A Non-Radiating, Accelerating Charge
As specified in a previous article, the relativistically correct equation of motion for a tiny spherical shell of charge is:
Presumably no radiation is emitted when
One obvious solution to Eq. 2 is:
A second solution can be computed by imposing initial conditions, say:
and by then applying the following algorithm (where dt<<1 sec):
dax/dt(t) = -(3g(t)2vx(t)ax(t)2)/c2
The loop can be terminated when vx reaches some desired value.
Figs. 1 and 2 plot the computed vx(t) and ax(t) respectively. Note that ax asymptotically approaches zero as vx asymptotically approaches c.
vx(t), Radiation Reaction Force = 0
ax(t), Radiation Reaction Force = 0
Let us stipulate that mo=1 kg and substitute the computed values of vx(t) (org) and ax(t) into the equation of motion (Eq. 1), which in this case simplifies to:
Fig. 3 plots the computed Fx(t). Evidently a charge subjected to a constant force does not radiate.
Fx(t), No Radiation
It is noteworthy in Fig. 2 that ax is not zero. Here then is an example of a non-radiating, accelerating charge. Clearly this is at odds with the Larmor theorem, which theorizes that radiated power is proportional to ax2. Indeed in the case of periodic motion (say x=A sin(wt)), ax is maximum when vx (and hence Fxvx) equals zero.
Interestingly enough, in the case of periodic motions Larmor and Abraham-Lorentz produce the same radiated energy per cycle time. This owes to the fact that sin2(wt) and cos2(wt) have the same definite integral over a cycle time. But of course ax2 and vx(dax/dt) are p/2 out of phase.
An experiment might settle this disconnect between Larmor and Abraham-Lorentz. For example, a charge could be inserted through a pinhole in one of the plates of a large, parallel plate capacitor. The charge is subject to a constant force while between the plates, and will certainly accelerate. Yet according to Abraham-Lorentz it should not radiate. However, short pulses of radiation might be expected in the pinhole(s), where the radiation term in Eq. 1 is nonzero.