# Chapter 6 Maxwell’s Equations for Time-Varying Fields

# Objectives

Upon learning the material presented in this chapter, you should be able to:

1. Apply Faraday’s law to compute the voltage induced by a stationary coil placed in a time-varying magnetic field or moving in a medium containing a magnetic field.

2. Describe the operation of the electromagnetic generator.

3. Calculate the displacement current associated with a time-varying electric field.

4. Calculate the rate at which charge dissipates in a material with known ε and σ.

# Dynamic Fields

Electric charges induce electric fields and electric currents induce magnetic fields. As long as the charge and current distributions remain constant in time, so will the fields they induce. If the charges and currents vary in time, the electric and magnetic fields vary accordingly. Additionally, however, the electric and magnetic fields become coupled and travel through space in the form of electromagnetic waves. Examples of such waves include light, x-rays, infrared, gamma rays, and radio waves (see Fig. 1-16).

To study time-varying electromagnetic phenomena, we need to consider the entire set of Maxwell’s equations simultaneously. These equations, first introduced in the opening section of Chapter 4, are given in both differential and integral form in Table 6-1. In the static case (∂/∂t=0), we use the first pair of Maxwell’s equations to study electric phenomena (Chapter 4) and the second pair to study magnetic phenomena (Chapter 5). In the dynamic case (∂/∂t≠0), the coupling that exists between the electric and magnetic fields, as expressed by the second and fourth equations in Table 6-1, prevents such decomposition. The first equation represents Gauss’s law for electricity, and it is equally valid for static and dynamic fields. The same is true for the third equation, which is Gauss’s law for magnetism. By contrast, the second and fourth equations—Faraday’s and Ampère’s laws—are of a totally different nature. Faraday’s law expresses the fact that a time-varying magnetic field gives rise to an electric field. Conversely, Ampère’s law states that a time-varying electric field must be accompanied by a magnetic field.

# Table 6-1 Maxwell’s equations.

Table 6 1 Full Alternative Text

Some statements in this and succeeding chapters contradict conclusions reached in Chapters 4 and 5 as those pertained to the special case of static charges and dc currents. The behavior of dynamic fields reduces to that of static ones when ∂/∂t is set to zero.

We begin this chapter by examining Faraday’s and Ampère’s laws and some of their practical applications. We then combine Maxwell’s equations to obtain relations among the charge and current sources, ρ v and J; the scalar and vector potentials, V and A; and the electromagnetic fields, E, D, H, and B. We do so for the most general time-varying case and for the specific case of sinusoidal-time variations.