Direct Energy, 2018a

12 RELATING ENERGY CONVERSION PROCESSES 273
Q
v
Ψ
i
Circuit Quantity
Charge in C
Voltage in V
Magnetic ux in Wb
Current in A
Electromagnetic Field
−→ D Displacement ux density in C m
2
−→ E
−→ B
−→ H
Electric eld intensity in V m
magnetic ux density in Wb
m 2
Magnetic eld intensity in A m
Table 12.2: Quantities used to describe circuits and electromagnetic elds.
Using electromagnetics language, four vector elds describe systems:
−→ D displacement ux density in C m
2 , −→ E electric eld intensity in V m , −→ B
magnetic ux density in Wb m , and −→ H magnetic eld intensity in A 2
m . These
electromagnetic elds are generalizations of the circuit parameters charge
Q, voltage v, magnetic ux Ψ, and current i respectively as shown in Table
12.2. However, the electromagnetic elds are functions of position x, y, and
z in addition to time, and they are vector instead of scalar quantities. More
specically, displacement ux density is the charge built up on a surface per
unit area, and magnetic ux density is the magnetic ux through a surface.
Similarly, electric eld intensity is the negative gradient of the voltage, and
magnetic eld intensity is the gradient of the current. We encountered
these electromagnetic elds when discussing antennas in Chapter 4.
A capacitor can store energy in the charge built up between the capacitor
plates. Analogously, an insulating material with permittivity greater
than the permittivity of free space, ɛ>ɛ 0 , can store energy in the distributed
charge separation throughout the material. We can describe the
energy conversion processes occurring in a capacitor using the language
of calculus of variations by choosingeither charge Q or voltage v as the
generalized path. Parameters resulting from these choices are shown in the
second and third column of Table 12.1. Analogously, we can describe the
energy conversion processes occurring in an insulating material with ɛ>ɛ 0
using the language of calculus of variations by choosing either −→ D or −→ E as
the generalized path. Parameters resulting from these choices are shown in
the second and third column of Table 12.3. The equation of motion that
results in either case is Gauss's law for the electric eld,
−→ ∇·−→ D = ρch (12.3)
where ρ ch is charge density. The derivation is beyond the scope of this text,
however, because it involves applyingcalculus of variations to quantities
with multiple independent and dependent variables. Gauss's law is one of