In electrodynamics, Poynting's theorem is a statement of conservation of energy for electromagnetic fields developed by British physicist John Henry Poynting. It states that in a given volume, the stored energy changes at a rate given by the work done on the charges within the volume, minus the rate at which energy leaves the volume. It is only strictly true in media which is not dispersive, but can be extended for the dispersive case. The theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation. Definition Poynting's theorem states that the rate of energy transfer per unit volume from a region of space equals the rate of work done on the charge distribution in the region, plus the energy flux leaving that region. Mathematically: where: is the rate of change of the energy density in the volume. ∇•S is the energy flow out of the volume, given by the divergence of the Poynting vector S. J•E is the rate at which the fields do work on charges in the volume (J is the current density corresponding to the motion of charge, E is the electric field, and • is the dot product). Integral Form Using the divergence theorem, Poynting's theorem can also be written in integral form: where S is the energy flow, given by the Poynting Vector. is the energy density in the volume. is the boundary of the volume. The shape of the volume is arbitrary but fixed for the calculation. Continuity Equation Analog In an electrical engineering context the theorem is sometimes written with the energy density term u expanded as shown. This form resembles the continuity equation: , where ε0 is the vacuum permittivity and μ0 is the vacuum permeability. is the density of reactive power driving the build-up of electric field, is the density of reactive power driving the build-up of magnetic field, and is the density of electric power dissipated by the Lorentz force acting on charge carriers. Derivation For an individual charge in an electromagnetic field, the rate of work done by the field on the charge is given by the Lorentz Force Law as: Extending this to a continuous distribution of charges, moving with current density J, gives: By Ampère's circuital law: (Note that the H and D forms of the magnetic and electric fields are used here. The B and E forms could also