On Oscillating Particles and the Forces that Drive Them
Abstract. Discusses and plots the forces needed to sinusoidally drive uncharged and charged particles.
In this article two types of oscillating particles are discussed: (1) uncharged particles, and (2) charged particles. The first might be said to be the domain of Newtonian mechanics, which concerns itself with "neutral matter" and where it is axiomatic that no particle can exert a force on itself. The second introduces the added complications of electric and magnetic fields and what might be construed to be violations of this Newtonian axiom. For when a charge has nonzero a (acceleration) and nonzero da/dt, then the charge experiences an electric force in its own fields. A fundamental question is: how do these "self" forces fit into the dynamic paradigm?
Unless explicitly specified otherwise, particle motion is assumed to be one-dimensional:
Non-relativistic cases are defined to be those with maximum particle speed (wA) much less than c. Cases where wA > .9c are defined to be relativistic.
Certain concepts, well grounded in theory and experiment, are accepted without proof. In particular the dependence of a particle’s inertial mass on the particle’s speed is specified by:
In this equation "mo" is the particle’s rest mass. Also, Newton’s second law theoretically applies to all uncharged particle cases when stated in the form
Experiment indicates that all particles have so-called mechanical mass. A defining characteristic of mechanical mass is that the particle’s kinetic energy and momentum are local to the particle. Any power expended by a driving force is instantly manifest as changes in the particle’s kinetic energy and momentum.
Particles with an excess of electric charge, of either sign, have electromagnetic mass in addition to their mechanical mass. In the case of electromagnetic mass, the kinetic energy and momentum reside in the particle’s electromagnetic field. Consequently these quantities ebb back and forth between the particle and its fields as the particle accelerates and decelerates.
The net electric force experienced by a charge in its own, a- and/or da/dt-induced fields is dubbed a reaction force. It is convenient to break this reaction force out into (a) an inertial reaction force, say FInertReact, and (b) a radiation reaction force, FRadReact. The power expended by either of these is the dot product of the force and the particle’s velocity. The term "PRad" is used to denote the negative of the power expended by FRadReact:
"W" is generally defined to be the net work per cycle done by a force:
An important result is the relation between (a) the work per cycle done by the negative of the radiation reaction force, and (b) the integral of S, the Poynting vector, over an enclosing surface, integrated w.r.t. time. It is always found that
A reasonable inference is that the negative of the dot product of FRadReact and v equates to the rate at which radiant energy is emitted into the electromagnetic field (hence the name "PRad").
2. An Uncharged, Oscillating Particle.
An uncharged particle is defined to be one whose mass is purely mechanical. Externally there is no electromagnetic field. Rigorously speaking it is now believed that many real-world particles (including "uncharged" ones such as neutrons) are composites of charged quarks. It must therefore be assumed in many cases that there are "internal" fields and associated Lorentz forces even in the case of "uncharged" particles. For present purposes, however, it is assumed that electromagnetic theory need not be considered in the case of uncharged particles.
The general equation of motion is then Newton’s second law in its relativistic form:
Here mmech denotes the particle’s rest mechanical mass. (It is set equal to 1E-6 kilograms, and A is set equal to 1 meter in the programs that compute F when the uncharged particle has the motion specified by Eq. 1_1.) In non-relativistic cases g varies little from unity and an alternate, acceptable form for the equation of motion is
Figs. 2_1 and 2_2 plot F(t) in non-relativistic and relativistic cases respectively. Note in the non-relativistic case that F(t), like a(t), is sinusoidal. In the relativistic case a(t) is again (by definition) sinusoidal, but F(t) clearly is not.
F(t), wA = 25 m/sec
F(t), wA = .95c
Newton’s third law stipulates that every force is accompanied by an equal and oppositely directed reaction force. In Eq. 2_1 the reaction force is an inertial reaction force,
The physical basis for this reaction force is something of a mystery in Newtonian mechanics (cases involving "neutral" matter). But assuming that F is a "contact" force, FInertReact is a force that the particle exerts on the driving agent (or source of F). Thus Newton’s third law implies the conservation of momentum and energy:
In words, the total change of momentum and energy in any given interaction is zero.
F in both the non-relativistic and relativistic cases might be said to be "conservative" in the following sense: W, the net work done per cycle by F, is zero. This is obvious on inspection when v is proportional to cos(wt) and F is proportional to sin(wt). It is less obvious, algebraically, when F(t) is as depicted in Fig. 2_2. Fig. 2_3 plots the dot product of F and v over the course of an oscillation when wA = .95c. Here it seems plausible that W is again zero, and indeed the integral of P(t) over a cycle time is computed to be practically zero.
P(t), wA = .95c
3. A Charged, Oscillating Particle.
3.1 Electromagnetic Mass, Momentum and Kinetic Energy.
As previously mentioned, a charged particle generally has both mechanical and electromagnetic mass. For example, we might begin by considering a tiny spherical shell of charge, of radius R<<1 meter. (The point charge can then be considered to be the limit as R goes to zero.) If the charge’s velocity has always been a constant v, with |v| << c, then it is readily shown that the electromagnetic momentum in the charge’s field is
For obvious reasons the electromagnetic mass, mElecMag, is defined to be
Noteworthy is the fact that mElecMag increases without bound as R is decreased toward zero. Evidently the electromagnetic mass of a true point charge would be infinite. Despite this obvious objection, the concept of a point charge has proven to be useful in many theoretical discussions. Indeed the general solutions for the fields of a point charge (at points other than that occupied by the charge itself) have been derived from Maxwell’s equations. The infinite electromagnetic mass issue can often be sidestepped by supposing that the charge in a charged "particle" is actually distributed in a volume of very small (but nonzero) dimensions.
Just as the momentum in the electromagnetic field (Eq. 3.1_1) can be derived for a spherical shell of charge moving with constant velocity, so can the energy in the magnetic field be determined. It turns out (again with v<<c) that
It is readily shown that momentum and energy flow from the charge into its fields when the charge is accelerated, and that they flow from the fields back to the charge when the charge is decelerated.
To the extent every charged particle has both mechanical and electromagnetic mass, the total momentum at any moment is
Similarly the total energy at any moment is
T = (mmech + melectromag)c2(g-1) (3.1-5)
Worth repeating is the fact that the electromagnetic momentum and kinetic energy reside in the fields, whereas the mechanical quantities are local to the particle.
In the case of a charged particle, the inertial reaction force technically has two parts: (a) the mechanical inertial reaction force, and (b) the electromagnetic inertial reaction force. An important feature of the electromagnetic inertial reaction force is that it can be shown to be part of the net electric force that the charge experiences in its own, acceleration-induced electric field:
Here, dq is an increment of charge in the distribution, and fE is part of the electric field at that point. The total integral of dqE usually contains another part, the radiation reaction force (discussed next). In any case, the total inertial reaction force is the sum of the mechanical and the electromagnetic inertial reaction forces. And as previously mentioned, for present purposes the mechanical inertial reaction force must remain something of a mystery (although it too might ultimately prove to be an electric force experienced by the "neutral" particle’s quarks in their own, acceleration-induced, "internal" fields).
3.2 The Driving Force for an Oscillating Charged Particle.
It can be demonstrated that the net electric force, experienced by a tiny spherical shell of charge when a is nonzero and da/dt is momentarily zero, is:
Thus the electromagnetic inertial reaction force has the same formula as the inertial reaction force of an uncharged particle (Eq. 2_3). The non-relativistic formula for the radiation reaction force (as first derived by Abraham and Lorentz) is:
Note that this force is independent of the charge distribution specifics. For example, It is theoretically true for point charges, spherical shells of charge of every size, solid spheres of charge, etc. The relativistic formula for FRadReact, for the one-dimensional motion considered herein, is
A key requirement is that, if a charged particle is to move periodically (Eq. 1_1), then the driving agent must counteract both the inertial and radiation reaction forces. That is, in the case of a charged particle moving periodically, the relativistic equation of motion is
Note that this reduces to Eq. 1_3 when q = 0.
3.2.1 A Non-Relativistically Oscillating Charged Particle.
Let us begin by considering a non-relativistic case. In such cases the equation of motion (Eq. 3.2_4) simplifies to:
(Here the sum of mmech and mElecMag has been abbreviated as "m".) For the sinusoidal motion specified by Eq. 1_1, we have
The dot product of a and v (and hence of F and v) integrates to zero over a complete cycle time. But note that the dot product of da/dt and v (and hence PRad) does not:
Fig. 3.2.1_1 plots FRadReact(t) over the course of a cycle time. Fig. 3.2.1_2 plots PRad(t) ( the dot product of –FRadReact(t) and v(t)). Note that PRad appears to be positive or zero at all times.
FRadReact(t), wA = 25m/sec
PRad(t), wA = 25m/sec
It is clear in Fig. 3.2.1_2 that a net, positive amount of work is done by the driving agent each cycle in the course of counteracting the radiation reaction force. And as mentioned, this work equals the flux of the Poynting vector through an enclosing surface, integrated over a cycle time. In short, the oscillating charge radiates. The work per cycle done by the ma part of F in Eq. 3.2.1_1 is zero. Thus the net work per cycle done by the driving force is
This is the quantity of radiant energy emitted every cycle.
3.2.2 A Relativistically Oscillating Charged Particle.
Fig. 3.2.2_1 plots FRadReact(t) over the course of a cycle time when wA = .95c. Fig. 3.2.2_2 plots PRad(t) ( the dot product of –FRadReact(t) and v(t)).
FRadReact(t), wA = .95c
PRad(t), wA = .95c
Here again it is found that the net work done per cycle by PRadis positive and equal to the Poynting vector flux over an enclosing surface, integrated over a cycle time. The notable feature of Fig. 3.2.2_2 is that PRad is not positive at all times. In any given cycle FRadReact actually does positive work on the charge (and its driving agent) part of the time! Negative values of PRad may at first seem illogical, particularly when a net amount of radiant energy is being emitted every cycle. Indeed one cannot help wondering if perhaps the relativistic expression for FRadReact might be wrong (notwithstanding the agreement between PRad, integrated over a cycle time, and the energy flux through a surrounding surface in a cycle time). It turns out, however, that a different, highly non-relativistic system, again oscillatory but with a motion other than that specified by Eq. 1_1, can also exhibit negative PRad for part of each oscillation. The implications are interesting.