Page 153 - Electromagnetics

Figure 3.10: Computation of electrostatic stored energy via the assembly energy of a
charge distribution.

Electrostatic stored energy. In § 2.9.5 we considered the energy relations for the
electromagnetic field. Those relations remain valid in the static case. Since our interpre-
tation of the dynamic relations was guided in part by our knowledge of the energy stored
in a static field, we must, for completeness, carry out a study of that effect here.
The energy of a static configuration is taken to be the work required to assemble the
configuration from a chosen starting point. For a configuration of static charges, the
stored electric energy is the energy required to assemble the configuration, starting with
all charges removed to infinite distance (the assumed zero potential reference). If the
assembled charges are not held in place by an external mechanical force they will move,
thereby converting stored electric energy into other forms of energy (e.g., kinetic energy
and radiation).
By (3.62), the work required to move a point charge q from a reservoir at infinity to
a point P at r in a potential field is

W = q (r).

If instead we have a continuous charge density ρ present, and wish to increase this to
ρ + δρ by bringing in a small quantity of charge δρ, a total work

δW = δρ(r) (r) dV (3.84)
V ∞
is required, and the potential field is increased to + δ . Here V ∞ denotes all of space.
(We could restrict the integral to the region containing the charge, but we shall find it
helpful to extend the domain of integration to all of space.)
Now consider the situation shown in Figure 3.10. Here we have charge in the form of
both volume densities and surface densities on conducting bodies. Also present may be
linear material bodies. We can think of assembling the charge in two distinctly different

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