A Stable, Maxwellian 2-charge System
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 mo. 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:
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
The work done by Fo, as q- moves from x=D to any x>R, is
But by the Work-Energy theorem we also have
where g at x=D is unity (since q- is at rest there). Thus
And, since g increases as v increases,
By definition, g = (1-v2/c2)-1/2, and
According to Newton’s second law, in one dimension
Now the relativistic formula for the radiation reaction force is usually expressed as a function of time:
By the Chain Rule, however,
Similarly for dg/dt. Thus |FRad| can be expressed as a function of x:
In the outer zone, a steadily increases in magnitude during the first quarter cycle, so that da/dt points toward the origin. Consequently FRad and v point in the same direction, and the differential of work done by FRad, in any given displacement of magnitude dx and toward the origin, is
The total work done by FRad, as q- goes from x=D to x=R, is
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 WRad 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 WRad implies that FRad is affecting the motion of q- and ultimately doing work on the field. This being the case, a positive WRad 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 da/dt points away from the origin when q- is inside of q+. Consequently FRad and v point in opposite directions inside q+, and WRad is negative … a result that (in the present case) suggests radiant energy in the field is being depleted. The relevant parameters inside q+ are:
where g1 is the value of g at x=R,
Let us denote the radiation created in the first quarter cycle, over the range R<x<D, as WRad(o). And we shall denote the radiation depleted, over the range 0<x<R, as WRad(i). Fig. 3_1 plots the ratio of WRad(o) / -WRad(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.
-WRad(o) / WRad(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
The time for an entire cycle is
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
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 WRad(o) / -WRad(i) is to equal unity, q- must spend part of each cycle time outside of q+!