Jefimenko's Equations

The Causal Structure of Fields

Imagine a charge distribution that suddenly changes. In a static world, the field would update instantly everywhere. In reality, information travels at a finite speed, \( c \). Jefimenko's equations encode this causality by evaluating the sources not at the current time \( t \), but at the retarded time \( t_r \). This means the field at a point \( \mathbf{r} \) depends on what the sources were doing at an earlier moment, specifically when the light signal from the source reached the observation point. The equations separate the field into two conceptual parts: a "near field" term that looks like the static Coulomb or Biot-Savart law but evaluated at the retarded time, and a "radiation field" term that depends on the time derivatives of the sources. This distinction is fundamental to understanding how energy is carried away from accelerating charges.

Deriving the Electric Field

The electric field is derived from the potentials via \( \mathbf{E} = -\nabla\varphi - \partial_t \mathbf{A} \). When we substitute the retarded potentials and perform the differentiation, we must carefully account for the time dependence of the retarded time \( t_r \). The gradient \( \nabla \) acts on both the spatial separation \( R \) and the retarded time \( t_r \). This leads to a term involving \( \dot{\rho} \) (the time derivative of charge density) and a term involving \( \dot{\mathbf{J}} \) (the time derivative of current density).

The first term is the familiar Coulomb-like contribution. The second term, proportional to \( \dot{\rho}/cR \), arises from the time variation of the charge density. The third term, proportional to \( \dot{\mathbf{J}}/c^2 R \), is the crucial radiation term. Note that the radiation terms fall off as \( 1/R \), whereas the near-field terms fall off as \( 1/R^2 \). This difference in scaling is why radiation dominates at large distances.

Deriving the Magnetic Field

The magnetic field is derived from the vector potential via \( \mathbf{B} = \nabla\times\mathbf{A} \). Similar to the electric field, the curl operation introduces dependencies on the retarded time. The resulting expression mirrors the structure of the electric field but involves the current density \( \mathbf{J} \) and its time derivative \( \dot{\mathbf{J}} \).

Here, the first term is the Biot-Savart-like near field. The second term is the magnetic radiation field. Notice the cross product with \( \hat{\mathbf{R}} \), which ensures that \( \mathbf{B} \) is perpendicular to the direction of propagation. The symmetry between the \( \dot{\rho} \) term in \( \mathbf{E} \) and the \( \dot{\mathbf{J}} \) term in \( \mathbf{B} \) reflects the underlying consistency of Maxwell's equations and the conservation of charge.

Connection to Conservation Laws

Jefimenko's equations are not just mathematical solutions; they embody the physical principle of local conservation of charge. The appearance of \( \dot{\rho} \) and \( \dot{\mathbf{J}} \) is a direct consequence of the continuity equation \( \nabla \cdot \mathbf{J} + \partial_t \rho = 0 \). If charge were not conserved, the terms involving time derivatives would not combine in this specific way, and the fields would not satisfy the homogeneous Maxwell equations (\( \nabla \cdot \mathbf{B} = 0 \) and \( \nabla \times \mathbf{E} = -\partial_t \mathbf{B} \)). Thus, the structure of Jefimenko's equations is a direct manifestation of the interplay between source dynamics and field propagation.