Analogies Between Heat and Radiation

** Abstract.**
Some useful analogies between heat and radiation are discussed.

__1.
Friction.__

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.

Figure 1-1

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:

, (1-1)

and the motion will be:

(1-2)

where

. (1-3)

If b>0 then the rubber band force must be augmented somehow in order for Eq. 1-2 to be satisfied:

, (1-4a)

. (1-4b)

W,
the work per cycle done by F_{Aug}, is __>__0:

. (1-5)

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

(1-6)

or

(1-7)

where

, (1-8)

etc. For adequately small values of b the solution of Eq. 1-7 is

(1-9a)

where

. (1-9b)

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.

Figure 1-2

x(t), F_{Aug}=0, b>0

__2. Radiation.__

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

. (2-1)

When the charge is accelerated it experiences an inertial reaction force (which is an electric force) in its own, acceleration-induced fields:

. (2-2)

Furthermore, if d**a**/dt is
nonzero, then the charge will also experience a *radiation* reaction
force (also electric) in its own fields:

. (2-3)

If m_{em} 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

(2-4a)

or

. (2-4b)

Owing to the c^{3} term
in the coefficient of x’’’, this coefficient may be much less than m_{em} 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

, (2-5)

then the charge’s motion will
indeed be sinusoidal (or periodic), and it is readily shown that W, the work
done per cycle by **F**_{Aug},
equates to the field energy flux per cycle through an enclosing surface:

, (2-6)

where __S__ is
the Poynting Vector.

__3. More on the Dual Roles of the Radiation
Reaction Force.__

When –kx is augmented by F_{Aug},
as specified in Eq. 2-5, then F_{Aug}and the radiation reaction force
sum to zero. Eq. 2-4a then becomes

(3-1)

or more simply

. (3-2)

The charge oscillates
sinusoidally, with a set amount of radiant energy released each cycle. In this
case the reaction to F_{Aug} is
provided by the radiation reaction force, and the reaction to –kx is provided
by the inertial reaction force (or vice versa). When F_{Aug} is
zero, da/dt will still vary in time. In these cases Eq. 3-1 becomes

. (3-3)

To the extent d**a**/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 (-m_{em}a). 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=A_{i}, the
system’s total energy is

. (3-4)

When the charge passes through
x=0, the spring force is zero and the radiation reaction force is the sole
counteraction to –m_{em}a (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=m_{em}v^{2}/2 is the
kinetic field energy. When the charge is at x=A_{i}, the radiation
reaction force is zero and the spring force is the sole counteraction to –m_{em}a
(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.

__4. Conclusions.__

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.