Ponderomotive force
Encyclopedia
In physics
, a ponderomotive force is a nonlinear
force
that a charged particle experiences in an inhomogeneous oscillating electromagnetic
field.
The ponderomotive force Fp is expressed by
where e is the electrical charge
of the particle, m is the 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 but also drifts toward the weak field area. It is noteworthy that this is one rare case where the sign of the particle charge does not change the direction of the force, unlike the Lorentz force
.
The mechanism of the ponderomotive force can be easily understood by considering the motion of the charge in an oscillating electric field. In the case of a homogeneous field, the charge returns to its initial position after one cycle of oscillation. In contrast, in the case of an inhomogeneous field, the position that the charge reaches after one cycle of oscillation shifts toward the lower field-amplitude area since the force imposed onto the charge at the turning point with a higher field amplitude is larger than that imposed at the turning point with a lower field amplitude, thus producing a net force that drives the charge toward the weak field area.
Consider a particle under the action of a non-uniform oscillating field. The equation of motion is given by:
neglecting the effect of the associated oscillating magnetic field.
If the length scale of variation of
is small enough, then the particle trajectory can be divided into a slow time motion and a fast time motion:
where
is the slow drift motion and 
represents fast oscillations. Now, let us also assume that 
. Under this assumption, we can use Taylor expansion on the force equation about 
to get,
, and because 
is small, 
, so
On the time scale on which
oscillates, 
is essentially a constant. Thus, the above can be integrated to get,
Substituting this in the force equation and averaging over the
timescale, we get,
Thus, we have obtained an expression for the drift motion of a charged particle under the effect of a non-uniform oscillating field.
. 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:
where
is the ponderomotive potential and is given by
To solve the above equation, we can make a similar assumption as we did for the case when
. This gives a generalized expression for the drift motion of the particle:
1) Quadrupole ion trap
2) Combined rf trap
3) Plasma acceleration
especially the Electrodeless plasma thruster
4) High Harmonic Generation
The ponderomotive force also plays an important role in laser induced plasmas as a major density lowering factor.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a ponderomotive force is a nonlinear
Nonlinearity
In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...
force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
that a charged particle experiences in an inhomogeneous oscillating electromagnetic
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
field.
The ponderomotive force Fp is expressed by
where e is the electrical charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
of the particle, m is the mass, ω is the angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
of oscillation of the field, and E is the amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
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 but also drifts toward the weak field area. It is noteworthy that this is one rare case where the sign of the particle charge does not change the direction of the force, unlike the Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...
.
The mechanism of the ponderomotive force can be easily understood by considering the motion of the charge in an oscillating electric field. In the case of a homogeneous field, the charge returns to its initial position after one cycle of oscillation. In contrast, in the case of an inhomogeneous field, the position that the charge reaches after one cycle of oscillation shifts toward the lower field-amplitude area since the force imposed onto the charge at the turning point with a higher field amplitude is larger than that imposed at the turning point with a lower field amplitude, thus producing a net force that drives the charge toward the weak field area.
Derivation
The derivation of the ponderomotive force expression is as follows:Consider a particle under the action of a non-uniform oscillating field. The equation of motion is given by:
neglecting the effect of the associated oscillating magnetic field.
If the length scale of variation of
is small enough, then the particle trajectory can be divided into a slow time motion and a fast time motion:where
is the slow drift motion and represents fast oscillations. Now, let us also assume that . Under this assumption, we can use Taylor expansion on the force equation about to get,, and because is small, , soOn the time scale on which
oscillates, is essentially a constant. Thus, the above can be integrated to get,Substituting this in the force equation and averaging over the
timescale, we get,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 plasmaPlasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
. 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
Vlasov equation
The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction...
. But, it is usually assumed that the time averaged density of the plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
can be directly obtained from the expression for the force expression for the drift motion of individual charged particles:
where
is the ponderomotive potential and is given byGeneralized Ponderomotive Force
Instead of just an oscillating field, there could also be a permanent field present. In such a situation, the force equation of a charged particle becomes:To solve the above equation, we can make a similar assumption as we did for the case when
. This gives a generalized expression for the drift motion of the particle:Applications
The idea of a ponderomotive description of particles under the action of a time varying field has immense applications in areas like:1) Quadrupole ion trap
Quadrupole ion trap
A quadrupole ion trap exists in both linear and 3D varieties and refers to an ion trap that uses constant DC and radio frequency oscillating AC electric fields to trap ions. It is commonly used as a component of a mass spectrometer...
2) Combined rf trap
Combined rf trap
A combined RF trap is a combination of a Quadrupole ion trap and a Penning trap. One of the main bottlenecks of a QIT is that it can confine only single-charged species or multiple species with similar masses. But in certain applications like antihydrogen production it is important to confine two...
3) Plasma acceleration
Plasma acceleration
Plasma Wakefield acceleration is a technique for accelerating charged particles, such as electrons, positrons and ions, using an electric field associated with an electron plasma wave. The wave is created either using electron pulses or through the passage of a very brief laser pulses, a technique...
especially the Electrodeless plasma thruster
Electrodeless plasma thruster
The electrodeless plasma thruster is a spacecraft propulsion engine. It was created by Mr. Gregory Emsellem based on technology developed by French Atomic Energy Commission scientist Dr Richard Geller and Dr...
4) High Harmonic Generation
High Harmonic Generation
- Perturbative Harmonic Generation :Perturbative Harmonic Generation is a process whereby laser light of frequency ω and photon energy ħω can be used to generate new frequencies of light. The newly generated frequencies are integer multiples nħω of the original light's frequency...
The ponderomotive force also plays an important role in laser induced plasmas as a major density lowering factor.
Journals
- J. R. Cary and A. N. Kaufman, Ponderomotive effects in collisionless plasma: A Lie transform approach, Phys. Fluids 24, 1238 (1981), http://dx.doi.org/10.1063/1.863527
- C. Grebogi and R. G. Littlejohn, Relativistic ponderomotive Hamiltonian, Phys. Fluids 27, 1996 (1984), http://dx.doi.org/10.1063/1.864855
- G. J. Morales and Y. C. Lee, Ponderomotive-Force Effects in a Nonuniform Plasma, Phys. Rev. Lett. 33, 1016 (1974), http://prola.aps.org/abstract/PRL/v33/i17/p1016_1
- B. M. Lamb and G. J. Morales, Ponderomotive effects in nonneutral plasmas, Phys. Fluids 26, 3488 (1983), http://dx.doi.org/10.1063/1.864132
- K. Shah and H. Ramachandran, Analytic, nonlinearly exact solutions for an rf confined plasma, Phys. Plasmas 15, 062303 (2008), http://link.aip.org/link/?PHPAEN/15/062303/1