A Real-Life Example of the Radiation Reaction Force
Ever since it was first suggested by Abraham and Lorentz, the radiation reaction force has been controversial. Many modern texts avoid mentioning it altogether. As this article will demonstrate, however, there is at least one case where it results directly from Maxwell’s equations and the Lorentz force law.
We exploit the fact that the fields of a point charge can always be computed, provided the charge’s past motion is known. More specifically, given knowledge of a point charge’s past motion, we can compute E and B at all points in space other than that momentarily occupied by the charge itself.
We shall consider a point charge with the following motion:
The only proviso is that the retarded time for each field evaluation point be sufficiently close to tr=0 that vr<<c.
At time t=0 the charge is at rest at the origin and has zero acceleration. Using the point charge field solutions, we can readily compute Ex(y) over the range 10-3 m < y <1 m. Fig. 1 plots the result. In this particular case Ex turns out to be single-valued and independent of y.
Ex(y), Constant dax/dt
The same result is obtained for other values of dax/dt. In every case Ex turns out to be proportional to dax/dt:
And this same Ex is found at points on the negative y-axis. Under these circumstances it seems reasonable to extrapolate to the origin. Doing so and applying the Lorentz force law, we find that the point charge experiences an electric force even though it is isolated from all other charges:
This is the force suggested by Abraham and Lorentz. More generally, they suggested that any charge, q, with nonzero da/dt, will experience a force proportional to da/dt:
It is interesting that this force depends only upon q and da/dt. That is, any distribution of charge q, with nonzero da/dt, will theoretically experience the same force.
It should be noted that this "radiation reaction" force might not be the only "self" force acting. In general there will be another part to the total self-force, and this other part will depend on the charge’s distribution.
But why did Abraham and Lorentz refer to the force in Eq. 4 as the "radiation reaction" force? The word "reaction" seems to imply (a la Newton) that the force is equal in magnitude and oppositely directed to an external driving force. And indeed we scarcely expect the motion of Eqs. 1a-d to occur in the absence of an external "agent" force. Note that, if the radiation reaction force is indeed a Newtonian reaction force, then we should not make the error of concluding that the total force "on the charge" is zero. It is sometimes helpful in this regard to think of the external agent as acting on the charge’s fields, and the fields as acting back on the agent. The charge then becomes the point of contact between agent and field.
With regard to the word "radiation," it is convenient to consider a point charge with motion:
Using the point charge solutions, we can compute the Poynting Vector (power) flux through an enclosing cylindrical surface, and integrate w.r.t. time. On doing so we find that there is a net positive (outward) flow of energy through the surface every cycle. Evidently oscillating charges radiate. Energy conservation dictates that the radiated energy per cycle equal the net work per cycle done by a driving agent. (Here again, Eq. 5 implies a driving agent force.)
According to Abraham-Lorentz and Newton’s third law, at least part of the driving agent force is:
Thus at least part of the agent power expenditure is:
This of course always integrates w.r.t. time to a positive amount of work.
The amount of radiant energy flowing out into space each cycle is:
where S is the computed Poynting Vector at points on the enclosing surface. For non-relativistic oscillations (i.e. for those withwA<<c), we always find that
Hence the word "radiation" in the radiation reaction force of Abraham-Lorentz.
The point charge field solutions are relativistically correct, and hence the right side of Eq. 9 can be computed correctly for allwA<c. But the radiation reaction force, as specified in Eq. 3, is correct only when v<<c. However, the relativistic form of Eq. 3 is known, and when it is plugged into the left side of Eq. 9 the demands of Energy Conservation are met at all wA<c.
It perhaps warrants emphasizing that the radiation reaction force is not generally the whole "self" force experienced by a charge (or by its driving agent). Usually there is also an "inertial" piece which, at low speeds, is proportional to –a. This inertial piece will be discussed in a separate article.
For now suffice it to say that the radiation reaction force can be shown, in the case of at least one motion (Eqs. 1a-d), to be a consequence of Maxwell’s equations and the Lorentz force law. Prudence accordingly suggests that this force not be disavowed on "philosophical" grounds. For example if, when an external driving force is suddenly removed, a charge continues to accelerate for a brief period of time under the influence of its own electric field, then that is simply what the field solutions indicate. Although such brief accelerations may offend the sensibilities of Newtonian purists, this might be the way charges sometimes actually behave.