Maxwell’s Equations: From 20 to 4

Source: ETHW website, date indeterminate

Maxwell’s Equations refer to a set of four relations that describe the properties and interrelations of electric and magnetic fields. The equations are shown in modern notation in Figure 2. The electric force fields are described by the quantities E (the electric field) and D = εE (the electric displacement), the latter including how the electrical charges in a material become polarized in an electric field. The magnetic force fields are described by H (the magnetic field) and B = µH (the magnetic flux density), the latter accounting for the magnetization of a material.

The equations can be considered in two pairs. The first pair consists of Equation 1 and Equation 2. Equation 1 describes the electric force field surrounding a distribution of electric charge ρ. It shows that the electric field lines diverge from areas of positive charge and converge onto areas of negative charge (Figure 3). Equation 2 shows that magnetic field lines curl to form closed loops (Figure 4), with the implication that every north pole of a magnet is accompanied by a south pole. The second pair, Equation 3 and Equation 4, describes how electric and magnetic fields are related. Equation 3 describes how a time-varying magnetic field will cause an electric field to curl around it. Equation 4 describes how a magnetic field curls around a time-varying electric field or an electric current flowing in a conductor.

Original 20 equations

Modern-Day 4 Equations

Heaviside championed the Faraday-Maxwell approach to electromagnetism and simplified Maxwell’s original set of 20 equations to the four used today. Importantly, Heaviside rewrote Maxwell’s Equations in a form that involved only electric and magnetic fields. Maxwell’s original equations had included both fields and potentials.

In an analogy to gravity, the field corresponds to the gravitational force pulling an object onto the Earth, while the potential corresponds to the shape of the landscape on which it stands. By configuring the equations only in terms of fields, Heaviside simplified them to his so-called Duplex notation, with the symmetry evident in the equations of Figure 2. He also developed the mathematical discipline of vector calculus with which to apply the equations. Heaviside analysed the interaction of electromagnetic waves with conductors and derived the telegrapher’s equations of Kirchhoff from Maxwell’s theory to describe the propagation of electrical signals along a transmission line.

Related Resources:

1.  “Maxwell’s Original Equations”, 2011

2.  Maxwell’s Scientific Papers, 8 Equations, date indeterminate

3. IEEE, Dec 2014

Maxwell’s own description of his theory was stunningly complicated. College students may greet the four Maxwell’s equations with terror, but Maxwell’s formulation was far messier. To write the equations economically, we need mathematics that wasn’t fully mature when Maxwell was conducting his work. Specifically, we need vector calculus, a way of compactly codifying the differential equations of vectors in three dimensions.

Maxwell’s theory today can be summed up by four equations. But his formulation took the form of 20 simultaneous equations, with 20 variables. The dimensional components of his equations (the x, y, and z directions) had to be spelled out separately. And he employed some counterintuitive variables. 

The net result of all of this complexity is that when Maxwell’s theory made its debut, almost nobody was paying attention.

It was Heaviside, working largely in seclusion, who put Maxwell’s equations in their present form. 

The key was eliminating Maxwell’s strange magnetic vector potential. “I never made any progress until I threw all the potentials overboard,” Heaviside later said. The new formulation instead placed the electric and magnetic fields front and center.

One of the consequences of the work was that it exposed the beautiful symmetry in Maxwell’s equations. One of the four equations describes how a changing magnetic field creates an electric field (Faraday’s discovery), and another describes how a changing electric field creates a magnetic field (the famous displacement current, added by Maxwell).

4. Mathematical Representations in Science