A Dash Of Maxwell's -- Chapter III -- The Difference A Del Makes

As part of our ongoing Technology 101 coverage, RF Globalnet is presenting the historic six-part article series on Maxwell's equations by Dr. Glen Dash. Written specifically for the Web, these articles are intended to serve as a primer on electromagnetic field theory. The first article introduces Maxwell's equations for static fields and can be found here. This third article covers using a differential form as opposed to an integral form when looking at Maxwell's equations. Additional installments will be featured in upcoming editions of RF Globalnet newsletter.

By Dr. Glen Dash, Ampyx LLC

In Chapter Two, I introduced Maxwell's Equations in their "integral form." Simple in concept, the integral form can be devilishly difficult to work with. To overcome that, scientists and engineers have evolved a number of different ways to look at the problem, including this, the "differential form of the Equations." The differential form makes use of vector operations.

A physical phenomena that has the both the attributes of magnitude and direction may be described by a vector. Velocity can be drawn in vector form; it has the attributes of both direction and speed. Vectors can be illustrated graphically as shown in Figure 1(a) – the length of the vector represents its magnitude and its angle from the x axis defines its direction. However, we will be using Cartesian coordinates. In this system, vectors are described as a sum of "unit" denominated vectors. A unit vector along the x axis is simply a vector aligned with the x axis that is one unit long (one meter long in the MKS system). We'll denote a unit vector in the x direction as i. Similarly, unit vectors in the y and z directions will be denoted j and k respectively.