**The following,
highly qualitative discussion, was featured on MySpace.com in response to a post
about inertia.**

Self-Forces and Inertia

G.R.Dixon, August 9, 2008

Consider a spherical shell, of charge q and radius R, whose center has always been at rest at the origin of inertial frame K. Every charge increment in the shell lies in the net, outward-pointing electric (**E**) field of the other increments. It accordingly experiences an outward-pointing, infinitesimal electric force. Assuming R remains constant, each such tiny force must be non-electromagnetically counteracted by some external agent. Owing to symmetry, all of the tiny forces sum to zero, and hence the net agent force (say **F _{A}**) is also zero.

If the shell is viewed from inertial frame K’, which moves in the negative x direction of K at speed v<<c, then at time t’=0 the charge will be at the origin of K’ and will be moving in the positive x’ direction at speed v. In frame K’ Maxwell’s equations indicate the existence of both an electric (**E’**) and magnetic (**B’**) field at points external to the shell.

Now in general an electromagnetic field has a momentum density that is proportional to **E** X **B** at each point in space. The momentum density in K’ can be integrated over all of space, and this yields a net momentum that is equal to q^{2}/(6pe_{o}Rc^{2})**v’** (where **v’** is the charge’s velocity relative to K’). For obvious reasons the q^{2}/(6pe_{o}Rc^{2}) term is referred to as the charge’s electromagnetic mass.

In K’ (as in K) each increment of charge experiences a tiny, outward-pointing electric force. But strictly speaking, Coulomb’s law cannot be invoked in the moving case when computing each tiny electric force. However, the point charge field solutions for arbitrarily moving point charges is now well established. And if each charge increment in the shell is approximated as an infinitesimal point charge, then in theory the electric force on any given increment (attributable to all the other increments) can still be computed. The net electric force, experienced by the spherical shell in its own electric field, could then again be found by adding up all the infinitesimal forces. And **F _{A}** would again equal the negative of that sum. It need only be mentioned that the net force in K’ (and hence

But what if the shell of charge is not at rest in *any* inertial frame? What if it accelerates relative to every inertial frame? The point charge field solutions can still be used to compute the tiny electric force experienced by each charge increment in the shell. And all of the tiny forces can again be summed to find the total electric force experienced by the shell in its own electric field. It turns out that this sum is *not* zero when the charge accelerates! The net force points opposite to the acceleration and equals the electromagnetic mass times the magnitude of the acceleration. Thus in this case **F _{A}** equals the electromagnetic mass times the acceleration!

It is noteworthy that this acceleration-opposing, electric "self" force is a consequence of Maxwell’s equations, and does not depend on anything in the shell’s environment. It would theoretically be manifest even if the charge were the sole entity in the universe.

Let us now consider "neutral matter," say a neutron. "External" to such a particle there is no electromagnetic field to speak of; whence we conclude that the particle has no excess charge. But it is now known that a neutron (for example) is composed of quarks which *are* charged. And when the neutron is accelerated, its internal quarks might be expected to experience acceleration-opposing electric forces quite as the charge increments in our spherical shell do. In brief, the inertia of "neutral" particles might also ultimately be traceable to Maxwell’s equations, quite as the electromagnetic inertia of charged particles is.

If you are interested in the general point charge field solutions, the following link might be a helpful starting point: Point Charge Book . These general solutions have been only recently derived (by Feynman and others), and may not have been known to Einstein. In any case, I find the characterization of inertia as a self-force to be preferable to Mach’s principle and similar hypotheses.