On a Needed Expansion of Newton’s Second Law G.R.Dixon, 4/22/2004
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: . (1) Its relativistic form is: . (2) In both cases the applied force is time-wise conservative in the following sense: any work, done by Newton’s third law states that any accelerated particle exerts an equal-magnitude, oppositely directed force on the accelerating external agent. This force is called the inertial reaction force. Non-relativistically, . (3) According to Newton, the applied force and the associated inertial reaction force always sum to zero. If a particle has an electric charge, q, then Newton’s second law (Eq. 1) must be expanded to: . (4) In this case . (5) 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: . (6) Using the point charge field solutions (which are based on Maxwell’s equations), the Poynting vector, . (7) 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, are . (8) For R<<1 meter all but the first two terms can usually be ignored. . `(9) 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 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 internal agent. For example, the radius of the spherical shell of charge mentioned above would not remain constant without such an agent. For 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 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 d 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 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 (i.e. It might at first seem confusing that the electric force found by Lorentz et al (Eq. 8) is a |