impossible. The intuitive interpretation is that one can only ever "see" the point source at one location/time at once unless it travels at least at the speed of light to another location. As the source moves forward in time, the cross section of its light cone at present time contracts faster than the source can approach, so it can never intersect the point r{displaystyle mathbf {r} }mathbf {r} again.


The conclusion is that, under certain conditions, the retarded time exists and is unique.



See also




  • Maxwell's equations which govern classical electromagnetism


  • Classical electromagnetism for the larger theory surrounding this analysis

  • Relativistic electromagnetism


  • Special relativity, which was a direct consequence of these analyses


  • Rydberg formula for quantum description of the EM radiation due to atomic orbital electrons

  • Jefimenko's equations

  • Larmor formula

  • Abraham–Lorentz force

  • Inhomogeneous electromagnetic wave equation


  • Wheeler–Feynman absorber theory also known as the Wheeler–Feynman time-symmetric theory

  • Paradox of a charge in a gravitational field

  • Whitehead's theory of gravitation



References




  1. ^ http://data.bnf.fr/10743554/alfred_lienard/ - A. Liénard, Champ électrique et magnétique produit par une charge concentrée en un point et animée d’un mouvement quelconque, L’éclairage Electrique ´ 16 p.5; ibid. p. 53; ibid. p. 106 (1898)


  2. ^ https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19013090403 - E. Wiechert, Elektrodynamische Elementargesetze, Arch. N´eerl. Sci. Exactes Nat. 5, pp. 549 (1900).


  3. ^ Some Aspects in Emil Wiechert


  • Griffiths, David. Introduction to Electrodynamics. Prentice Hall, 1999. .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
    ISBN 0-13-805326-X.



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