On the Non-Relativistic Electrodynamics of an Isolated Spherical Shell of Charge
Abstract. Discusses Newton's Laws and charge dynamics.
1. Electromagnetic Reaction Forces.
In Maxwellian theory a spherical shell of charge can persist in time only if an agent non-electromagnetically holds the charge together. Indeed this is true of all distributions of charge. Every increment of charge in a distribution repels every other increment, and an external agent must counteract each tiny repulsive force if the distribution is to maintain its shape. Although often not explicitly mentioned, these counteracting agent forces are always implied. The good news is that, like the electromagnetic forces they counteract, they sum to zero provided the distribution is at rest or moves with constant velocity.
Lorentz and others derived the sum of the internal, interactive Lorentz (electromagnetic) forces when a spherical shell of charge’s velocity is not constant. Owing to time delays inherent in electromagnetic interactions, these forces do not sum to zero. In one dimension Lorentz et al found that
For shells with radii R<<1 meter, the net internal force can be simplified to
Here again in order for the accelerated shell to maintain its shape, an agent must counteract each infinitesimal internal force; the total agent force must therefore be
It seems clear in Eqs. 1.2 and 1.3 that Fx(interactive) and Fx are examples of Newton’s third law at work. In words, when a nonzero Fx causes ax and/or dax/dt to be nonzero then the charge, in its own motion-induced fields, reacts with a force of Fx(interactive) = -Fx.
2. The Equation of Motion for Charged Particles.
Now there are certain interesting differences between charge and "neutral matter" so far as momentum and kinetic energy are concerned. The mechanical momentum and kinetic energy of an uncharged particle appear to be local attributes of the particle. The electromagnetic momentum and kinetic energy of a charged particle, on the other hand, are distributed in the particle’s electromagnetic field. Specifically, when our spherical shell of charge moves with constant velocity, then
Obviously there is motivation to define the shell’s "electromagnetic mass" to be
Eq. 1.3 can then be more compactly stated as
Or, since all known charged particles appear to have both mechanical and electromagnetic mass,
In the case of uncharged particles, with zero q and zero mem, Eq. 2.5 simplifies to Newton’s second law.
Quite often Eq. 2.5 is approximated as
To the extent the dax/dt term in Eq. 2.5 is greatly overshadowed by the ax term, solutions to Eq. 2.6 will approximate reality. (And, Eq. 2.6 is analytically more tractable than Eq. 2.5.) However, rigorously speaking Eq. 2.5 is the non-relativistic equation of motion for a charged particle.
3. Energy Conservation and Agent-Particle/Field Dynamics.
In view of Newton’s second and third laws, we might be motivated to dub (–memax) the electromagnetic inertial reaction force. The total force in Eq. 2.5 can then be considered to be the sum of three forces: (1) a counteraction to Newton’s somewhat mysterious mechanical inertial reaction force; (2) a counteraction to the charge’s electromagnetic inertial reaction force; and (3) a counteraction to the radiation reaction force of Abraham-Lorentz.
The agent’s counteraction to the radiation reaction force provides a dynamic basis for the energy that flows to/from the particle’s radiation field:
Similarly the agent’s counteraction to the electromagnetic inertial reaction force does the same for the energy that flows to/from the particle’s electromagnetic "inertial" field:
Of particular interest is the agent’s expended work per cycle in the case of periodic motion. For example, our spherical shell might have motion
Clearly the net work per cycle, expended to counteract the mechanical and electromagnetic inertial reaction forces, is zero. In the case of the electromagnetic part, "inertial" energy flows from agent to field part of each cycle, and back from field to agent the remainder of the cycle.
The work expended each cycle to counteract the radiation reaction force is quite another matter. It is positive. Indeed a cylindrical surface, enclosing the oscillation site, can be constructed in space. Approximating the (tiny) spherical shell’s fields on such a surface with the fields of a point charge, the Poynting vector, S, can readily be computed at points on the surface. The integral over the surface gives the power flux through the surface. And the integral of this power flux over one cycle time gives the energy flux per cycle out to/in from infinite space. For periodic motion this flux is always outward; oscillating charges radiate. Indeed it is always found that
4. Newton’s Inertial Reaction Force … Also an Electric Force?
Although the (mechanical) inertial reaction force is somewhat mysterious in the case of uncharged particles (such as atoms), it is interesting to speculate whether these reaction forces might also be electromagnetic. For example, the electromagnetic field outside an atom is nominally zero. But there are significant fields within. Might the atom’s resistance to being accelerated ultimately be explained by electric forces on the nucleus and orbital electrons? It is a question worthy of investigation. Suffice it to say here that an external agent will also generally have to counteract such interactive forces in order to maintain a given motion. (Special case: two charge distributions at rest or moving with a common, constant velocity. Clearly some agent must counteract the attractive/repulsive forces between the distributions.)
5. In Conclusion.
The equation of motion for a tiny (R<<1 m) spherical shell of charge is, in one dimension,
If the charge’s motion is given, then the agent force needed to drive the charge is readily calculated. Solving for the motion when Fx(t) is given is usually more difficult.
Eq. 5.1 (the spherical shell of charge version of Newton’s second law) applies only to isolated spherical shells of charge. If two or more charges are in proximity to one another, then Fxmust nominally counteract the interactive forces as well as the individual reaction forces.
The energy flow out of/into a surface that contains a dimensionally small oscillating charge is readily computed using the point charge field solutions. The flow per cycle is always positive (away into infinite space), and it equates to the agent work per cycle expended to counteract the radiation reaction force. In the case of uncharged particles, Newton’s second law applies. Among other things, the agent work per cycle, expended to counteract the mechanical inertial reaction force, is zero. Oscillating uncharged particles do not radiate.
It is important to bear in mind what the net internal electromagnetic force determined by Lorentz et al (Eq. 1.1) acts upon. It is a reaction force that acts upon the agent that maintains the charge’s shape. The external agent’s role does not vanish if and when the net external force (and the charge’s acceleration) goes to zero. Whether explicitly mentioned or not, the non-electromagnetic action of an external agent is always required if a charge distribution’s shape is to persist in time.