Page 21 - Electromagnetics Handbook

Macroscopic effects as averaged microscopic effects. Macroscopic electromag-
netics can hold in a world of discrete charges because applications usually occur over
physical scales that include vast numbers of charges. Common devices, generally much
larger than individual particles, “average” the rapidly varying fields that exist in the
spaces between charges, and this allows us to view a source as a continuous “smear” of
charge. To determine the range of scales over which the macroscopic viewpoint is valid,
we must compare averaged values of microscopic fields to the macroscopic fields we mea-
sure in the lab. But if the effects of the individual charges are describable only in terms
of quantum notions, this task will be daunting at best. A simple compromise, which
produces useful results, is to extend the macroscopic theory right down to the micro-
scopic level and regard discrete charges as “point” entities that produce electromagnetic
fields according to Maxwell’s equations. Then, in terms of scales much larger than the
classical radius of an electron (≈ 10 −14 m), the expected rapid fluctuations of the fields
in the spaces between charges is predicted. Finally, we ask:over what spatial scale must
we average the effects of the fields and the sources in order to obtain agreement with the
macroscopic equations?
In the spatial averaging approach a convenient weighting function f (r) is chosen, and

is normalized so that f (r) dV = 1.
An example is the Gaussian distribution
2 −3/2 −r /a
f (r) = (πa ) e 2 2 ,
where a is the approximate radial extent of averaging. The spatial average of a micro-
scopic quantity F(r, t) is given by

F(r, t) = F(r − r , t) f (r ) dV . (1.1)
The scale of validity of the macroscopic model can be found by determining the averaging
radius a that produces good agreement between the averaged microscopic fields and the
macroscopic fields.

The macroscopic volume charge density. At this point we do not distinguish
between the “free” charge that is unattached to a molecular structure and the charge
found near the surface of a conductor. Nor do we consider the dipole nature of polarizable
materials or the microscopic motion associated with molecular magnetic moment or the
magnetic moment of free charge. For the consideration of free-space electromagnetics,
we assume charge exhibits either three degrees of freedom (volume charge), two degrees
of freedom (surface charge), or one degree of freedom (line charge).
In typical matter, the microscopic fields vary spatially over dimensions of 10 −10 m
or less, and temporally over periods (determined by atomic motion) of 10 −13 s or less.
At the surface of a material such as a good conductor where charge often concentrates,
averaging with a radius on the order of 10 −10 m may be required to resolve the rapid
variation in the distribution of individual charged particles. However, within a solid or
liquid material, or within a free-charge distribution characteristic of a dense gas or an
6
electron beam, a radius of 10 −8 m proves useful, containing typically 10 particles. A
diffuse gas, on the other hand, may have a particle density so low that the averaging
radius takes on laboratory dimensions, and in such a case the microscopic theory must
be employed even at macroscopic dimensions.
Once the averaging radius has been determined, the value of the charge density may
be found via (1.1). The volume density of charge for an assortment of point sources can

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