Poynting Vector - Formulation in Terms of Microscopic Fields

Formulation in Terms of Microscopic Fields

In some cases, it may be more appropriate to define the Poynting vector S as

where μ₀ is the magnetic constant. It can be derived directly from Maxwell's equations in terms of total charge and current and the Lorentz force law only.

The corresponding form of Poynting's theorem is

where J is the total current density and the energy density u is

where ε₀ is the electric constant.

The two alternative definitions of the Poynting vector are equivalent in vacuum or in non-magnetic materials, where B = μ₀ H. In all other cases, they differ in that

and the corresponding u are purely radiative, since the dissipation term, (−J • E) covers the total current, while the definition in terms of H has contributions from bound currents which then lack in the dissipation term.

Since only the microscopic fields E and B are needed in the derivation of

assumptions about any material possibly present can be completely avoided, and Poynting's vector as well as the theorem in this definition are universally valid, in vacuum as in all kinds of material. This is especially true for the electromagnetic energy density, in contrast to the case above.