For the past year or so I have been reading Introduction to Electrodynamics by David J. Griffiths, and slowly working my way through the chapters. The book is an upper division university textbook, intended for serious students and Physics majors, but I’d like to share something that I just learned. We take an awful lot of science for granted these days, and much of it is beyond us – General Relativity, Quantum Mechanics, for example, and I talked about the motions of the planets in an earlier blog. In my undergraduate Physics courses many years ago I would often hear the professors talk about James Clerk Maxwell and the beauty of Maxwell’s Equations, and how he had explained the theory of electromagnetism so elegantly but I never understood them until now.

To fully understand Maxwell’s Equations, you have to understand Differential and Integral Calculus, and some advanced mathematics, particularly partial differentiation and gradients, Green’s and Gauss’ theorems, but to just appreciate and understand their importance you really don’t need all that. Suffice it to know that Calculus is the mathematics of things that are functions of variables, such as x, y, and z that change in some predictable way. Functions can be scalar or vector, and the gradient of a function is a vector that points in the direction of greatest change. The gradient, which is formed by combining the rates of change of each of the three directional variables x, y, and z with time, i.e. their partial differentials , is so important that it has been given a special symbol called del, and del can operate on both scalar and vector functions in several different ways. One is called the divergence, and the other is called the curl. I tell you this only because Maxwell’s Equations are traditionally written in the differential form with del, though each is a generalized representation of an earlier and perhaps more familiar law.

Therein lies the beauty of Maxwell’s Equations.

For example, Coulomb’s Law which leads directly to Gauss’ Law and the first of Maxwell’s Equations, states that the electric force (F) between two charges (q) is proportional to the amount of charge, and inversely proportional to the square of the distance between them. The value of the proportionality constant, which has traditionally been represented as 1/4пє(ǒ), could even in Coulomb’s time be somewhat precisely determined from experiments where little charged balls were dangled from wires, and the forces could be measured by measuring the torsion in the wire. Likewise was the situation with the third equation, which is derived from Faraday’s Law of electromagnetic induction, and Ampere’s Law (4th Equation) which contained another important constant μ(ǒ).

In regions of space where there is no electric charge or current, Maxwell’s Equations look like this:

Note that two of the equations involve del dotted with a function (divergence), and two have del crossed with a function (curl). Note also that the equations in this form are coupled, specifically equations (iii) and (iv) contain both E and B, where E is the Electric Field vector and B is the Magnetic Field vector. They can be de-coupled mathematically by performing another curl operation on each of them. It’s rather simple, and the result leads to two decoupled equations in the following form:

When Maxwell saw this, he must have been flabbergasted. The equations for the Electric Field and the Magnetic Field were both practically identical in form to Isaac Newton’s equation for a wave moving on a piece of string, implying that electricity and magnetism are both waves, but even more astounding was the fact that the two constants, є(ǒ) and μ(ǒ) appeared in the same position as the factor 1 over the velocity-squared in the wave equation.

When calculated, the speed of both fields were found to be precisely the speed of light. Whoa! Could light be an electromagnetic wave? The implication of course that light was an electromagnetic wave was astounding, especially in lieu of the way that the two constants had been determined from experiments with pith balls, batteries and wires, none of which seemed to have anything even remotely to do with light – or so the scientists of that day thought.

But what a great bit of detective work.