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 LLCother, 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 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 — ±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 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 validity for engineers