Poynting's theorem

Poynting's theorem is the local statement of energy conservation for the electromagnetic field, derivable in two lines from Maxwell's equations. Start with u = (ε₀/2)E² + B²/(2μ₀), the electromagnetic energy density. Take its time derivative, substitute Maxwell's equations for ∂E/∂t and ∂B/∂t, and rearrange. The result is ∂u/∂t + ∇·S = −J·E, where S = E×B/μ₀ is the Poynting vector. This says: the rate of change of field energy at a point equals the net flow of field energy into the point (via −∇·S) plus the rate at which the field does negative work on the charges there (via −J·E, which is negative when the current is flowing with E, i.e., when charges are gaining kinetic energy from the field).

Integrated over a volume Ω with surface ∂Ω, the theorem reads d/dt ∫Ω u dV = −∮∂Ω S·dA − ∫Ω J·E dV. The first term on the right is the power flowing out through the surface; the second is the rate at which the field is giving up energy to charges inside. Energy is conserved — none disappears, none appears from nowhere. The theorem explicitly identifies S as the energy-flux vector and u as the energy density, closing the bookkeeping Maxwell's equations had opened.

The theorem's practical payoff: every electromagnetic energy-flow analysis — radiation from antennas, absorption in solar cells, heating in resistive cookware, thermal radiation from hot objects, power flow through coaxial cables — uses Poynting's theorem as the framework. The antenna engineer computes directivity by integrating |S| over a far-field sphere. The solar-cell designer computes efficiency by comparing absorbed S (the photon flux) to output electrical power. The waveguide designer computes modal power by integrating S across the cross-section. It is the exact conservation statement that makes energy a tractable quantity in electromagnetism.