Page 22 - Electromagnetics

be written in terms of the three-dimensional Dirac delta as

o
ρ (r, t) = q i δ(r − r i (t)),
i
where r i (t) is the position of the charge q i at time t. Substitution into (1.1) gives

o
ρ(r, t) =
ρ (r, t) = q i f (r − r i (t)) (1.2)
i
as the averaged charge density appropriate for use in a macroscopic field theory. Because
the oscillations of the atomic particles are statistically uncorrelated over the distances
used in spatial averaging, the time variations of microscopic fields are not present in the
macroscopic fields and temporal averaging is unnecessary. In (1.2) the time dependence
of the spatially-averaged charge density is due entirely to bulk motion of the charge
aggregate (macroscopic charge motion).
With the definition of macroscopic charge density given by (1.2), we can determine
the total charge Q(t) in any macroscopic volume region V using

Q(t) = ρ(r, t) dV. (1.3)
V
We have

Q(t) = q i f (r − r i (t)) dV = q i .
i V r i (t)∈V
Here we ignore the small discrepancy produced by charges lying within distance a of
the boundary of V . It is common to employ a box B having volume V :

f (r) = 1/ V, r ∈ B,
0, r /∈ B.
In this case
1
ρ(r, t) = q i .
V
r−r i (t)∈B
The size of B is chosen with the same considerations as to atomic scale as was the
averaging radius a. Discontinuities at the edges of the box introduce some difficulties
concerning charges that move in and out of the box because of molecular motion.

The macroscopic volume current density. Electric charge in motion is referred
to as electric current. Charge motion can be associated with external forces and with
microscopic fluctuations in position. Assuming charge q i has velocity v i (t) = dr i (t)/dt,
the charge aggregate has volume current density
o
J (r, t) = q i v i (t)δ(r − r i (t)).
i
Spatial averaging gives the macroscopic volume current density

o
J(r, t) =
J (r, t) = q i v i (t) f (r − r i (t)). (1.4)
i

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