Fundamentals of Plasma Physics 2025, 30 - Wave-Energy Equation

For EM energy density, as usual consult Poynting's theorem, E∇×B - B∇×E = E(μ₀J+ε₀μ₀∂_tE) + B∂_tB, or equivalently, ∂_tw + ∇P = 0, and P the pointing vector. By integration, w(t) = w(t₀) + ∫_t0^t dt {EJ + ε₀E∂_t + (μ₀)^-1B∂_tB}, with EJ is the rate of change in kinetic energy density of the particles. If positive, it corresponds to work going into the particles.

The maths allow for a negative energy densities, in the change of kinetic energy density to create the wave, which may be larger in magnitude than the energy densities for the positive definite electric and magnetic contributions. The wave energy density is the change in the total system energy density in going from a situation where there is no wave to a situation where there is a wave. For example, consider unmagnetized cold electron streams with v₀ through a background of infinitely massive ions.