Poynting's theorem

Energy is conserved, but for electromagnetic fields you want the local version — a statement, valid at every point, of where field energy goes when charges do work. Poynting's theorem is that local energy-conservation law, the electromagnetic analogue of the continuity equation for charge.

The theorem reads du/dt + div S = -J·E, where u = (epsilon_0/2)E^2 + (1/(2 mu_0))B^2 is the field energy density and S = E x B/mu_0 is the Poynting vector. Read it as a balance sheet at a point: the field energy stored there can decrease either because energy flows away (div S) or because the fields do work on charges (the term J·E, the rate the field delivers energy to matter). Integrated over a volume, it says the rate of change of field energy inside plus the power radiated out through the surface equals minus the mechanical work done on the charges within.

Poynting's theorem is how we assign a definite energy to the electromagnetic field and track it as it flows and converts. The term J·E is the crucial bridge to mechanics — positive when the field accelerates charges (as in a load), negative when charges pump energy into the field (as in an antenna or a battery). Set J = 0 in empty space and the theorem becomes pure field energy conservation, du/dt + div S = 0, exactly the continuity equation for radiant energy.