One can derive an electric radiation reaction force for an accelerating charged particle caused by the particle emitting electromagnetic radiation.
Likewise, a magnetic radiation reaction force can be derived for an accelerating magnetic moment emitting electromagnetic radiation.
Similar to the electric radiation reaction force, three conditions must be met in order to derive the following formula for the magnetic radiation reaction force.
First, the motion of the magnetic moment must be periodic, an assumption used to derive the force.
Second, the magnetic moment is traveling at non-relativistic velocities (that is, much slower than the speed of light).
Finally, this only applies this force is proportional to the fifth derivative of the position as a function of time (sometimes somewhat facetiously referred to as the "Crackle").
Mathematically, the magnetic radiation reaction force is given by, in SI units:
Physically, a time changing magnetic moment emits radiation similar to the Larmor formula of an accelerating charge.
In classical electrodynamics, problems are typically divided into two classes: In some fields of physics, such as plasma physics and the calculation of transport coefficients (conductivity, diffusivity, etc.
The reason for this is twofold: This conceptual problems created by self-fields are highlighted in a standard graduate text.
[Jackson] The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle.
Although partial solutions, workable within limited areas, can be given, the basic problem remains unsolved.
One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties.
While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the classical ones.
It is one of the triumphs of comparatively recent years (~1948–50) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment.
The magnetic radiation reaction force is the result of the most fundamental calculation of the effect of self-generated fields.
It arises from the observation that accelerating non-relativistic particles with associated magnetic moment emit radiation.
The self-fields in quantum electrodynamics generate a finite number of infinities in the calculations that can be removed by the process of renormalization.
This has led to a theory that is able to make the most accurate predictions that humans have made to date.
The renormalization process fails, however, when applied to the gravitational force.
The infinities in that case are infinite in number, which causes the failure of renormalization.
String theory is a current attempt to resolve these problems for all forces.
We begin with the Larmor formula for radiation of the second derivative of a magnetic moment with respect to time:
In the case that the magnetic moment is produced by an electric charge moving along a circular path is
If we assume the motion of a charged particle is periodic, then the average work done on the particle by the Abraham–Lorentz force is the negative of the Larmor power integrated over one period from
If we assume that there is periodic motion, the boundary term in the integral by parts disappears:
Below is an illustration of how a classical analysis can lead to surprising results.
See the quote from Rohrlich [2] in the introduction concerning "the importance of obeying the validity limits of a physical theory".
Thus future values of the force affect the acceleration of the particle in the present.
sec, which is the time it takes for a light wave to travel across the "size" of an electron.