On a Needed Expansion of Newton’s Second Law
Abstract. In the case of charged particles, Newton's 2nd Law requires additional terms. The formulas for these additional terms are stated, and the amended Newton 2 is derived.
Newton’s second law specifies the relationship between an externally applied force and the resulting motion of an uncharged particle relative to any inertial frame. Its non-relativistic form is:
Its relativistic form is:
In both cases the applied force is time-wise conservative in the following sense: any work, done by F to increase the particle’s kinetic energy, can be recouped by returning the kinetic energy to its initial value.
Newton’s third law states that any accelerated particle exerts an equal-magnitude, oppositely directed force on the accelerating external agent. In the case of uncharged particles this force is called the inertial reaction force. Non-relativistically,
According to Newton, the applied force and the associated inertial reaction force always sum to zero.
If a particle with a periodic motion has an electric charge, q, then Newton’s second law (Eq. 1) must be expanded to:
In this case F is not generally time-wise conservative. The rate at which the second term does work is always positive, and in fact equals the rate at which radiant field energy flows away into space each cycle:
For example, given a charged particle with motion x=A sin(wt), the work per cycle done by the second term in Eq. 4 is:
Using the point charge field solutions (which are based on Maxwell’s equations), the Poynting vector, S, can readily be computed at points on an enclosing surface and numerically integrated over a cycle time. For non-relativistic values of wA this exercise always satisfies
Abraham and Lorentz dubbed the negative of the second term in Eq. 4 the radiation reaction force. Lorentz and others had earlier found that this force, and at least part of the inertial reaction force in the case of charged particles, are electric forces experienced by a charge in its own electromagnetic field. More specifically, given a uniform spherical shell, of charge q and radius R, they found that the shell experiences a net electric force in itsown field when a and/or da/dt, etc. are nonzero:
For R<<1 meter all but the first two terms can usually be ignored. Fself is then just the negative of F in Eq. 4, provided the "electromagnetic mass" equals q2/6peoRc2. This value for the electromagnetic mass is consistent with other results. For example, the momentum, in the field of a spherical shell of charge, whose velocity has always been constant, is (q2/6peoRc2)v. In the case of most (if not all) real charged particles, there is also "mechanical" inertia or mass, and the "m" in Eq. 4 is the sum of the electromagnetic and mechanical masses:
Note that if the particle is uncharged (q=0), then Eq. 4 reduces back to Newton’s second law for uncharged particles.
Newton’s third law is valid for both uncharged and charged particles. Thus the "zero total force" rule applies in both cases. When an external agent applies a force to a charged particle the particle applies a force of [-ma + (q2/6peoc3)da/dt] back on the external agent.
An important, if often unmentioned, limitation of Maxwellian theory is that no static distribution of charge can exist without the non-electromagnetic action of some agent. For example, the radius of the spherical shell of charge mentioned above would not remain constant without such an agent. Every increment of charge in the shell experiences an incremental electric force outward. And this force must be non-electromagnetically counteracted if R is to remain constant.
This rarely mentioned (and mysterious) internal agent may play a role in answering an objection often raised against the idea that the electric forces derived by Lorentz et al (Eq. 8) are reaction forces. It is sometimes pointed out that, owing to time delays inherent in the electromagnetic field, these "reaction" forces would persist in time if the external force in Eq. 4 were suddenly removed. The question then becomes: what opposes the "reaction" forces under such circumstances? The answer might be found in the often-neglected internal, non-electromagnetic agent force increments that hold the charge together.
Another question has to do with systems of charged particles whose net charge is zero (e.g. atoms). A driving agent’s counteraction to the radiation reaction force points opposite to da/dt regardless of the charge’s sign. How is it, then, that an atom, forced to oscillate, doesn’t radiate? The answer lies in the fact that the induced "reaction" fields (both inertial and radiation) do not exist solely at points in an accelerated charge; they extend into space beyond the charge. For example, we might model a neutral atom as a tiny spherical shell of positive charge, at the center of a larger spherical shell of negative charge. The positive charge’s radiation reaction field points in the same direction as da/dt. And it does this not only at the positive charge but also at points in the surrounding negative charge. The negative charge’s radiation reaction field, on the other hand, points opposite to da/dt. Thus the positive charge’s radiation reaction field tends to cancel the negative charge’s radiation reaction field. Indeed since the atom does not radiate, this cancellation must be total. The driving agent (causing the atom to oscillate) must expend no power to counteract the negative charge’s radiation reaction force plus the interactive force it experiences in the positive charge’s field.
Rigorously speaking the expanded version of Newton’s second law, cited in Eq. 4, applies only when v<<c. A more general, relativistic version applies for all speeds less than c. Regarding the force transformation between inertial frames, it is the relativistically correct version of Eq. 4 that should be subjected to a Lorentz transformation when F acts on a charged particle. (In the case of uncharged particles, or of charged particles whose velocities are held constant, only d(mv)/dt need be transformed.)
In many cases the second (radiation) term in Eq. 4 will be much smaller than the first (inertial) term, and Newton’s second law for uncharged particles is used to predict a charged particle’s motion. (Given some force, Eq. 1 is more easily solved than Eq. 4.) However, Eq. 4 is the more rigorous (and correct) non-relativistic law. If nothing else, it provides for the radiation emitted by an accelerated, charged particle.
It might at first seem confusing that the electric force found by Lorentz et al (Eq. 8) is a reaction force that ultimately acts on the external agent causing a, da/dt, etc. Perhaps the Newtonian axiom that no particle can exert a net force on itself will prove helpful in this regard. And it warrants mentioning again: no static distribution of charge can exist without the non-electromagnetic constraint of some agent. This long-neglected, necessary feature of electromagnetic theory may resolve some of the conceptual difficulties and objections often raised against the phenomenon of electric reaction forces.