The Stimulated Genesis of Radiant Energy from Electromagnetic Field Energy
In this article we consider a 1-coulomb charge that, up until t=0, moves along the negative x-axis, toward the origin, at constant speed vo=1E3 m/sec. At t=0 it reaches the origin where some external agent interacts non-electromagnetically, bringing it to a sudden halt at x1=1E-4 meters. The motion is completely specified by the following 3 equations:
At t1=1.57E-7 sec, when the charge has come to permanent rest, all interaction between the agent and the charge ceases.
We shall construct a spherical surface of radius ct1=47.09 meters and centered on x1. dF/dt(t) is defined to be the rate at which electromagnetic field energy flows through this surface:
where S(t), the Poynting vector at a point on the surface, is related to E and B by
F(ta,tb), the energy fluxing through the surface in the time interval ta<t<tb, is thus
In order to avoid infinite electromagnetic mass issues, we shall suppose that the charge is distributed in a spherical shell of radius R=1E-3 meters. The electromagnetic mass of such a distribution is
Since ct1>>R, we may safely use the point charge field solutions to compute S at points on the spherical surface.
"Conventional wisdom" suggests that, at a distance D from the charge, any interaction between agent and charge at time t cannot be manifest in the fields until time t+D/c. Since x1=.1R, the charge comes to a halt almost instantaneously. The intense deceleration that occurs in the time interval 0<t<t1 should accordingly not be manifest at points on the surface until t1. And at time 2t1 the fields within and at points on the surface should have become electrostatic, with zero magnetic field.
2. Agent/Charge Interaction.
Since v(t)<1E3 m/sec, we shall use non-relativistic equations. In the time interval 0<t<t1, the charge exerts a 2-part reaction force on the decelerating agent:
And the external agent counteracts this force:
The work done by the first part of this force is
And the work done by the second part is
WInert is thus the negative of the magnetic field energy in all of space at/prior to t=0. And WRad presumably equates to the energy of emitted radiation.
3. Two Paradoxes.
According to conventional wisdom, all of the negative work, expended by the agent to counteract the inertial reaction force, must have come from the fields in the spherical volume of radius ct1 and centered on x1. Yet at time t1 there is still magnetic field energy outside of this volume, all the way out to infinity! And if we wait for an infinite time, this energy vanishes; B=0 everywhere (except in the radiant pulse spreading out "beyond infinity"). Here, then, is the first paradox: How can the driving agent have (by time t1) absorbed an amount of energy equal to all of the initial magnetic field energy, when there is still magnetic field energy out in the fields at that moment?
The second paradox has to do with an inequality between WRad, the work expended to counteract the radiation reaction force, and the energy flux through the spherical surface in the time interval t1<t<2t1. Fig. 3_1 plots dF/dt over this interval. The time integral is computed to be
Since dF/dt is zero for all t>2t1, how can WRad be greater than F(t1,2t1)?
4. A Suggested Resolution.
At time t1 the magnetic field energy remaining in all of space outside the sphere of radius ct1 is
And from t1 to 2t1 energy fluxes through the sphere’s surface in an amount
After an infinite time the B field energy has vanished and there is radiant energy in an amount
Suppose that the magnetic field energy converts to radiant energy as the original pulse, of energyF(t1,2t1), expands out into space. In this event, after an infinite time has elapsed, the radiant energy would be emag(t1)+F(t1,2t1)=1.13E-3 joules, which does not differ appreciably from the value of WRad=1.75E-3 joules. ((emag(t1)+F(t1,2t1)) / WRad=.65)
Better results might be obtained by using the relativistic expressions for E and B. For example, outside the sphere of radius ct1 (and at time t1) the electric field is that of a charge moving at constant velocity, whereas after an infinite time it is electrostatic. Thus some of the electric field energy might also be converted to radiant energy as the pulse emitted between t1 and 2t1 passes through.
We shall not pursue the matter with relativistic rigor here. Suffice it to say that the stimulated conversion of field energy to radiant energy may offer a solution to our two paradoxes. If this is the case, then there are many interesting implications. For example, it is not necessarily true that all of the photons, emitted when an agent causes a charge to accelerate, originate at the charge! Some of them may "materialize" in empty space (or, if you prefer, in Maxwell’s stressed ether).