Chapter 1
Background

This chapter provides a review of Maxwell's equations in integral and differential forms. The capacitor and inductor are used to demonstrate and interpret the integral forms. The Poynting theorem, Lorentz reciprocity theorem, Friis transmission formula and radar range equation are also described. Some of the properties of high-frequency asymptotic techniques are reviewed.

1.1 Field Laws

Maxwell's equations in integral form are

We will use the MKS system of units. The Volt, Ampere, Coulomb, Weber, and Tesla are abbreviated as V, A, C, Wb, and T. The electric field is in ; the magnetic field is in ; the electric flux density is in , and the magnetic flux density is in (equivalent to ). The electric current density is in ; charge is in , and charge density is in .

The surface and volume integrals are associated with the mathematical surfaces shown in Figure 1.1. The first equation is Faraday's law. The second one is credited to Ampère and Maxwell, and the third one is Gauss's law. The fourth equation is called Gauss's law for magnetism. The group of four equations is usually referred to as Maxwell's equations.

Figure 1.1 Mathematical surfaces associated with the field laws. (a) Closed surface and volume, (b) open surface and contour C, (c) boundary between regions 1 and 2.

If a region has fields , , and a charge is moving through those fields with a velocity u, the charge will experience a force, in accordance with the Lorentz force law

Charge cannot be created or destroyed. Any increase or decrease of charge occurs because there is a current. This is stated mathematically as the continuity equation. In integral form, the outflux of current across a closed surface S equals the time rate of decrease of the charge that is inside S

The point-form equivalent is

By integrating both sides of (1.7) over a volume V and applying the divergence theorem to the left-hand side, the integral form (1.6) is obtained.

From the electric field, the voltage is

An electric field having is said to be irrotational. This occurs in electrostatics and in the transverse cross section of a transmission line. In these cases, the line integral becomes path independent, and hence, the voltage is uniquely defined by the endpoints a and b.

From the magnetic field, the current is

where is in the direction of the right-hand thumb and C is a closed contour in the direction of the fingers. This relationship is strictly true for steady (DC) currents. It is still true in the AC case if there is no component perpendicular to the surface bounded by C.

1.2 Properties of Materials

The electrical properties of materials are governed by their physical makeup. In this book, the physics and chemistry of these topics will not be covered, and the reader is referred to the references at the end of the chapter. It will be adequate for our purposes to describe the mathematical models that account for the presence of materials.

In free space, we have the constitutive relations

Interestingly, in any system of units, the value of or can be arbitrarily chosen. However, must equal the speed of light. In the MKS system, in (1.13) is chosen as an exact value. Then, in (1.12) is determined.

In dielectric materials,

The term is what we have in free space. If a dielectric is present, the applied electric field will push its atomic charges, positive towards one direction and negative in the opposite direction, forming dipoles. These dipoles contribute an additionalelectric flux density , the polarization in . Generally, the relation between and can be complicated, that is, non-linear. In the special case of linear materials, is linearly proportional to the applied field. More precisely, where the constant of proportionality is called the electric susceptibility. In this case,

Therefore, in linear materials, we can use the simple relation where the permittivity is .

In magnetic materials,

The term is what we have in free space. If a magnetic material is present, the applied magnetic field will reorient the material's electronic orbits (which act as current loops) and contribute an additional magnetic flux density , the magnetization in . In non-linear materials, the relation between and can be complicated. In the special case of linear materials, where the constant of proportionality is called the magnetic susceptibility. In this case,

Therefore, in linear materials, we can use the simple relation where the permeability is .

In a good conductor, when an electric field is applied, the charges move immediately. Dipoles (as in a dielectric) do not have a chance to form. Therefore, and consequently . In non-magnetic materials, . Such approximations are good for non-magnetic conductors such as aluminium or copper.

1.3 Types of Currents

The convection current is associated with charges that are moving with a velocity u

Such a 'stream' of charged particles occurs, for example, in a vacuum tube, a cathode ray tube or a scanning electron microscope.

Inside a conductor, an electric field will push on the charges and cause a conduction current

The conductivity is in S/m (Siemens/m, or equvalently, mho/m). The main difference between a convection current and a conduction current is that the latter type occurs in an electrically neutral material. For example, in a wire, for every charge that enters at one end, a charge leaves at the other end. Therefore there is no net charge and .

An impressed current is independent of the field around it, but the field around it depends on the impressed current. An example of an impressed current is a dipole antenna. An induced current comes from the interaction of a field with any surrounding media and/or boundaries. As an example, if a dipole antenna illuminates a metal body, it will cause surface currents to flow on the body; these are induced currents. The purpose of induced currents is that they adjust themselves in just the right way so that their field, when added to the impressed field, will give a total field that satisfies the boundary conditions, that is, on the metal. Inside dielectrics there are volume-equivalent induced currents; these are discussed in Chapter 4.

1.4 Capacitors, Inductors

To gain a better understanding of Maxwell's equations in the integral form, this section demonstrates their application to the fields inside capacitors and inductors.

First, the Ampere-Maxwell equation

will be applied to a capacitor, in Figure 1.2. The capacitor supports an electric field in the region . With in case (a), the current density pierces . Because is zero outside the capacitor, will be zero on , so that (1.20) becomes

With in case (b), the current density is zero on and so that

The right-hand side of (1.21) is the total current . Since C is the same in both cases, the left-hand side of (1.21) and (1.22) are equal. This leads to

or

Recognizing the capacitance , we see that

Figure 1.2 (a) Contour C bounds the disk . (b) Contour C bounds the open surface .

If is right at the surface of the plate, then , and we can say that from which we obtain the well-known result . Equating this with (1.23) implies that , which gives us the capacitance .

Next, we apply Faraday's law to a wire loop and a toroidal inductor. Figure 1.3(a) shows a wire loop in the plane. The integration path C is tangent to the wire and crosses the gap at the terminals 1-2. Because on the wire, the line integral is zero everywhere except at the gap, and

The direction of C implies that . Let us denote the flux through the loop as . If there are N turns, the flux is , and Faraday's law (1.24) becomes

We can apply this result to the toroidal inductor in Figure 1.3(b). To better understand the relationships between , i and , it is helpful to consider what happens if a positive step of voltage is applied to the terminals. Equation (1.25) indicates that a ramp of flux will occur in the indicated direction. Also, by Ampere's law, if the right-hand thumb points in the direction of , the fingers give the direction of i.

Figure 1.3 (a)...