# Jefimenko's equations

In electromagnetism, Jefimenko's equations (named after Oleg D. Jefimenko) describe the behavior of the electric and magnetic fields in terms of the charge and current distributions at retarded times.

Jefimenko's equations[1] are the solution of Maxwell's equations for an assigned distribution of electric charges and currents, under the assumption that there is no electromagnetic field other than the one produced by those charges and currents.

## Equations

### Electric and magnetic fields

Position vectors r and r′ used in the calculation.

Jefimenko's equations give the E-field and B-field produced by an arbitrary charge or current distribution, of charge density ρ and current density J:[2]

${\displaystyle \mathbf {E} (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \left[\left({\frac {\rho (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|^{3}}}+{\frac {1}{|\mathbf {r} -\mathbf {r} '|^{2}c}}{\frac {\partial \rho (\mathbf {r} ',t_{r})}{\partial t}}\right)(\mathbf {r} -\mathbf {r} ')-{\frac {1}{|\mathbf {r} -\mathbf {r} '|c^{2}}}{\frac {\partial \mathbf {J} (\mathbf {r} ',t_{r})}{\partial t}}\right]\mathrm {d} ^{3}\mathbf {r} '}$
${\displaystyle \mathbf {B} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \left[{\frac {\mathbf {J} (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|^{3}}}+{\frac {1}{|\mathbf {r} -\mathbf {r} '|^{2}c}}{\frac {\partial \mathbf {J} (\mathbf {r} ',t_{r})}{\partial t}}\right]\times (\mathbf {r} -\mathbf {r} ')\mathrm {d} ^{3}\mathbf {r} '}$

where r' is a point in the charge distribution, r is a point in space, and

${\displaystyle t_{r}=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}$

is the retarded time. There are similar expressions for D and H.[3]

These equations are the time-dependent generalization of Coulomb's law and the Biot–Savart law to electrodynamics, which were originally true only for electrostatic and magnetostatic fields, and steady currents.

### Origin from retarded potentials

Jefimenko's equations can be found[4] from the retarded potentials φ and A:

{\displaystyle {\begin{aligned}&\varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi \epsilon _{0}}}\int {\dfrac {\rho (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '\\&\mathbf {A} (\mathbf {r} ,t)={\dfrac {\mu _{0}}{4\pi }}\int {\dfrac {\mathbf {J} (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '\\\end{aligned}}}

which are the solutions to Maxwell's equations in the potential formulation, then substituting in the definitions of the electromagnetic potentials themselves

${\displaystyle \mathbf {E} =-\nabla \varphi -{\dfrac {\partial \mathbf {A} }{\partial t}}\,,\quad \mathbf {B} =\nabla \times \mathbf {A} }$

and using the relation

${\displaystyle c^{2}={\frac {1}{\epsilon _{0}\mu _{0}}}}$

replaces the potentials φ and A by the fields E and B.

## Discussion

There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave[5] (electromagnetism). However, Jefimenko's equations show an alternative point of view.[6] Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents."[7]

As pointed out by McDonald,[8] Jefimenko's equations seem to appear first in 1962 in the second edition of Panofsky and Phillips's classic textbook.[9] Essential features of these equations are easily observed which is that the right hand sides involve "retarded" time which reflects the "causality" of the expressions. In other words, the left side of each equation is actually "caused" by the right side, unlike the normal differential expressions for Maxwell's equations where both sides take place simultaneously. In the typical expressions for Maxwell's equations there is no doubt that both sides are equal to each other, but as Jefimenko notes, "... since each of these equations connects quantities simultaneous in time, none of these equations can represent a causal relation."[10] The second feature is that the expression for E does not depend upon B and vice versa. Hence, it is impossible for E and B fields to be "creating" each other. Charge density and current density are creating them both.