Analogies Between Heat and Radiation G.R.Dixon, 1/17/2005
1. Friction. G.R.Dixon, 1/17/2005 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 in order for Eq. 1-2 to be satisfied: , (1-4a) . (1-4b) W, the work per cycle done by F . (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 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 . (2-3) If m (2-4a) or . (2-4b) Owing to the c 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 , (2-6) where 3. More on the Dual Roles of the Radiation Reaction Force. When –kx is augmented by F (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 . (3-3) 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 (-m At the beginning of any given cycle "i", with the charge momentarily at rest at x=A . (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 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 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. |