8
B. Sutcliffe and R.G. Woolley
it quickly became apparent that Rutherford’s solar system model of the atom (planetary electrons moving about a central nucleus) cannot avoid eventual collapse if
classical electrodynamics applies to it. This is because of Earnshaw’s theorem which
states that it is impossible for a collection of charged particles to maintain a static
equilibrium purely through electrostatic forces [16]. This is the classical result that
Bohr alluded to in his 1922 Nobel lecture [17] to rule out an electrostatical explanation for the stability of atoms and molecules.
The theorem may be proved by demonstrating a contradiction. Suppose the
charges are at rest and consider the motion of a particular charge e n in the electric
field, E, generated by all of the other charged particles. Assume that this particular
charge has e n > 0. The equilibrium position of this particle is the point x 0
n where
E(x 0
n ) = 0, since the force on the charge is e n E(x n ) (the Lorentz force for this static
case). Obviously, x 0
n cannot be the equilibrium position of any other particle. However, in order for x 0
n to be a stable equilibrium point, the particle must experience
a restoring force when it is displaced from x 0
n in any direction. For a positively
charged particle at x 0
n , this requires that the electric field points radially towards x 0
n
at all neighbouring points. But from Gauss’s law applied to a small sphere centred
on x 0
n , this corresponds to a negative flux of E through the surface of the sphere, implying the presence of a negative charge at x 0
n , contrary to our original assumption.
Thus E cannot point radially towards x 0
n at all neighbouring points, that is, there
must be some neighbouring points at which E is directed away from x 0
n . Hence,
a positively charged particle placed at x 0
n will always move towards such points.
There is therefore no static equilibrium configuration. According to classical electrodynamics accelerated charges must radiate electromagnetic energy, and hence
lose kinetic energy, so even a dynamical model cannot be stable according to purely
classical theory.
Molecular models which can be represented in terms of the (phase-space) variables of classical dynamics had a far-reaching influence on the interpretation of
molecular spectra after the dissemination of Bohr’s quantum theory of atoms and
molecules based on transitions between stationary states [18]. An important feature
of his new theory was that classical electrodynamics should be deemed to be still
operative when transitions took place, but not when the system was in a stationary
state, by fiat. Bohr had originally used the fact that two particles with Coulombic
interaction lead to a Hamiltonian problem that is completely soluble by separation
of variables. With more particles and Coulombic interactions this is no longer true;
however by largely qualitative reasoning he was able to develop a quantum theory of
the atom and the Periodic Table (reviewed in [17]). Furthermore by the introduction
of Planck’s constant h through the angular momentum quantization condition, Bohr
solved another problem of the classical theory. In classical electrodynamics the only
characteristic length available is the classical radius r o for a charged particle. This is
obtained by equating the rest-mass energy for the charge to the electrostatic energy
of a charged sphere of radius r o
r o =
e 2
4πε 0 mc 2
.
B. Sutcliffe and R.G. Woolley
it quickly became apparent that Rutherford’s solar system model of the atom (planetary electrons moving about a central nucleus) cannot avoid eventual collapse if
classical electrodynamics applies to it. This is because of Earnshaw’s theorem which
states that it is impossible for a collection of charged particles to maintain a static
equilibrium purely through electrostatic forces [16]. This is the classical result that
Bohr alluded to in his 1922 Nobel lecture [17] to rule out an electrostatical explanation for the stability of atoms and molecules.
The theorem may be proved by demonstrating a contradiction. Suppose the
charges are at rest and consider the motion of a particular charge e n in the electric
field, E, generated by all of the other charged particles. Assume that this particular
charge has e n > 0. The equilibrium position of this particle is the point x 0
n where
E(x 0
n ) = 0, since the force on the charge is e n E(x n ) (the Lorentz force for this static
case). Obviously, x 0
n cannot be the equilibrium position of any other particle. However, in order for x 0
n to be a stable equilibrium point, the particle must experience
a restoring force when it is displaced from x 0
n in any direction. For a positively
charged particle at x 0
n , this requires that the electric field points radially towards x 0
n
at all neighbouring points. But from Gauss’s law applied to a small sphere centred
on x 0
n , this corresponds to a negative flux of E through the surface of the sphere, implying the presence of a negative charge at x 0
n , contrary to our original assumption.
Thus E cannot point radially towards x 0
n at all neighbouring points, that is, there
must be some neighbouring points at which E is directed away from x 0
n . Hence,
a positively charged particle placed at x 0
n will always move towards such points.
There is therefore no static equilibrium configuration. According to classical electrodynamics accelerated charges must radiate electromagnetic energy, and hence
lose kinetic energy, so even a dynamical model cannot be stable according to purely
classical theory.
Molecular models which can be represented in terms of the (phase-space) variables of classical dynamics had a far-reaching influence on the interpretation of
molecular spectra after the dissemination of Bohr’s quantum theory of atoms and
molecules based on transitions between stationary states [18]. An important feature
of his new theory was that classical electrodynamics should be deemed to be still
operative when transitions took place, but not when the system was in a stationary
state, by fiat. Bohr had originally used the fact that two particles with Coulombic
interaction lead to a Hamiltonian problem that is completely soluble by separation
of variables. With more particles and Coulombic interactions this is no longer true;
however by largely qualitative reasoning he was able to develop a quantum theory of
the atom and the Periodic Table (reviewed in [17]). Furthermore by the introduction
of Planck’s constant h through the angular momentum quantization condition, Bohr
solved another problem of the classical theory. In classical electrodynamics the only
characteristic length available is the classical radius r o for a charged particle. This is
obtained by equating the rest-mass energy for the charge to the electrostatic energy
of a charged sphere of radius r o
r o =
e 2
4πε 0 mc 2
.