1.2.2 Formalization of field theory Before we can invoke physical laws, we must find a way to describe the state of the ystem we intend to study. We generally begin by identifying a set of state variable that can depict the physical nature of the system. In a mechanical theory such as Newtons law of gravitation, the state of a system of point masses is expressed in terms of the instantaneous positions and momenta of the individual particles. Hence 6N state variables are needed to describe the state of a system of N particles, each particle having three position coordinates and three momentum components. The time evolution of the system state is determined by a supplementary force function(e. g, gravitation attraction), the initial state(initial conditions), and Newtons second law F=dP/dr Descriptions using finite sets of state variables are appropriate for action-at-a-distance interpretations of physical laws such as Newtons law of gravitation or the interaction of charged particles. If Coulomb,s law were taken as the force law in a mechanical description of electromagnetics, the state of a system of particles could be described completely in terms of their positions, momenta, and charges. Of course, charged particle interaction is not this simple. An attempt to augment Coulombs force law with Amperes force law would not account for kinetic energy loss via radiation. Hence we abandon the mechanical viewpoint in favor of the field viewpoint, selecting a different set of state variables. The essence of field theory is to regard electromagnetic phenomena as affecting all of space. We shall find that we can describe the field in terms of the four vector quantities E, D, B, and H. Because these fields exist by definition at each point in space and each time t, a finite set of state variables cannot describe the system Here then is an important distinction between field theories and mechanical theories the state of a field at any instant can only be described by an infinite number of state variables. Mathematically we describe fields in terms of functions of continuous variables however, we must be careful not to confuse all quantities described as"fields"with those fields innate to a scientific field theory. For instance, we may refer to a temperature "field"in the sense that we can describe temperature as a function of space and time However, we do not mean by this that temperature obeys a set of physical laws analogous to those obeyed by the electromagnetic field What special character, then, can we ascribe to the electromagnetic field that has meaning beyond that given by its mathematical implications? In this book, E, D, B and H are integral parts of a field-theory description of electromagnetics. In any field theory we need two types of fields: a mediating field generated by a source, and a field describing the source itself. In free-space electromagnetics the mediating field consists of E and B, while the source field is the distribution of charge or current. An important consideration is that the source field must be independent of the mediating field that it"sources. "Additionally, fields are generally regarded as unobservable: they can only be measured indirectly through interactions with observable quantities. We need a link to mechanics to observe e and B: we might measure the change in kinetic energy of a particle as it interacts with the field through the lorentz force. The Lorentz force becomes the force function in the mechanical interaction that uniquely determines the (observable)mechanical state of the particle A field is associated with a set of field equations and a set of constitutive relations. The field equations describe, through partial derivative operations, both the spatial distribu- tion and temporal evolution of the field. The constitutive relations describe the effect TAttempts have been made to formulate electromagnetic theory purely in action-at-a-distance terms but this viewpoint has not been generally adopted 69 @2001 by CRC Press LLC1.2.2 Formalization of field theory Before we can invoke physical laws, we must find a way to describe the state of the system we intend to study. We generally begin by identifying a set of state variables that can depict the physical nature of the system. In a mechanical theory such as Newton’s law of gravitation, the state of a system of point masses is expressed in terms of the instantaneous positions and momenta of the individual particles. Hence 6N state variables are needed to describe the state of a system of N particles, each particle having three position coordinates and three momentum components. The time evolution of the system state is determined by a supplementary force function (e.g., gravitational attraction), the initial state (initial conditions), and Newton’s second law F = dP/dt. Descriptions using finite sets of state variables are appropriate for action-at-a-distance interpretations of physical laws such as Newton’s law of gravitation or the interaction of charged particles. If Coulomb’s law were taken as the force law in a mechanical description of electromagnetics, the state of a system of particles could be described completely in terms of their positions, momenta, and charges. Of course, charged particle interaction is not this simple. An attempt to augment Coulomb’s force law with Ampere’s force law would not account for kinetic energy loss via radiation. Hence we abandon1 the mechanical viewpoint in favor of the field viewpoint, selecting a different set of state variables. The essence of field theory is to regard electromagnetic phenomena as affecting all of space. We shall find that we can describe the field in terms of the four vector quantities E, D, B, and H. Because these fields exist by definition at each point in space and each time t, a finite set of state variables cannot describe the system. Here then is an important distinction between field theories and mechanical theories: the state of a field at any instant can only be described by an infinite number of state variables. Mathematically we describe fields in terms of functions of continuous variables; however, we must be careful not to confuse all quantities described as “fields” with those fields innate to a scientific field theory. For instance, we may refer to a temperature “field” in the sense that we can describe temperature as a function of space and time. However, we do not mean by this that temperature obeys a set of physical laws analogous to those obeyed by the electromagnetic field. What special character, then, can we ascribe to the electromagnetic field that has meaning beyond that given by its mathematical implications? In this book, E, D, B, and H are integral parts of a field-theory description of electromagnetics. In any field theory we need two types of fields:a mediating field generated by a source, and a field describing the source itself. In free-space electromagnetics the mediating field consists of E and B, while the source field is the distribution of charge or current. An important consideration is that the source field must be independent of the mediating field that it “sources.” Additionally, fields are generally regarded as unobservable:they can only be measured indirectly through interactions with observable quantities. We need a link to mechanics to observe E and B:we might measure the change in kinetic energy of a particle as it interacts with the field through the Lorentz force. The Lorentz force becomes the force function in the mechanical interaction that uniquely determines the (observable) mechanical state of the particle. A field is associated with a set of field equations and a set of constitutive relations. The field equations describe, through partial derivative operations, both the spatial distribu￾tion and temporal evolution of the field. The constitutive relations describe the effect 1Attempts have been made to formulate electromagnetic theory purely in action-at-a-distance terms, but this viewpoint has not been generally adopted [69]