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Jefimenko's equations

Equations of electromagnetism

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Equations

Electric and magnetic fields

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

E(r,t)=14πε0∫[r−r′|r−r′|3ρ(r′,tr)+r−r′|r−r′|21c∂ρ(r′,tr)∂t−1|r−r′|1c2∂J(r′,tr)∂t]dV′, B(r,t)=−μ04π∫[r−r′|r−r′|3×J(r′,tr)+r−r′|r−r′|2×1c∂J(r′,tr)∂t]dV′,

where r′ is a point in the charge distribution, r is a point in space, and tr=t−|r−r′|c is the retarded time. There are similar expressions for D and H.

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 from the retarded potentials φ and A: φ(r,t)=14πε0∫ρ(r′,tr)|r−r′|dV′,A(r,t)=μ04π∫J(r′,tr)|r−r′|dV′, which are the solutions to Maxwell's equations in the potential formulation, then substituting in the definitions of the electromagnetic potentials themselves: E=−∇φ−∂A∂t,B=∇×A and using the relation c2=1ε0μ0 replaces the potentials φ and A by the fields E and B.

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