A Stable, Maxwellian 2-charge System

G.R.Dixon

1. Introduction

This article investigates the radiant energy emitted/absorbed in the case of a particular 2-charge system. The results suggest that Newtonian/Maxwellian theory predicts a "stationary" (non-radiating) state.

The system consists of a positive, solid sphere of charge with radius R, and a negative point charge with finite rest mass m_{o}. The positive charge is centered on the origin and remains at rest at all times. The negative charge is initially at rest at x = D, but generally accelerates toward the origin. It is assumed that the negative charge can move freely through the positive charge without resistance. R is varied from run to run, the objective being to determine whether some value of R results in zero emitted radiant energy. The following values are used in
the modeling software:

, (1_1a)

, (1_1b)

. (1_1c)

, (1_1d)

. (1_1e)

Standard practice is to use Newton’s second law to determine the negative charge’s motion. To the extent the energy radiated per cycle is small compared to the total system energy, this approach usually produces a close approximation to the actual motion. In the present case the motion of q_{-} is determined in this way, and the radiated energy is then computed as the work done by the (relatively small) radiation reaction force.

2. The Motion of q_{-}

The magnitude of the force experienced by q_{-}, in the electric field of q_{+}, has different formulas in the ranges R<x__<__D and 0<x__<__R. But the force points toward the origin in both cases. Using subscripts "o" and "i" for outside and inside of q_{+}, the formulas are

, (2_1a)

. (2_1b)

The work done by F_{o}, as q_{-} moves from x=D to any x__>__R, is

. (2_2)

But by the Work-Energy theorem we also have

, (2_3)

where g at x=D is unity (since q_{-} is at rest there). Thus

. (2_4)

And, since g increases as v increases,

. (2_5)

By definition, g = (1-v^{2}/c^{2})^{-1/2}, and

. (2_6)

According to Newton’s second law, in one dimension

. (2_7)

Thus

. (2_8)

And

. (2_9)

Now the relativistic formula for the radiation reaction force is usually expressed as a function of time:

. (2_10)

By the Chain Rule, however,

. (2_11)

Similarly for dg/dt. Thus |__F___{Rad}| can be expressed as a function of x:

. (2_12)

In the outer zone, __a__ steadily increases in magnitude during the first quarter cycle, so that d__a__/dt points toward the origin. Consequently __F___{Rad} and __v__ point in the same direction, and the differential of work done by __F___{Rad}, in any given displacement of magnitude dx and toward the origin, is

. (2_13)

The __total__ work done by F_{Rad}, as q_{-} goes from x=D to x=R, is

. (2_14

This integral can be numerically computed.

In previous articles a non-electromagnetic agent drove the oscillating charge, and the agent __counteracted__ the radiation reaction force. In such cases a positive value of W_{Rad} implied that the radiation reaction force was doing work on the driving agent, and said agent was __absorbing__ radiant energy. In the present case it is the electromagnetic field that drives q_{-}, and a positive value of W_{Rad} implies that F_{Rad} is affecting the motion of q_{-} and ultimately doing work on the field. This being the case, a positive W_{Rad} implies that radiant energy is being __created__ in the field. That is, as q_{-} accelerates toward the origin outside of q_{+}, radiant energy is hypothetically created in the field.

It is noteworthy that d__a__/dt points __away__ from the origin when q_{-} is __inside of__ q_{+}. Consequently __F___{Rad} and __v__ point in __opposite__ directions inside q_{+}, and W_{Rad} is __negative__ … a result that (in the present case) suggests radiant energy in the field is being __depleted__. The relevant parameters __inside__ q_{+} are:

, (2_15a)

, (2_15b)

where g_{1} is the value of g at x=R,

, (2_15c)

, (2_15d)

, (2_15e)

, (2_15f)

. (2_15g)

3. Results

Let us denote the radiation __created__ in the first quarter cycle, over the range R<x__<__D, as W_{Rad(o)}. And we shall denote the radiation __depleted__, over the range 0<x__<__R, as W_{Rad(i)}. Fig. 3_1 plots the ratio of W_{Rad(o)} / -W_{Rad(i)} over the range 9.5E-15 m __<__ R __<__ 1E-14 m. (5 values of R were computed.) Note that __the curve has a value of unity__ within this range, indicating that no __net__ radiation is created or depleted at that particular value of R.

Figure 3_1

-W_{Rad(o)} / W_{Rad(i)} vs. R

At R=9.8E-15 m (approximately where the net radiated energy is zero), the computed speed of q_{-} at x=0 is 2.13E8 m/sec, or slightly more than 2/3 the speed of light. The computed time for the first quarter cycle, at this value of R, is

. (3_1)

The time for an entire cycle is

, (3_2)

which is somewhat less than the value of 1.52E-16 sec for the Bohr Hydrogen atom. But of course in the Bohr model q_{-} orbits q_{+}; it does not dive through q_{+}. The frequency of the oscillation is

. (3_3)

4. Concluding Remarks

The value of D = .529E-10 meters is motivated by the Bohr radius of a Hydrogen atom. And the system modeled, with q_{-} diving through q_{+}, is a suggested ground state.

The modeling of q_{+} as a solid sphere of positive charge, while perfectly Maxwellian, is objectionable for at least two reasons. First, theoretically __no__ distribution of charge can persist in time without the imposition of non-electromagnetic constraints. (See, for example, Sands’ discussion in Chap 5, Vol. 2, __The Feynman Lectures on Physics__.) Secondly, present theory holds that the proton is composed of quarks … point charges for present purposes.

The objection that a __point__ q_{-} would theoretically have infinite electromagnetic (and hence total) inertial mass can be met by specifying that q_{-} is also a spherical distribution of some sort, but with a radius <<R.

Despite such objections, it is interesting that there can, according to purely Maxwellian theory, be a stable state for a 2-charge system. When Bohr advanced his model of the Hydrogen atom it was generally agreed that the electron would __inevitably__ (according to Maxwell) radiate. And it is not difficult to show that Maxwellian theory does in fact predict radiation when the electron __orbits__ the proton. Bohr seems to have bought into the conventional wisdom of the time (that the classical theory predicts radiation in every imaginable case), and concluded that Maxwellian theory cannot accurately describe reality in the small spaces occupied by non-radiating atoms. The results obtained herein, however, seem to indicate that such a generalization may not be warranted.

Perhaps one of the most interesting implications of the present model is the hypothesis that radiant energy is __depleted__ from the system’s field when q_{-} is inside q_{+}. The hypothesis that this energy temporarily transforms to negative particle __kinetic__ energy suggests that there can never be a stable state where D__<__R. That is, although q_{-} might spend part of the time inside q_{+}, it can never spend __all__ of the time in there. For if the ratio of W_{Rad(o)} / -W_{Rad(i)} is to equal unity, q_{-} must spend part of each cycle time outside of q_{+}!