# 4.1 Maxwell’s Equations

The modern theory of electromagnetism is based on a set of four fundamental relations known as Maxwell’s equations:

Here E and D are the electric field intensity and flux density interrelated by D=ε E where ε is the electrical permittivity; H and B are magnetic field intensity and flux density interrelated by B=μ H where μ is the magnetic permeability; ρ v is the electric charge density per unit volume; and J is the current density per unit area. The fields and fluxes E, D, B, H were introduced in Section 1.3, and ρ v and J will be discussed in Section 4-2. Maxwell’s equations hold in any material, including free space (vacuum). In general, all of the above quantities may depend on spatial location and time t. In the interest of readability, we will not, however, explicitly reference these dependencies (as in E(x, y, z, t)) except when the context calls for it. By formulating these equations, published in a classic treatise in 1873, James Clerk Maxwell established the first unified theory of electricity and magnetism. His equations are deduced from experimental observations reported by Coulomb, Gauss, Ampère, Faraday, and others; they not only encapsulate the connection between the electric field and electric charge and between the magnetic field and electric current but also capture the bilateral coupling between electric and magnetic fields and fluxes. Together with some auxiliary relations, Maxwell’s equations comprise the fundamental tenets of electromagnetic theory.

Under these circumstances, the time derivatives of B and D in Eqs. (4.1b) and (4.1d) vanish, and Maxwell’s equations reduce to the following pairs.

Electrostatics

Magnetostatics

Maxwell’s four equations separate into two uncoupled pairs with the first pair involving only the electric field and flux E and D and the second pair involving only the magnetic field and flux H and B.

This allows us to study electricity and magnetism as two distinct and separate phenomena as long as the spatial distributions of charge and current flow remain constant in time. We refer to the study of electric and magnetic phenomena under static conditions as electrostatics and magnetostatics, respectively. Electrostatics is the subject of the present chapter, and we learn about magnetostatics in Chapter 5. The experience gained through studying electrostatic and magnetostatic phenomena will prove invaluable in tackling the more involved material in subsequent chapters that deal with time-varying fields, charge densities, and currents.

We study electrostatics not only as a prelude to the study of time-varying fields but also because it is an important field in its own right. Many electronic devices and systems are based on the principles of electrostatics. They include x-ray machines, oscilloscopes, ink-jet electrostatic printers, liquid crystal displays, copy machines, micro-electromechanical switches and accelerometers, and many solid-state–based control devices. Electrostatic principles also guide the design of medical diagnostic sensors, such as the electrocardiogram, which records the heart’s pumping pattern, and the electroencephalogram, which records brain activity, as well as the development of numerous industrial applications.