When a charge is placed in an alternating electric field it will oscillate according to the alternating force on it. If the time spent in each field direction is equal it should return to the same position after one cycle. This assumption is only true under the condition that the electric field is perfectly homogeneous and so there is no change of electric field strength as a function of distance. If the field is inhomogeneous, it will clearly have a gradient of field strength over the distance the charge oscillates. While the charge performs its half cycle within the stronger field it will experience greater returning force than the half of the cycle in the weaker section of field. This leads to, over one complete oscillation, an average force transporting the particle from strong to weak field. This is called the ponderomotive force and has the mathematical form:
Where e and m are the charge and mass of the particle, ω is the angular frequency of the fields oscillations and the final term is the gradient of the electric field strength, E, squared (if E is small enough the action of the magnetic field can be ignored).
One of the key things to note about this equation is that the force direction is independent on the sign of the charge (unlike many situations in electrostatics) and both positive and negative particles will move down the field gradient. This effect is used to study free electrons in liquids. The problem is that individual electrons act as the most potent reducing agents for chemical reactions making them hard to study with very variable lifetimes in water. High energy lasers or electron beams are normally used to produce the electrons although their behaviour after production is complicated and the models elusive. Recently it was shown that after the production of the electrons they could be accelerated using the ponderomotive force to reach kinetic energies of over ten electron volts. These energies are enough to excite and ionise the water which was shown experimentally through the observation of cavitation bubbles (a sign of the ponderomotive action). The method described offers a way of reliably producing a stream of high energy wet electrons that can be used for various research purposes.