The Equations of Motion for n Charged Particles
With Periodic Motion
This article continues to develop the equations of motion for oscillating charges. In a previous article it was shown that the energy, radiated per cycle by a single, non-relativistically oscillating charged particle, equates to the work per cycle done by the force
In the present article we begin by considering two charged particles with charges q1 = q2, and with motions
Again,wA<<c. L, the distance between the charges, will be expressed in units of l, where
We shall assume that q1 = q2 = 1 coulomb, andl = .1 meters. w will then be 1.885E10/second. If wA = .1c, then A = .0016 meters.
2. Erad(L), the Energy Radiated per Cycle as a Function of the Charge Separation.
Let the particles be enclosed by a spherical surface, centered on the origin and with radius R>(L/2+A). The relativistically correct, individual point charge field solutions can be used to compute Ei and Bi at points on that surface. Having done this, we can compute the Poynting vector:
And, knowing S, we can numerically integrate to find Erad(L), the field energy flux per cycle through the surface:
Fig. 2-1 plots the computed values of Erad(L) over the range .5l<L<3l. Note that the energy radiated per cycle is sensitive to the charge separation. It drops off with increasing L until L is slightly less than one wavelength. At that separation it begins to increase again, and reaches a second maximum at slightly less than 1.5 wavelengths. The process continues for greater separations.
Erad(L), Two Oscillating Charges
3. Subtracting Out the Reaction Forces.
The equation of motion for a single, charged particle with periodic motion is
ma is the driving agent’s counteraction to the inertial reaction force, -ma. And the da/dt term is the agent’s counteraction to the radiation reaction force, (q2/6peoooc3)(da/dt).
The "ma" part of the agent force does no net work per cycle, and thus contributes nothing to the energy flux per cycle plotted in Fig. 2-1. And the work expended to counteract the radiation reaction force depends only upon a given charge's motion. In the present case of two charges the total work per cycle, expended to counteract both charges’ radiation reaction forces, is independent of the charge separations:
Fig. 3-1 repeats Fig. 2-1 with the constant work per cycle in Eq. 3-2 subtracted out.
Erad(L) Less the Radiation Reaction Work
Our task is to determine what term(s) must be added to Eq. 3-1 to result in a net work per cycle equal to Erad(L) in Fig. 2-1. That is, what part of the total agent force does an amount of work per cycle equal to Fig. 3-1?
4. Interactive Forces.
Let us momentarily assume that q1 and q2 are at rest at x1 = -L/2 and x2 = L/2. In order to maintain this state of rest, some agent must counteract the repulsive electric force that each charge experiences in the other’s electrostatic field.
When the charges oscillate, the interactive force that each charge experiences in the other’s electric field is easily computed using the point charge field solutions. Fig. 4-1 plots these two forces and their sum over a cycle time, at a separation of L=l/2. Note that, owing to time delays inherent in Maxwell’s equations (and thus in the field solutions), F1(interactive) and F2(interactive), the interactive forces on q1 and q2, are not generally equal and oppositely directed. (The middle trace is the sum of F1(interactive) and F2(interactive)).
Interactive Forces, Separation = l/2
Let us assume that, in order to maintain the oscillations, the driving agent must counteract the interactive forces (quite as it must do when the charges are at rest). The work per cycle in doing so is
Fig. 4-2 plots Winteractive(L) over the range of separations .5l<<L<3l. Note that the curve is identical to Fig. 3-1. Here, then, is the mechanical basis for the form of Erad(L) when two charges oscillate in phase. Adding the work expended per cycle to counteract the radiation reaction forces of course reproduces Fig. 2-1.
Winteractive(L), Two Charges
5. The Total External (Agent) Force on q1 and q2.
We can now write down the equation for the total external agent force that must be applied to the two identical charges moving in accordance with Eqs. 1-2a and b:
Dot multiplied by the common velocity and integrated over a cycle time, this net agent force produces an amount of work per cycle equal to the radiated energy per cycle:
6. Newton’s Second Law for Multiple, Charged Particles that Move Periodically.
When none of the particles in a system is charged, then there are no radiation reaction forces and no (electromagnetic) interactions to counteract. Thus the equation of motion (Eq. 5-1) simplifies to
This is of course Newton’s second law for an uncharged particle. It is completely general and not limited to periodic motions. When a single, isolated charged particle moves periodically, then Newton’s second law must be expanded to
And when there are n charged particles moving periodically, then Newton’s second law must be expanded to
where Fi(interact) is the interactive force experienced by qi in the net electromagnetic field of all the other (n-1) charged particles. (We are ignoring other types of interaction, such as gravity.) The net work per cycle time, expended to drive all n particles, is (assuming all particle motions have the common period 2p/w)
And this net work per cycle will equal the net radiant energy flux per cycle out through any enclosing surface:
7. Some Caveats.
It is worth emphasizing that Eqs. 6-2 and 6-3 apply only to charged particles that move periodically. Theoretically such particles almost always radiate some net amount of energy per cycle. An interesting question is: what equation governs if there is no driving agent force? Setting Fi in Eq. 6-3 equal to zero and rearranging produces
In this scenario the net interactive force on qi counteracts the inertial and radiation reaction forces. In general there may still be radiation, but the particles will not move periodically; the energy lost to radiation comes at the expense of particle kinetic energy, which decays with the passage of time.
An interesting exception to the rule, that charged particles with periodic motion radiate, is provided by a spinning, circular line charge. This charge can be modeled as an infinite number of infinitesimal point charges, arrayed bead-like around the circular contour. Each point charge moves periodically; yet collectively there is no radiation. The physical basis for this will be discussed in a follow-on article.
Eqs. 6-3 thru 6-5 might be of only limited practical utility. From the theoretical perspective, however, it is nothing less than fascinating how the net work per cycle, done by all of the Fi in Eq. 6-3, equates to the field energy flux per cycle out through any enclosing surface. While required by energy conservation on the one hand, it is uncanny how such equalities can be traced back to Maxwell’s equations.
One might "push the envelope," so to speak, and inquire what happens if there is no interactive force in Eq. 6-3, and if Fi suddenly goes to zero. (This would correspond to a single, isolated charged particle that suddenly loses its external agent force.) The simplest and long-term correct answer is that a and da/dt are zero. However, owing to inherent time delays, there may be brief intervals of time during which the "interactive" forces between different parts of the "particle’s" charge do not sum to zero. The charge would seem to experience a net, nonzero "self" force during such intervals … a prospect that many theorists find intolerable. In such cases it should perhaps be borne in mind that there can never be any such thing as a static charge distribution with no agent action. The persistence of any charge distribution in time requires the non-electromagnetic action of some internal agent. This rarely mentioned (and mysterious) action may counteract the reaction (or "self") forces during such brief intervals. Consequently the particle’s velocity may become constant the moment the more traditional external force drops to zero.
All of the present discussion assumes that particle speeds are at all times non-relativistic. Different, relativistically correct expressions for the inertial and radiation reaction forces must be used when particle speeds are relativistic for at least part of the time. (The interactive forces are correct at all speeds, being computed using the point charge field solutions and the Lorentz force law … both relativistically rigorous. Similar remarks apply to the computed energy flux through an enclosing surface.)
The approach, in this article and in numerous others featured on this site, is that the particle motions are a given. In such cases the energy flux per cycle through an enclosing surface, etc., can readily be computed. Practically speaking, it may be the particle motions that we wish to compute, given a known set of initial conditions. Given no knowledge of past motions, however, a precise determination of the initial conditions (particularly of E(r,0) and B(r,0)) may not be feasible. And even where such initial conditions are stipulated, a solution of future motions may be challenging in cases involving many interacting particles.
Despite these and other limitations, the approach in this and other articles is not entirely without merit. We can, for example, mechanically drive a charged object in a pre-specified manner. For example, we do not require an antenna many kilometers in length to generate extremely low frequency radiation (although the intensity may be low). Isolated oscillating charges always (classically) radiate. They always exert inertial and radiation reaction forces on whatever agent causes them to oscillate. The equations in this article specify a priori what the driving force must be in such cases, and what patterns of radiation can be expected.