My project will be based on the theoretical content of Chapter 10 of Griffiths, which deals with the previous equations for the electric and magnetic potentials (and fields) of charges, but with the additional dimension of time. My goal is not to rewrite Griffiths’ chapter, but rather to gain a deeper understanding of the equations he presents through graphical analysis. Central to the difference between the old equations and the new time-dependent ones is the concept of retarded time, which is defined as

(1)  

(David J. Griffiths, Introduction to Electrodynamics)

Where t is the actual time, the r vector represents the position of a given point in the system, the r’ vector represents the position of the charge, and c is the speed of light a vacuum. The second term represents a delay- how long it takes for the electromagnetic information to travel from the charge to a given point in space. Thus retarded time embodies the idea that electromagnetic messages take time to propagate through space, and in fact propagate at the speed of light. This means that the present position of a particle becomes incidental. At any given time there exists an effective position, illustrated below:

For a more in depth discussion of effective position (in three dimensions), see page 430 of Griffiths. In any case, our definition of retarded time leads to a time-dependent generalization of the potential equations:

(2)  

(3)  

(David J. Griffiths, Introduction to Electrodynamics)

Which are nearly identical to previous definitions of the potentials, with the exception that both the charge density and the current density now depend on the retarded time. These potentials can be used to derive Jefimenko’s Equations, which describe the non-static electric and magnetic fields of an arbitrary continuous charge distribution. I have omitted them here; since they rely so heavily on vector notation, at this point I’m unsure whether Mathematica’s plotting capabilities will be useful in obtaining graphical representations of them or the above potentials. I intend to explore manipulating them into a form easier to graph, or perhaps examine them in the plane z = 0. Additionally, choosing a simplistic charge distribution may result in an expression that could be graphed more readily.

In any case, I want to focus more on the electrodynamics of moving point charges. Their potentials, which can be expressed without an integral, are referred to as the Liénard-Wiechert Potentials:

(4)  

(David J. Griffiths, Introduction to Electrodynamics)

I believe these equations will prove easier manipulate and graph, and may apply restrictions such as the assumption that the dot product in the denominator is zero. Additionally I’ll examine these equations in only two spacial dimensions (various vertical and horizontal planes).

The following equation can be derived from Jefimenko’s (see Griffiths’ Example 10.3), but I have made a few assumptions of my own: that the velocity is constant and that the position vector r is perpendicular to the velocity vector v. Additionally, I’ve expressed the position r in terms of x and y, and let z = 0.

(5)  

Using Mathematica and plugging in a few values, I was able to generate a preliminary animation:

This is the kind of result I’d like to obtain for a number of other cases, magnetic as well as electric. I think it will be easy to formulate constraints for linear and quadratic charge accelerations, as well as circular motion, and applying them to the Liénard-Wiechert Potentials will hopefully yield workable results. Ideally I’ll have a number of equations and animations and be able to select and pursue the most interesting for presentation.