Direct Energy, 2018a

274 12.2 Electrical Energy Conversion
Maxwell's equations, and it was introduced in Section 1.6.1. In Chapter 2,
piezoelectric energy conversion devices were discussed, and in Chapter 3,
pyroelectric and electro-optic energy conversion devices were discussed. All
of these devices involved converting electrical energy to and from energy
stored in a material polarization of an insulating material with ɛ>ɛ 0 .
Calculus of variations can be used to describe energy conversion in all of
these devices with either displacement ux density or electric eld intensity
as the generalized path. For a device made from a material of permittivity
ɛ withan external electric eld intensity across it given by −→ E, the energy
density stored is 1ɛ|−→ E | 2 in
J 2 m . The energy stored in a volume V is found
3
by integrating this energy density with respect to volume, and this energy
stored in a volume is listed in the second to last row of Table 12.3. Notice
the similarity of the equation for the energy stored in a capacitor (second
column, second to last box of Table 12.1) and this equation for the energy
density stored in a material with ɛ>ɛ 0 (second column second to last box
of the Table 12.3).
Energy can also be stored in materials withpermeability greater than
the permeability of free space, μ>μ 0 . Hall eect devices and magnetohydrodynamic
devices were discussed in Chapter 5. These devices are all
inductor-like, and the parameters used to describe inductive energy conversion
processes in the language of calculus of variations are summarized
in the last two columns of the Table 12.3. Calculus of variations can be
used to describe energy conversion processes in these devices with either
magnetic ux density or magnetic eld intensity as the generalized path
and the other choice as the generalized potential. The equation of motion
resulting from using calculus of variations to describe inductive systems
corresponds to Gauss's law for the magnetic eld,
−→ ∇·−→ B =0. (12.4)
The physics of antennas is described by electric and magnetic elds, and
any of the columns of Table 12.3 can be used to describe energy conversion
between electricity and electromagnetic waves in antennas using the
language of calculus of variations.