PoyntingTheorem

The Poynting theorem is a statement of local conservation of electromagnetic energy in classical electrodynamics. It expresses how the energy stored in and carried by electric and magnetic fields changes in time and space as the fields perform work on charges and as energy flows through space.

The theorem uses the Poynting vector S, defined as S = E × H, which represents the directional energy flux of the electromagnetic field. The electromagnetic energy density is u = 1/2 (E · D + B · H). In vacuum, D equals ε0E and H equals B/μ0, so the energy density becomes u = 1/2 ε0 E^2 + B^2/(2μ0). The local energy balance is given by the continuity equation ∂u/∂t + ∇·S = -E · J, where J is the current density (in full generality, J may be taken as the total current, with a common macroscopic form distinguishing free and bound currents).

In materials, D and H account for polarization and magnetization, and the same form of the balance equation holds with u = 1/2 (E · D + B · H) and S = E × H. The term -E · J represents the rate at which field energy is converted to other forms, such as heat, in the medium or absorbed by charges. The Poynting theorem thus unifies energy storage, transport by field energy flow, and energy transfer to matter.

Historically named after John Henry Poynting, the theorem underpins the analysis of energy transfer in antennas, waveguides, optical media, and many other electromagnetic systems.