APPENDIX
PLANCK’S DIPOLE RESONATOR
EQUATIONS
In the same year the electron was discovered, the first
theoretical expression for radiation damping was derived by Max Planck in his
research on the relation between the energy and entropy of the electromagnetic
field as it interacts with small “dipole resonators”.[i] Using the assumption of incident
electromagnetic plane waves polarized parallel to the dipole axis, Planck found
an equation relating the incident electric field to the electric dipole moment
of the resonators:
K p(x,t) + L d2p(x,t)/dt2
– (2/3c3) d3p(x,t)/dt3
= E(x,t)
where, in Gaussian units, E(x,t) is the incident electric field, K and
L are resonator-dependent constants,
and p(x,t) is the dipole
moment of the resonator. Planck showed that when the resonator’s conditions are
such that its energy changes slowly in comparison with variations in the
incident field--when damping is much slower than the frequency of the incident
electromagnetic wave--the third time derivative can be replaced with a first
time derivative. (This is simple harmonic
oscillation approximation
K p(x,t) + L d2p(x,t)/dt2 + (2K/3c3L) dp(x,t)/dt = E(x,t)
This equation is of the form
used in the classical model of the atom.
Planck was searching for an explanation via electrodynamics
for the increase in entropy of closed systems, the observed macroscopic
irreversibility in time not predicted by mechanical equations of motion, which
are time reversal invariant.
The appearance of the damping terms in the resonator
equations implies a partial breaking of the forward and backward symmetry in
time, since first derivatives
change sign when t (time) changes sign. Planck called radiation damping “conservative
damping” in order to distinguish it from the dissipative effects of
non-conservative damping forces such as friction.[ii] Planck’s work, however, did not lead him to
an explanation of entropy based on electrodynamics. As Kuhn[iii]
explains:
Through
most of the year 1897, Planck continued to believe that he could prove irreversibility
directly, without the aid of any statistical or other special hypotheses. That proof had been his initial objective in
taking up the black-body problem at all.
But by the spring of 1898 he had recognized that that goal could not
possibly be achieved, and the concepts deployed in his subsequent papers came
more and more to resemble those developed by Boltzmann for gas theory.
Though
Planck was not successful at explaining entropy from fundamental
electrodynamics, his research resulted in his postulate that a charged
oscillator must emit and absorb radiation only in discrete amounts--the idea
that gave birth to quantum mechanics in 1900.
