be written in terms of the three-dimensional dirac delta as p(r,)=∑ here ri (t)is the position of the charge qi at time 1. Substitution into(1.1)gives p(r,1)=p°r,1)=∑qf(r-r1(t) (1.2) 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 th 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 e()= p(r, t)dv (1.3) We have 0=∑%-;o)y=∑ o Here we ignore the small discrepancy produced by charges lying within distance a of e boundary of v. It is common to employ a box b having volume AV (r)=1/△V,r∈B E B In this case A∑ r-r()∈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 qi has velocity v;(t)=dri (t)/dt the charge aggregate has volume current density T(r, t) qv()8(r-r;(t) Spatial averaging gives the macroscopic volume current densit J(r,)=o(r, d)=2qivi(f(r-r() @2001 by CRC Press LLCbe written in terms of the three-dimensional Dirac delta as ρo (r, t) =  i qi δ(r − ri(t)), where ri(t) is the position of the charge qi at time t. Substitution into (1.1) gives ρ(r, t) = ρo (r, t) =  i qi f (r − ri(t)) (1.2) 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) = V ρ(r, t) dV. (1.3) We have Q(t) =  i qi V f (r − ri(t)) dV =  ri(t)∈V qi . 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 ρ(r, t) = 1 V  r−ri(t)∈B qi . 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 qi has velocity vi(t) = dri(t)/dt, the charge aggregate has volume current density Jo (r, t) =  i qivi(t)δ(r − ri(t)). Spatial averaging gives the macroscopic volume current density J(r, t) = Jo (r, t) =  i qivi(t) f (r − ri(t)). (1.4)