In the previous chapters we saw several cases of the central role played by magnetic flux through open or closed surfaces. In this chapter we use magnetic flux and Faraday’s Law to derive a very general expression for the energy stored within a magnetic field. We will also return to the inductance of current loops in Section 6.2, with several examples and illustrations, and then find the relation between the inductance of a circuit element and the energy stored within the magnetic field produced by the current in that element. Finally, we will explore magnetic forces and torques in Section 6.3.

We are now ready to introduce magnetic fields, which are generated by electrical currents and which apply forces on moving charges and current-carrying wires. Historically, magnetic effects in lodestones, an iron ore that can be magnetized, have been known for a long time. The first magnetic compasses date back to about 1000 BCE, and the ancient Chinese are believed to have used such devices for navigation as early as 1100 CE. The properties of magnetic fields can be derived from a number of observations of magnetic effects that have been recorded over many years. One of the earliest such observations, by Hans Christian Oersted in 1820, was that a current-carrying wire exerts a torque on a permanent magnet (such as a compass). Current-carrying wires can also exert forces on each other, as first observed by Biot and Savart and more fully characterized by Ampère. Finally, beams of charged particles, such as electrons in a cathode ray tube (see TechNote 3.4), are deflected when in the presence of current-carrying wires. Each of these phenomena can be described quantitatively in terms of a magnetic field produced by current distributions, as we will discuss throughout this chapter.

With the introduction in the previous chapter of the electric field and electric potential, and their properties in materials, we are now ready to examine the energy stored in electric fields, the electric forces that can be exerted on objects, and the capacitance between conductors. We cover these topics in this chapter. We also introduce additional methods that can be used for determining electric fields and potentials.

As we have been discussing electric and magnetic effects throughout this text, we have been developing a set of equations, known collectively as Maxwell’s Equations, that describe the properties of these fields in a very general sense. These equations are named after James Clerk Maxwell, whose contributions are discussed in Biographical Note 7.1. The development of Maxwell’s Equations has been critical to our understanding and application of electromagnetic effects, as they govern such diverse effects as are present in capacitors, transformers, and electric generators, which we have already examined, and free-wave propagation, transmission lines, waveguides, and antennas, which we have not yet discussed. Before we can undertake our study of these new topics, we must first complete the development of Maxwell’s Equations, which are not quite finished. As we will show shortly, there is an inconsistency in these equations as they stand to this point, an inconsistency that can be rectified by introducing a new term, known as the displacement current, to Ampère’s Law. This additional term is the final piece of the puzzle, and with its inclusion Maxwell’s Equations can be used to describe wave propagation, allowing us to understand (at an overview level, at least) the principles that govern our wireless routers, microwave ovens, and cable and satellite TV systems. In this chapter, we will introduce the displacement current, redefine the potential functions for time-varying fields, and re-examine the boundary conditions that must be satisfied at the interface between two different materials.