other, and so first described the principle of charge conservation. Twentieth century physics has added dramatically to the understanding of charge 1. Electric charge is a fundamental property of matter, as is mass or dimension 2. Charge is quantized: there exists a smallest quantity (quantum) of charge that can be associated with matter. No smaller amount has been observed, and large amounts always occur in integral multiples of this quantity. 3. The charge quantum is associated with the smallest subatomic particles, and these particles interact through electrical forces. In fact, matter is organized and arranged through electrical interactions; for example, our perception of physical contact is merely the macroscopic manifestation of countless charges in our fingertips pushing against charges in the things we touch 4. Electric charge is an invariant: the value of charge on a particle does not depend on the speed of the particle. In contrast, the mass of a particle increases with speed 5. Charge acts as the source of an electromagnetic field; the field is an entity that can carry energy and momentum away from the charge via propagating waves We begin our investigation of the properties of the electromagnetic field with a detailed examination of its source 1.3.1 Macroscopic electromagnetics We are interested primarily in those electromagnetic effects that can be predicted by classical techniques using continuous sources(charge and current densities ). Although macroscopic electromagnetics is limited in scope, it is useful in many situations en countered by engineers. These include, for example, the determination of currents and voltages in lumped circuits, torques exerted by electrical machines, and fields radiated by antennas. Macroscopic predictions can fall short in cases where quantum effects are im- portant: e. g, with devices such as tunnel diodes. Even so, quantum mechanics can often be coupled with classical electromagnetics to determine the macroscopic electromagnetic properties of important materials Electric charge is not of a continuous nature. The quantization of atomic charge te for electrons and protons, te/3 and +2e/ 3 for quarks - is one of the most precisely established principles in physics(verified to 1 part in 10-). The value of e itself is known to great accuracy e=1.60217733 x 10-19 Coulombs(C) However, the discrete nature of charge is not easily incorporated into everyday engineer ing concerns. The strange world of the individual charge - characterized by particle spin, molecular moments, and thermal vibrations is well described only by quantum theory. There is little hope that we can learn to describe electrical machines using such concepts. Must we therefore retreat to the macroscopic idea and ignore the discretization of charge completely? A viable alternative is to use atomic theories of matter to estimate the useful scope of macroscopic electromagnetics Remember, we are completely free to postulate a theory of nature whose scope may be limited. Like continuum mechanics, which treats distributions of matter as if they were continuous, macroscopic electromagnetics is regarded as valid because it is verified by experiment over a certain range of conditions. This applicability range generally corresponds to dimensions on a laboratory scale, implying a very wide range of validit or engineers @2001 by CRC Press LLC
other, and so first described the principle of charge conservation. Twentieth century physics has added dramatically to the understanding of charge: 1. Electric charge is a fundamental property of matter, as is mass or dimension. 2. Charge is quantized:there exists a smallest quantity (quantum) of charge that can be associated with matter. No smaller amount has been observed, and larger amounts always occur in integral multiples of this quantity. 3. The charge quantum is associated with the smallest subatomic particles, and these particles interact through electrical forces. In fact, matter is organized and arranged through electrical interactions; for example, our perception of physical contact is merely the macroscopic manifestation of countless charges in our fingertips pushing against charges in the things we touch. 4. Electric charge is an invariant:the value of charge on a particle does not depend on the speed of the particle. In contrast, the mass of a particle increases with speed. 5. Charge acts as the source of an electromagnetic field; the field is an entity that can carry energy and momentum away from the charge via propagating waves. We begin our investigation of the properties of the electromagnetic field with a detailed examination of its source. 1.3.1 Macroscopic electromagnetics We are interested primarily in those electromagnetic effects that can be predicted by classical techniques using continuous sources (charge and current densities). Although macroscopic electromagnetics is limited in scope, it is useful in many situations encountered by engineers. These include, for example, the determination of currents and voltages in lumped circuits, torques exerted by electrical machines, and fields radiated by antennas. Macroscopic predictions can fall short in cases where quantum effects are important:e.g., with devices such as tunnel diodes. Even so, quantum mechanics can often be coupled with classical electromagnetics to determine the macroscopic electromagnetic properties of important materials. Electric charge is not of a continuous nature. The quantization of atomic charge — ±e for electrons and protons, ±e/3 and ±2e/3 for quarks — is one of the most precisely established principles in physics (verified to 1 part in 1021). The value of e itself is known to great accuracy: e = 1.60217733 × 10−19 Coulombs (C). However, the discrete nature of charge is not easily incorporated into everyday engineering concerns. The strange world of the individual charge — characterized by particle spin, molecular moments, and thermal vibrations — is well described only by quantum theory. There is little hope that we can learn to describe electrical machines using such concepts. Must we therefore retreat to the macroscopic idea and ignore the discretization of charge completely? A viable alternative is to use atomic theories of matter to estimate the useful scope of macroscopic electromagnetics. Remember, we are completely free to postulate a theory of nature whose scope may be limited. Like continuum mechanics, which treats distributions of matter as if they were continuous, macroscopic electromagnetics is regarded as valid because it is verified by experiment over a certain range of conditions. This applicability range generally corresponds to dimensions on a laboratory scale, implying a very wide range of validity for engineers
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. re 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( 10-4 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 normalized so that(r)dv=l An example is the Gaussian distribution f(r)=(xra2)-32e-r2 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,of(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 10-0 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 electron beam, a radius of 10-8m 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 @2001 by CRC Press LLC
Macroscopic effects as averaged microscopic effects. Macroscopic electromagnetics 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 measure 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 microscopic 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 f (r) = (πa2 ) −3/2 e−r 2/a2 , where a is the approximate radial extent of averaging. The spatial average of a microscopic 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 electron beam, a radius of 10−8 m proves useful, containing typically 106 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