ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 38

which can be generalized into the following double summation to account for n
charges:

n n

1
4 πε 0

q i q j
r ij

W tot = (2-68)

i = 1

j 1

j>i

=

The limit on the j term exists to ensure that terms are not counted twice. For
example, without the limit the term 12 would be counted twice because 12 =

21 , which simply means that it takes the same amount of work to move q 1 into
the vicinity of q 2 as it does to move q 2 into the vicinity of q 1 . Equation (2-68)
can be simplified by allowing the terms to be counted twice and dividing by 2
to compensate for the double counting of terms:

=

n n n n

1
2

1
4 πε 0

q i q j
r ij

1
2

W tot = q i ij (2-69)

i = 1 i = 1

j = 1 j = 1

j = i j = i

Equation (2-69) is the work it takes to assemble n charges. It is interesting to
consider where the energy is stored in an accumulation of charges. It is analogous
to a mechanical system of a compressed spring with a weight on each end. If
the weights are forced together, the energy is stored in the stressed state of the
spring. Therefore, just like the spring example, the charges, which will tend to
repel each other, will have stored energy that is a function of the proximity of
the charges and the properties of an electric field.
To calculate the energy stored in a continuous charge distribution, and thus
an electric field, it is more convenient to express the charge in terms of the unit
volume, the potential in terms of a continuous function, and to take the limit as
n →∞ , which allows us to write (2-69) in terms of an integral,

2 V ρ(r)(r)dV

1

W tot = W e = joules (2-70)

where ρ (r) is the charge density in units of C/m 3 .
It is useful to express (2-70) in terms of the electric field. To do this, Gauss's
law is used to express the charge density in terms of the electric field:

2 V ( ∇· ε E)(r) dV

∇· D =∇· ε E = ρ → W e =

1

(2-71)

This equation can be simplified using the flowing vector identity (Appendix
A):

∇· ψ a = a ·∇ ψ + ψ( ∇· a) → ψ( ∇· a) =∇· ψ a − a ·∇ ψ