Figure 1. 1: Intersection of the averaging function of a point charge with a surface S, as the charge crosses S with velocity v:(a) at some time t= tI, and(b)at t=f2>t1. The averaging function is represented by a sphere of radius a Spatial averaging at time t eliminates currents associated with microscopic motions that are uncorrelated at the scale of the averaging radius (again, we do not consider the magnetic moments of particles). The assumption of a sufficiently large averaging radius leads to J(r, t)=p(r, t)v(r, t) The total Hux /(t) of current through a surface S is gi I(r)=J(r, t)ndS where n is the unit normal to S. Hence, using(4), we have ∑qaco)/rr-ro)ds if f stays approximately constant over the extent of the averaging function and S is not in motion. We see that the integral effectively intersects S with the averaging function su rounding each moving point charge (Figure 1.1). The time derivative of ri. f represents the velocity at which the averaging function is"carried across"the surface. Electric current takes a variety of forms, each described by the relation J= pv. Isolated charged particles(positive and negative)and charged insulated bodies moving through space comprise convection currents. Negatively-charged electrons moving through the positive background lattice within a conductor comprise a conduction current. Empirical evidence suggests that conduction currents are also described by the relation J= oE known as Ohm's law. a third type of current, called electrolytic current, results from the How of positive or negative ions through a fuid 1.3.2 Impressed vS secondary sources In addition to the simple classification given above we may classify currents as primary or secondary, depending on the action that sets the charge in motion @2001 by CRC Press LLCFigure 1.1:Intersection of the averaging function of a point charge with a surface S, as the charge crosses S with velocity v:(a) at some time t = t1, and (b) at t = t2 > t1. The averaging function is represented by a sphere of radius a. Spatial averaging at time t eliminates currents associated with microscopic motions that are uncorrelated at the scale of the averaging radius (again, we do not consider the magnetic moments of particles). The assumption of a sufficiently large averaging radius leads to J(r, t) = ρ(r, t) v(r, t). (1.5) The total flux I(t) of current through a surface S is given by I(t) = S J(r, t) · nˆ d S where nˆ is the unit normal to S. Hence, using (4), we have I(t) =  i qi d dt (ri(t) · nˆ) S f (r − ri(t)) d S if nˆ stays approximately constant over the extent of the averaging function and S is not in motion. We see that the integral effectively intersects S with the averaging function sur￾rounding each moving point charge (Figure 1.1). The time derivative of r i ·nˆ represents the velocity at which the averaging function is “carried across” the surface. Electric current takes a variety of forms, each described by the relation J = ρv. Isolated charged particles (positive and negative) and charged insulated bodies moving through space comprise convection currents. Negatively-charged electrons moving through the positive background lattice within a conductor comprise a conduction current. Empirical evidence suggests that conduction currents are also described by the relation J = σE known as Ohm’s law. A third type of current, called electrolytic current, results from the flow of positive or negative ions through a fluid. 1.3.2 Impressed vs. secondary sources In addition to the simple classification given above we may classify currents as primary or secondary, depending on the action that sets the charge in motion.