Nonlinear force experienced by a charged particle
In physics, a ponderomotive force is a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field. It causes the particle to move towards the area of the weaker field strength, rather than oscillating around an initial point as happens in a homogeneous field. This occurs because the particle sees a greater magnitude of force during the half of the oscillation period while it is in the area with the stronger field. The net force during its period in the weaker area in the second half of the oscillation does not offset the net force of the first half, and so over a complete cycle this makes the particle move towards the area of lesser force.
The ponderomotive force Fp is expressed by
formula_1formula_2formula_3formula_4
which has units of newtons (in SI units) and where "e" is the electrical charge of the particle, "m" is its mass, "ω" is the angular frequency of oscillation of the field, and "E" is the amplitude of the electric field. At low enough amplitudes the magnetic field exerts very little force.
This equation means that a charged particle in an inhomogeneous oscillating field not only oscillates at the frequency of "ω" of the field, but is also accelerated by Fp toward the weak field direction. This is a rare case in which the direction of the force does not depend on whether the particle is positively or negatively charged.
Etymology.
The term "ponderomotive" comes from the Latin "ponder-" (meaning weight) and the english "motive" (having to do with motion).
Derivation.
The derivation of the ponderomotive force expression proceeds as follows.
Consider a particle under the action of a non-uniform electric field oscillating at frequency formula_5 in the x-direction. The equation of motion is given by:
formula_6
neglecting the effect of the associated oscillating magnetic field.
If the length scale of variation of formula_7 is large enough, then the particle trajectory can be divided into a slow time (secular) motion and a fast time (micro)motion:
formula_8
where formula_9 is the slow drift motion and formula_10 represents fast oscillations. Now, let us also assume that formula_11. Under this assumption, we can use Taylor expansion on the force equation about formula_9, to get:
formula_13
formula_14, and because formula_10 is small, formula_16, so
formula_17
On the time scale on which formula_10 oscillates, formula_9 is essentially a constant. Thus, the above can be integrated to get:
formula_20
Substituting this in the force equation and averaging over the formula_21 timescale, we get,
formula_22
formula_23
Thus, we have obtained an expression for the drift motion of a charged particle under the effect of a non-uniform oscillating field.
Time averaged density.
Instead of a single charged particle, there could be a gas of charged particles confined by the action of such a force. Such a gas of charged particles is called plasma. The distribution function and density of the plasma will fluctuate at the applied oscillating frequency and to obtain an exact solution, we need to solve the Vlasov Equation. But, it is usually assumed that the time averaged density of the plasma can be directly obtained from the expression for the force expression for the drift motion of individual charged particles:
formula_24
where formula_25 is the ponderomotive potential and is given by
formula_26
Generalized ponderomotive force.
Instead of just an oscillating field, a permanent field could also be present. In such a situation, the force equation of a charged particle becomes:
formula_27
To solve the above equation, we can make a similar assumption as we did for the case when formula_28. This gives a generalized expression for the drift motion of the particle:
formula_29
Applications.
The idea of a ponderomotive description of particles under the action of a time-varying field has applications in areas like:
The quadrupole ion trap uses a linear function formula_30 along its principal axes. This gives rise to a harmonic oscillator in the secular motion with the so-called trapping frequency formula_31, where formula_32 are the charge and mass of the ion, the peak amplitude and the frequency of the radiofrequency (rf) trapping field, and the ion-to-electrode distance respectively. Note that a larger rf frequency lowers the trapping frequency.
The ponderomotive force also plays an important role in laser induced plasmas as a major density lowering factor.
Often, however, the assumed slow-time independency of formula_33 is too restrictive, an example being the 
ultra-short, intense laser pulse-plasma(target) interaction. Here a new ponderomotive effect comes into play, 
the ponderomotive memory effect. The result is a weakening of the ponderomotive force and the generation of wake fields and ponderomotive streamers. In this case the fast-time averaged density becomes for a Maxwellian plasma:
formula_34,
where formula_35 and formula_36.
References.
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