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1. The Fields of a Point Charge.

Provided a point charge’s past motion is known, its electric and magnetic fields can be computed at any arbitrary time, t, and at any point in space other than that occupied by the charge itself. The solutions are relativistically rigorous and are expressed in terms of retarded position, velocity and acceleration. These in turn follow from knowledge of t_r, the retarded time.

The retarded time is defined in the following way. Let us suppose that at some time t we wish to calculate the retarded time at some point P that is not momentarily occupied by the charge itself. The retarded time is defined to be the moment in the past when a light signal, originating at the charge, would arrive at point P at time t. Provided the charge’s speed is less than c at all times, t_r is unique. The retarded position r_r is then the charge’s position at time t_r, with similar remarks applying to the retarded velocity (v_r) and the retarded acceleration (a_r). Fig. 1_1 illustrates for a point charge whose motion has always been confined to the x-axis. Note that the field evaluation point, P, is not necessarily on the x-axis. The crucial relation is:

(1_1)

Figure 1_1

Retarded Time and Retarded Position Example

Note that t_r (or the time interval (t–t_r)) generally varies from field evaluation point to field evaluation point. Sometimes a formula can be found and t_r can be calculated exactly. In other cases it must be approximated, typically using a computer. In these instances the following algorithm can be implemented in one’s programming language of choice. The number 2^-30 assumes a 32 bit double precision processor. P_x is the x-component of the field evaluation point P at time t, etc. D_x is the x-component of D, which is defined to be the displacement vector from the retarded position (r_r) to the field evaluation point (P), and similarly for D_y and D_z if applicable.

Let dtmin = 0 and dtmax = an adequately large time interval into the past.

Do

dt= ( dtmin + dtmax ) / 2

Compute a trial t_r = t – dt

Compute trial retarded coordinates using the trial t_r

D_x = P_x – x_r

D_y = P_y – y_r

D_z = P_z – z_r

D = (D_x² + D_y² + D_z²)^1/2

If | c dt – D | < 2^-30 then Exit Do

If ( c dt – D ) > 0 then dtmax = dt

If ( c dt – D ) < 0 then dtmin = dt

Loop

Once t_r has been determined, x_r , etc., follow directly from the known past motion.

In terms of the retarded position and kinematic quantities, the formulas for the electric and magnetic fields are:

, (1_2)

. (1_3)

The vector u is defined to be

. (1_4)

These completely general point charge field solutions are derived in INTRODUCTION TO ELECTRODYNAMICS, Second Edition, David J. Griffiths, Prentice Hall, Chapter 9. (The notation has been altered slightly.)

In theory a point charge’s past motion is limited only by the analyst’s imagination. The sole constraints are (1) |v(t_r)| must be less than c, and (2) P(t) must not be the same as r(t). In practice it may not always be obvious what environment (or driving force) will produce the desired motion, or indeed even what that force should be. Such matters will be investigated in due course. For present purposes, certain interesting motions will simply be specified and the fields will be computed using Eqs. 1_2 thru 1_4, the assumption being that the environment is "whatever it takes" to produce the specified motion.