Analogies Between Heat and Radiation
Fig. 1-1 depicts a rigid sphere of mass m, suspended and at rest in a viscous medium. The "ideal" rubber band that holds the sphere at rest is stretched an amount L. (By "ideal" it is meant that the rubber band does not heat up when it is stretched, and it has no mass.) The rubber band’s spring constant is k. When the sphere moves up or down through the medium, the medium exerts a frictional (or drag) force of magnitude bv, said force being oppositely directed to the sphere’s velocity.
Rigid Sphere Suspended in Viscous Medium
If the sphere is displaced downward an amount A<L and then released at time t=0, then it will accelerate upward. If b=0 (vacuum) then the equation of motion is:
and the motion will be:
If b>0 then the rubber band force must be augmented in order for Eq. 1-2 to be satisfied:
W, the work per cycle done by FAug, is >0:
By energy conservation, W equals the net gain per cycle of heat energy by the medium (assuming the sphere is perfectly rigid and its temperature remains constant).
If the rubber band force is not augmented, then the equation of motion (Eq. 1-4a) reduces to
etc. For adequately small values of b the solution of Eq. 1-7 is
Fig. 1-2 plots a solution of Eq. 1-7 for m=1 kg, k=1 nt/meter, b=.1 nt sec/meter and A=.5 meter. In agreement with experience, the motion attenuates as heat is generated.
x(t), FAug=0, b>0
Let us now imagine that a spherical shell of charge, q, is driven by a spring "contact" (or non-electromagnetic) force. If the charge’s radius is R, then its electromagnetic mass is
When the charge is accelerated it experiences an inertial reaction force (which is an electric force) in its own, acceleration-induced fields:
Furthermore, if da/dt is nonzero, then the charge will also experience a radiation reaction force (also electric) in its own fields:
If mem constitutes the whole inertial mass of the spherical shell of charge, and if the spring force counteracts both the inertial and radiation reaction forces, then the equation of motion is
Owing to the c3 term in the coefficient of x’’’, this coefficient may be much less than mem and k. As in the frictional case where b~0, the motion in such cases will be approximately sinusoidal … but not quite. Indeed since x’’’ is proportional to –x’ in the case of true sinusoidal motion, the frictional and radiation cases are mathematically similar. In the frictional case, heat energy is generated. In the radiation case it is radiant energy that is generated.
If the spring force is augmented, such that
then the charge’s motion will indeed be sinusoidal (or periodic), and it is readily shown that W, the work done per cycle by FAug, equates to the field energy flux per cycle through an enclosing surface:
where S is the Poynting Vector.
3. More on the Dual Roles of the Radiation Reaction Force.
When –kx is augmented by FAug, as specified in Eq. 2-5, then FAug and the radiation reaction force sum to zero. Eq. 2-4a the becomes
or more simply
The charge oscillates sinusoidally, with a set amount of radiant energy released each cycle. In this case the reaction to FAug is provided by the radiation reaction force, and the reaction to –kx is provided by the inertial reaction force (or vice versa). When FAug is zero, da/dt will still vary in time. In these cases Eq. 3-1 becomes
To the extent da/dt points opposite to (and is proportional to) v, Eqs. 3-3 and 1-6 are mathematically the same. The attenuated motion plotted in Fig. 1-2 can therefore again be expected, the difference being that radiant (rather than heat) energy is generated in the spherical shell of charge case. Note that In Eq. 3-3 the radiation reaction force joins the spring force in counteracting the inertial reaction force (-mema). Or, one could say that the driving spring force is modulated by the radiation reaction force, with an attendant release of radiant energy.
At the beginning of any given cycle "i", with the charge momentarily at rest at x=Ai, the system’s total energy is
When the charge passes through x=0, the spring force is zero and the radiation reaction force is the sole counteraction to –mema (which will not be zero in the attenuating case). At this moment the rate at which radiant energy is generated will equate to dT/dt, where T=memv2/2 is the kinetic field energy. When the charge is at x=Ai, the radiation reaction force is zero and the spring force is the sole counteraction to –mema (which will have its maximum absolute value).
If we opt to define the start of each cycle as the instant when the charge passes through x=0, traveling in the positive x-direction, then its kinetic energy at the start of the next cycle will be less than it is at the beginning of the current cycle by the amount of radiant energy released in the quasi-cycle time. In effect a portion of the kinetic field energy, which could be said to be "closely held" to the charge, is transformed into radiant field energy, said radiant energy bleeding away into infinite space.
In certain cases many instructive parallels can be noted between a rigid sphere, driven through a viscous medium, and a non-deformable spherical shell of charge driven in a vacuum. The technique is to substitute the radiation reaction force for the viscous frictional force, and to view generated heat energy and generated radiant energy as analogous quantities.