Vfrdtyky

Part of a series of articles about
Electromagnetism
Electricity Magnetism
Electrostatics Conductor Coulomb's law Electric charge Electric dipole moment Electric field Electric flux / potential energy Electrostatic discharge Gauss's law Induction Insulator Polarization density Static electricity Triboelectricity
Magnetostatics Ampère's law Biot–Savart law Gauss's law for magnetism Magnetic field Magnetic flux Magnetic dipole moment Magnetization Magnetomotive force
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
Electrical network Alternating current Capacitance Direct current Electric current Electric potential Electromotive force Impedance Inductance Ohm's law Parallel circuit Resistance Resonant cavities Series circuit Voltage Waveguides
Covariant formulation Electromagnetic tensor (stress–energy tensor) Four-current Electromagnetic four-potential
Scientists Ampère Coulomb Faraday Gauss Heaviside Henry Hertz Lorentz Maxwell Tesla Volta Weber Ørsted
v t e

In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light c, so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution (the point of cause) to another point in space (where the effect is measured), see figure below.^[1]

Potentials in the Lorenz gauge

Position vectors r and r′ used in the calculation.

The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge:

◻φ=ρϵ0,◻A=μ0J{displaystyle Box varphi ={dfrac {rho }{epsilon _{0}}},,quad Box mathbf {A} =mu _{0}mathbf {J} }

where φ(r, t) is the electric potential and A(r, t) is the magnetic potential, for an arbitrary source of charge density ρ(r, t) and current density J(r, t), and ◻{displaystyle Box } is the D'Alembert operator ^[2]. Solving these gives the retarded potentials below (all in SI units).

Retarded and advanced potentials for time-dependent fields

For time-dependent fields, the retarded potentials are:^[3][4]

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

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} '|}},mathrm {d} ^{3}mathbf {r} ',.}

where r is a point in space, t is time,

tr=t−|r−r′|c{displaystyle t_{r}=t-{frac {|mathbf {r} -mathbf {r} '|}{c}}}

is the retarded time, and d³r' is the integration measure using r'.

From φ(r,t) and A(r, t), the fields E(r, t) and B(r, t) can be calculated using the definitions of the potentials:

−E=∇φ+∂A∂t,B=∇×A.{displaystyle -mathbf {E} =nabla varphi +{frac {partial mathbf {A} }{partial t}},,quad mathbf {B} =nabla times mathbf {A} ,.}

and this leads to Jefimenko's equations. The corresponding advanced potentials have an identical form, except the advanced time

ta=t+|r−r′|c{displaystyle t_{a}=t+{frac {|mathbf {r} -mathbf {r} '|}{c}}}

replaces the retarded time.

Comparison with static potentials for time-independent fields

In the case the fields are time-independent (electrostatic and magnetostatic fields), the time derivatives in the ◻{displaystyle Box } operators of the fields are zero, and Maxwell's equations reduce to

∇2φ=−ρϵ0,∇2A=−μ0J,{displaystyle nabla ^{2}varphi =-{dfrac {rho }{epsilon _{0}}},,quad nabla ^{2}mathbf {A} =-mu _{0}mathbf {J} ,,}

where ∇² is the Laplacian, which take the form of Poisson's equation in four components (one for φ and three for A), and the solutions are:

φ(r)=14πϵ0∫ρ(r′)|r−r′|d3r′{displaystyle mathrm {varphi } (mathbf {r} )={frac {1}{4pi epsilon _{0}}}int {frac {rho (mathbf {r} ')}{|mathbf {r} -mathbf {r} '|}},mathrm {d} ^{3}mathbf {r} '}

A(r)=μ04π∫J(r′)|r−r′|d3r′.{displaystyle mathbf {A} (mathbf {r} )={frac {mu _{0}}{4pi }}int {frac {mathbf {J} (mathbf {r} ')}{|mathbf {r} -mathbf {r} '|}},mathrm {d} ^{3}mathbf {r} ',.}

These also follow directly from the retarded potentials.

Potentials in the Coulomb gauge

In the Coulomb gauge, Maxwell's equations are^[5]

∇2φ=−ρϵ0{displaystyle nabla ^{2}varphi =-{dfrac {rho }{epsilon _{0}}}}

∇2A−1c2∂2A∂t2=−μ0J+1c2∇(∂φ∂t),{displaystyle nabla ^{2}mathbf {A} -{dfrac {1}{c^{2}}}{dfrac {partial ^{2}mathbf {A} }{partial t^{2}}}=-mu _{0}mathbf {J} +{dfrac {1}{c^{2}}}nabla left({dfrac {partial varphi }{partial t}}right),,}

although the solutions contrast the above, since A is a retarded potential yet φ changes instantly, given by:

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

A(r,t)=14πε0∇×∫d3r′∫0|r−r′|/cdtrtrJ(r′,t−tr)|r−r′|3×(r−r′).{displaystyle mathbf {A} (mathbf {r} ,t)={dfrac {1}{4pi varepsilon _{0}}}nabla times int mathrm {d} ^{3}mathbf {r'} int _{0}^{|mathbf {r} -mathbf {r} '|/c}mathrm {d} t_{r}{dfrac {t_{r}mathbf {J} (mathbf {r'} ,t-t_{r})}{|mathbf {r} -mathbf {r} '|^{3}}}times (mathbf {r} -mathbf {r} '),.}

This presents an advantage and a disadvantage of the Coulomb gauge - φ is easily calculable from the charge distribution ρ but A is not so easily calculable from the current distribution j. However, provided we require that the potentials vanish at infinity, they can be expressed neatly in terms of fields:

φ(r,t)=14π∫∇⋅E(r′,t)|r−r′|d3r′{displaystyle varphi (mathbf {r} ,t)={dfrac {1}{4pi }}int {dfrac {nabla cdot mathbf {E} ({r}',t)}{|mathbf {r} -mathbf {r} '|}}mathrm {d} ^{3}mathbf {r} '}

A(r,t)=14π∫∇×B(r′,t)|r−r′|d3r′{displaystyle mathbf {A} (mathbf {r} ,t)={dfrac {1}{4pi }}int {dfrac {nabla times mathbf {B} ({r}',t)}{|mathbf {r} -mathbf {r} '|}}mathrm {d} ^{3}mathbf {r} '}

Retarded potentials in linearized gravity

The retarded potential in linearized general relativity is closely analogous to the electromagnetic case. The trace-reversed tensor h~μν=hμν−12ημνh{displaystyle {tilde {h}}_{mu nu }=h_{mu nu }-{frac {1}{2}}eta _{mu nu }h} plays the role of the four-vector potential, the harmonic gauge h~μν,μ=0{displaystyle {tilde {h}}^{mu nu }{}_{,mu }=0} replaces the electromagnetic Lorenz gauge, the field equations are ◻h~μν=−16πGTμν{displaystyle Box {tilde {h}}_{mu nu }=-16pi GT_{mu nu }}, and the retarded-wave solution is

h~μν(r,t)=4G∫Tμν(r′,tr)|r−r′|d3r′{displaystyle {tilde {h}}_{mu nu }(mathbf {r} ,t)=4Gint {frac {T_{mu nu }(mathbf {r} ',t_{r})}{|mathbf {r} -mathbf {r} '|}}mathrm {d} ^{3}mathbf {r} '}.^[6]

Occurrence and application

A many-body theory which includes an average of retarded and advanced Liénard–Wiechert potentials is the Wheeler–Feynman absorber theory also known as the Wheeler–Feynman time-symmetric theory.

Example

The potential of charge with uniform speed on a straight line has inversion in a point that is in the recent position. The potential is not changed in the direction of movement.^[7]

See also

• Maxwell's Equations


• Liénard–Wiechert potential


• Lenz's law

References