The inertial reaction force, in the case of an uncharged particle (such as an atom), is simply –m_mecha. If the particle’s shape is kept constant, then a driving agent must counteract this force, and thus

. (1.6.1_1)

If the particle is charged, then there may be a second component to the total reaction force. This second component is called the radiation reaction force:

. (1.6.1_2)

And we have called the driving agent’s counteraction the radiative part of the total agent force.

The net force, required to counteract both the inertial and radiation reaction forces in cases of charged particles with some mechanical mass, is

. (1.6.1_3)

Let us, for discussion purposes, suppose that a charged particle with zero mechanical mass is being driven sinusoidally:

. (1.6.1_4)

If the charge is distributed in a spherical shell, then we have for the driving agent force:

. (1.6.1_5)

For the motion specified by Eq. 1.6.1_4,

.(1.6.1_6)

The rate at which the driving agent does work is

. (1.6.1_7)

And the net work per cycle done by F_inert is zero:

. (1.6.1_8)

Thus we need only concern ourselves with the work per cycle done by F_radiative. And, since this work is independent of the spherical shell’s radius, we can consider a point charge.

The work done per cycle by F_radiative is always positive:

. (1.6.2_1)

Energy conservation suggests that this should equal the net energy flux per cycle out through an enclosing surface. And a computer program indicates that this is indeed the case. For example, using a spherical surface that encloses the oscillation, the point charge field solutions can be used to compute S at points on the surface and at a sequence of epochs. Numerical integration then demonstrates that

. (1.6.2_2)