The Equations of Motion for n Charged Particles With Periodic Motion G.R.Dixon, 5/24/2004
1. Overview. 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 . (1-1) In the present article we begin by considering , (1-2a) . (1-2b) Again, wA<<c. L, the distance between the charges, will be expressed in units of l, where . (1-3) We shall assume that q 2. E 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 . (2-1) And, knowing . (2-2) Fig. 2-1 plots the computed values of E Figure 2-1 E 3. Subtracting Out the Reaction Forces. The equation of motion for a single, charged particle with periodic motion is . (3-1) m The "m . (3-2) Fig. 3-1 repeats Fig. 2-1 with the Figure 3-1 E 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 E 4. Interactive Forces. Let us momentarily assume that q 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), Figure 4-1 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 . (4-1) Fig. 4-2 plots W Figure 4-2 W 5. The Total External (Agent) Force on q We can now write down the equation for the . (5-1) 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: . (5-2) 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 . (6-1) 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 . (6-2) And when there are n charged particles moving periodically, then Newton’s second law must be expanded to , (6-3) where . (6-4) And this net work per cycle will equal the net radiant energy flux per cycle out through any enclosing surface: . (6-5) 7. Some Caveats. It is worth emphasizing that Eqs. 6-2 and 6-3 apply only to charged particles that . (7-1) In this scenario the net interactive force on q 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 One might "push the envelope," so to speak, and inquire what happens if there is no interactive force in Eq. 6-3, and if 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 Despite these and other limitations, the approach in this and other articles is not entirely without merit. We |