4.1 Energy and Power

The field concept is based upon the hypothesis that electromagnetic energy is distributed over the space. The electric and magnetic fields carry energy and changing electric and magnetic energy densities are related to power flow in space [1 6]. We introduce the electric energy density w _e ( x, t) and the magnetic energy density w _m ( x, t) with the corresponding three-forms

(4.1a)

(4.1b)

The energy densities are given by

(4.2a)

(4.2b)

Figure 4.1 visualizes the exterior product of the field one-form ? and the flux density two-form . The resulting three-form is visualized by the subdivision of the space into cells. The number of cells per unit of volume is proportional to the electric energy density.

Figure 4.1: The exterior product of the field form ? and the flux density form .

In order to investigate energy storage and power flow in the electromagnetic field, we start again with Maxwell's equations (2.114a) and (2.114b). Exterior multiplication of Amp re's law from the left with - ? and of Faraday's law from the right with yields

(4.3)

(4.4)

where the dot means the partial derivative with respect to t, that is, . This yields

(4.5)

This equation can be brought into the form

(4.6)

The power loss density p _L( x, t) with the corresponding differential form

(4.7)

is given by

(4.8)

Due to the impressed current density J ₀, a...