Ponderomotive force

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

$\mathbf {F} _{\text{p}}=$ $-{\frac {e^{2}}{4m\omega ^{2}}}$ $\nabla$ $(E^{2})$ 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 where the sign of the charge on the particle does not change the direction of the force ((-e)2=(+e)2).

The mechanism of the ponderomotive force can be understood by considering the motion of a 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 the case of an inhomogeneous field, the force exerted on the charge during the half-cycle it spends in the area with higher field amplitude points in the direction where the field is weaker. It is larger than the force exerted during the half-cycle spent in the area with a lower field amplitude, which points towards the strong field area. Thus, averaged over a full cycle there is a net force that drives the charge toward the weak field area.

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 $\omega$  in the x-direction. The equation of motion is given by:

${\ddot {x}}=g(x)\cos(\omega t),$

neglecting the effect of the associated oscillating magnetic field.

If the length scale of variation of $g(x)$  is large enough, then the particle trajectory can be divided into a slow time motion and a fast time motion:

$x=x_{0}+x_{1}$

where $x_{0}$  is the slow drift motion and $x_{1}$  represents fast oscillations. Now, let us also assume that $x_{1}\ll x_{0}$ . Under this assumption, we can use Taylor expansion on the force equation about $x_{0}$ , to get:

${\ddot {x}}_{0}+{\ddot {x}}_{1}=\left[g(x_{0})+x_{1}g'(x_{0})\right]\cos(\omega t)$
${\ddot {x}}_{0}\ll {\ddot {x}}_{1}$ , and because $x_{1}$  is small, $g(x_{0})\gg x_{1}g'(x_{0})$ , so
${\ddot {x}}_{1}=g(x_{0})\cos(\omega t)$

On the time scale on which $x_{1}$  oscillates, $x_{0}$  is essentially a constant. Thus, the above can be integrated to get:

$x_{1}=-{\frac {g(x_{0})}{\omega ^{2}}}\cos(\omega t)$

Substituting this in the force equation and averaging over the $2\pi /\omega$  timescale, we get,

${\ddot {x}}_{0}=-{\frac {g(x_{0})g'(x_{0})}{2\omega ^{2}}}$
$\Rightarrow {\ddot {x}}_{0}=-{\frac {1}{4\omega ^{2}}}\left.{\frac {d}{dx}}\left[g(x)^{2}\right]\right|_{x=x_{0}}$

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:

${\bar {n}}(x)=n_{0}\exp \left[-{\frac {e}{\kappa T}}\Phi _{\text{P}}(x)\right]$

where $\Phi _{\text{P}}$  is the ponderomotive potential and is given by

$\Phi _{\text{P}}(x)={\frac {m}{4\omega ^{2}}}\left[g(x)\right]^{2}$

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:

${\ddot {x}}=h(x)+g(x)\cos(\omega t)$

To solve the above equation, we can make a similar assumption as we did for the case when $h(x)=0$ . This gives a generalized expression for the drift motion of the particle:

${\ddot {x}}_{0}=h(x_{0})-{\frac {g(x_{0})g'(x_{0})}{2\omega ^{2}}}$

Applications

The idea of a ponderomotive description of particles under the action of a time-varying field has applications in areas like:

The ponderomotive force also plays an important role in laser induced plasmas as a major density lowering factor.