Consider a spatially inhomogeneous electromagnetic wave-field oscillating at frequency :

(2.121) |

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

where

in accordance with 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,

When this inequality is satisfied, Equation (2.123) implies that the magnetic force experienced by the particle is smaller than the electric force by one order in the expansion parameter.

Let us now apply the averaging technique. We make the substitution in the oscillatory terms, and seek a change of variables,

(2.125) | ||

(2.126) |

such that

(2.127) |

Moreover, the lowest order oscillating component of gives

(2.128) |

The solutions to the previous two equations, taking into account the constraints

(2.129) | ||

(2.130) |

respectively. Here, represents an oscillation average.

It follows that, to lowest order, there is no motion of the center of oscillation. To first order, the oscillation average of Equation (2.122) yields

(2.131) |

which reduces to

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

where

(2.134) |

It follows that the oscillation center experiences a force, known as the

The total energy of the oscillation center,

(2.135) |

is conserved by its equation of motion, Equation (2.133). However, it follows from Equation (2.129) that the ponderomotive potential energy is equal to the average kinetic energy of the oscillatory motion: that is,

(2.136) |

Thus, the force on the center 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 center 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 that acts as a wave-guide for the laser beam (Kruer 2003).