A Charged Particle on a Spring G.R.Dixon, 2/13/04
Given a nonrelativistic (v_{max}<<c) charged particle with motion restricted to the xaxis, the equation of motion is: . (1) In other words, the "action" force (F_{x}) and the "reaction" forces (ma_{x} and ada_{x}/dt) sum to zero (Newton’s third law). Let us define such a particle’s motion to be "oscillatory" when its velocity conforms to a pattern of: …v_{x} = 0; v_{x} > 0; v_{x} = 0; v_{x} < 0 … In the oscillator’s "rest" frame the charge passes through the origin twice in every "cycle." In this article we shall suppose that the particle oscillates on the end of an ideal spring. The spring force depends only upon x: . (2) And if this is the only external force acting, then Eq. 1 becomes: . (3) Any modulating effect of a nonzero a must be on a_{x}. Let us say that a cycle begins with the particle at rest at some x = A. In the ensuing first "quarter cycle" all three terms in Eq. 3 are positive. Thus a_{x} will be less than it would be if a were zero. Consequently v_{x} will be less at the origin than it would be if a were zero. In the second quarter cycle –kx and ma_{x} are negative, and –da_{x}/dt is positive. Thus a_{x} in Eq. 3 will have to be more negative than would be the case if q were zero. The particle will certainly come to rest at x<A. In the third quarter cycle all three terms in Eq. 3 will be negative. Thus a_{x} will be less negative than it would be if q equaled zero. Again v_{x} will be attenuated. Similarly with the fourth quarter cycle. The particle will come to rest at x=A’>A. Now the increment of radiated energy in any displacement dx is just the work done by the springmass counteraction to ada_{x}/dt (the radiation reaction force): . (4) And as Eq. 3 indicates, (5) But . (6) Thus . (7) Or, since an incremental change in the springmass total energy equals: , (8) every incremental loss of the system’s energy maps to an increment of radiated energy: . (9) The particle’s kinetic energy is zero at x=A and x=A’, and thus the radiated energy in the cycle is: . (10) The oscillator attenuates with each cycle. The maximum value of v_{x} asymptotically approaches zero. The charged particle on a spring constitutes a case where the inertial reaction force collaborates with the spring force, part of the time, to counteract the radiation reaction force. In order for the motion to be nonattenuating, the spring force must be augmented by another external force, say F_{x_Aug}. For example, if the particle’s motion is: , (11) then the equation of motion would be . (12) Like the spring force, the augmenting force would be sinusoidal: . (13) But this force is p/2 out of phase with the spring force. Its magnitude is maximum at x=0 and zero at x=+A. The radiated power is always positive: . (14)
