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We have seen that charged particles can be confined by a static magnetic field.
A somewhat more surprising fact is that charged
particles can also be confined by a rapidly
oscillating, inhomogeneous electromagnetic wavefield. In order to demonstrate this,
we again make use of our averaging technique. To lowest order, a particle
executes simple harmonic motion in response to an oscillating wavefield.
However, to higher order, any weak inhomogeneity in the field causes the restoring force
at one turning point to exceed that at the other. On average, this yields a net force
which acts on the centre of oscillation of the particle.
Consider a spatially inhomogeneous electromagnetic wavefield oscillating at
frequency :

(151) 
The equation of motion of a charged particle placed in this field is written

(152) 
where

(153) 
according to Faraday's law.
In order for our averaging technique to be applicable, the electric field
experienced by the particle must remain approximately constant
during an oscillation. Thus,

(154) 
When this inequality is satisfied, Eq. (153) implies that the magnetic
force experienced by the particle
is smaller than the electric force by one order in the
expansion parameter. In fact, Eq. (154) is equivalent to the requirement,
, that the particle be unmagnetized.
We now apply the averaging technique. We make the substitution
in the oscillatory terms, and seek a change of variables,
such that
and are periodic functions of
with vanishing mean. Averaging
again yields
to all orders. To lowest order, the momentum evolution
equation reduces to

(157) 
The solution, taking into account the constraints
, is
Here,
represents
an oscillation average.
Clearly, there is no motion of the centre of oscillation to lowest order.
To first order, the oscillation average of Eq. (152) yields

(160) 
which reduces to

(161) 
The oscillation averages of the trigonometric functions are both equal to .
Furthermore, we have
. Thus, the equation of motion for
the centre of oscillation reduces to

(162) 
where

(163) 
It is clear that the oscillation centre experiences a force, called the
ponderomotive force, which is proportional to the gradient
in the amplitude of the wavefield. The ponderomotive force is
independent of the sign of the charge, so both electrons and
ions can be confined in the same potential well.
The total energy of the oscillation centre,

(164) 
is conserved by the equation of motion (161). Note that the ponderomotive
potential energy is equal to the average kinetic energy of the
oscillatory motion:

(165) 
Thus, the force on the centre of oscillation originates in a transfer of
energy from the oscillatory motion to the average motion.
Most of the important applications of the ponderomotive force occur in laser
plasma physics.
For instance, a laser beam can propagate in a plasma provided that
its frequency exceeds the plasma frequency. If the beam is sufficiently
intense then plasma particles are repulsed from the centre of the
beam by the ponderomotive force. The resulting variation in the plasma
density gives rise to a cylindrical well in the index of refraction
which acts as a waveguide for the laser beam.
Next: Plasma Fluid Theory
Up: Charged Particle Motion
Previous: Third Adiabatic Invariant
Richard Fitzpatrick
20110331