A Charged Particle on a Spring
Given a non-relativistic (vmax<<c) charged particle with motion restricted to the x-axis, the equation of motion is:
In other words, the "action" force (Fx) and the "reaction" forces (-max andadax/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: …vx = 0; vx > 0; vx = 0; vx < 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:
And if this is the only external force acting, then Eq. 1 becomes:
Any modulating effect of a nonzeroa must be on ax.
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 ax will be less than it would be ifa were zero. Consequently vx will be less at the origin than it would be if a were zero.
In the second quarter cycle –kx and max are negative, and –dax/dt is positive. Thus ax 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 ax will be less negative than it would be if q equaled zero. Again vx 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 spring-mass counteraction toadax/dt (the radiation reaction force):
And as Eq. 3 indicates,
Or, since an incremental change in the spring-mass total energy equals:
every incremental loss of the system’s energy maps to an increment of radiated energy:
The particle’s kinetic energy is zero at x=-A and x=-A’, and thus the radiated energy in the cycle is:
The oscillator attenuates with each cycle. The maximum value of |vx| 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 non-attenuating, the spring force must be augmented by another external force, say Fx_Aug. For example, if the particle’s motion is:
then the equation of motion would be
Like the spring force, the augmenting force would be sinusoidal:
But this force isp/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: