 Magnetic radiation reaction force

In the physics of electromagnetism, 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 nonrelativistic velocities (that is, much slower than the speed of light). Finally, this only applies to the realm of classical physics.
Since the magnetic moment is proportional to velocity, this force is proportional to the fifth derivative of the position as a function of time (sometimes somewhat facetiously referred to as the "Crackle"). Unlike the Abraham–Lorentz force, the force points in the direction opposite of the "Crackle".
Contents
Definition and description
Mathematically, the Abraham–Lorentz force is given by:

 (SI units)
where:
 F is the force,
 is the Pop (the third derivative of acceleration, or the sixth derivative of displacement),
 μ_{0} is the permeability of free space,
 c is the speed of light in free space^{[1]}
 q is the electric charge of the particle.
 R is the radius of the magnetic moment
Note that this formula applies only for nonrelativistic velocities.
Physically, a time changing magnetic moment emits radiation similar to the Larmor formula of an accelerating charge. Since momentum is conserved, the magnetic moment is pushed in the direction opposite the direction of the emitted radiation. In fact the formula above for radiation force can be derived from the magnetic version of the Larmor formula, as shown below.
Background
In classical electrodynamics, problems are typically divided into two classes:
 Problems in which the charge and current sources of fields are specified and the fields are calculated, and
 The reverse situation, problems in which the fields are specified and the motion of particles are calculated.
In some fields of physics, such as plasma physics and the calculation of transport coefficients (conductivity, diffusivity, etc.), the fields generated by the sources and the motion of the sources are solved selfconsistently. In such cases, however, the motion of a selected source is calculated in response to fields generated by all other sources. Rarely is the motion of a particle (source) due to the fields generated by that same particle calculated. The reason for this is twofold:
 Neglect of the "selffields" usually leads to answers that are accurate enough for many applications, and
 Inclusion of selffields leads to problems in physics such as renormalization, some of which still unsolved, that relate to the very nature of matter and energy.
This conceptual problems created by selffields 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 quantummechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantummechanical discussions are beset with even more elaborate troubles than the classical ones. It is one of the triumphs of comparatively recent years (~ 1948  1950) 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. From a fundamental point of view, however, the difficulties remain.
The magnetic radiation reaction force is the result of the most fundamental calculation of the effect of selfgenerated fields. It arises from the observation that accelerating nonrelativistic particles with associated magnetic moment emit radiation. The Abraham–Lorentz force is the average force that an accelerating charged particle feels in the recoil from the emission of radiation. The introduction of quantum effects leads one to quantum electrodynamics. The selffields 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. See precision tests of QED. 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. Therefore general relativity has unsolved selffield problems. String theory is a current attempt to resolve these problems for all forces.
Derivation
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
 ,
where is the position of the charge q relative to the center of the circle and is the instantaneous velocity of the charge.
The above Larmor formula becomes as follows:
 .
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 τ_{1} to τ_{2}:
 .
Notice that we can integrate the above expression by parts. If we assume that there is periodic motion, the boundary term in the integral by parts disappears:
 .
Integrating by parts a second time, we find
 .
Clearly, we can identify
 .
Signals from the future
Below is an illustration of how a classical analysis can lead to surprising results. The classical theory can be seen to challenge standard pictures of causality, thus signaling either a breakdown or a need for extension of the theory. In this case the extension is to quantum mechanics and its relativistic counterpart quantum field theory. See the quote from Rohrlich ^{[2]} in the introduction concerning "the importance of obeying the validity limits of a physical theory".
For a particle in an external force , we have
where
This equation can be integrated once to obtain
The integral extends from the present to infinitely far in the future. Thus future values of the force affect the acceleration of the particle in the present. The future values are weighted by the factor
which falls off rapidly for times greater than t_{0} in the future. Therefore, signals from an interval approximately t_{0} into the future affect the acceleration in the present. For an electron, this time is approximately 10 ^{− 24} sec, which is the time it takes for a light wave to travel across the "size" of an electron.
See also
 Max Abraham
 Hendrik Lorentz
 Cyclotron radiation
 Electromagnetic mass
 Radiation resistance
 Radiation damping
 Synchrotron radiation
 Wheeler–Feynman absorber theory
References
 ^ The symbol c_{0} is used by CIPM and NIST.
 ^ F. Rohrlich: The dynamics of a charged sphere and the electron Am J Phys 65 (11) p. 1051 (1997). "The dynamics of point charges is an excellent example of the importance of obeying the validity limits of a physical theory. When these limits are exceeded the predictions of the theory may be incorrect or even patently absurd. In the present case, the classical equations of motion have their validity limits where quantum mechanics becomes important: they can no longer be trusted at distances of the order of (or below) the Compton wavelength…Only when all distances involved are in the classical domain is classical dynamics acceptable for electrons."
Further reading
 Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 013805326X. See sections 11.2.2 and 11.2.3
 Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 047130932X.\
 Jose A. Heras, The Radiation Force of an Electron Reexamined, 2003, http://www.joseheras.com/jheras_papers/JAHPAPER_16.pdf.
 Donald H. Menzel, Fundamental Formulas of Physics, 1960, Dover Publications Inc., ISBN 0486605957, vol. 1, page 345.
External links
Categories: Electrodynamics
 Electromagnetic radiation

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