# Ponderomotive force and conservation of momentum

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• Philip Koeck
Philip Koeck
According to https://en.wikipedia.org/wiki/Ponderomotive_force a charge in an inhomogeneous, oscillating electric field will be accelerated in the direction where the field gets weaker.

One could think of a charged particle placed slightly off-center inside a laser beam.
If I read the Wikipedia page correctly this particle should be accelerated away from the center of the beam.

What would happen to the laser beam during this process?

There is a theorem like Poynting’s theorem, but for momentum. As the momentum of the particle changes, the momentum of the EM field changes in the equal and opposite direction.

vanhees71, berkeman and Philip Koeck

## What is ponderomotive force?

Ponderomotive force is a nonlinear force experienced by a charged particle in an oscillating electromagnetic field. It is typically observed in high-frequency fields, such as those produced by lasers, and acts to push the particle towards regions of lower field intensity.

## How is ponderomotive force related to conservation of momentum?

The ponderomotive force arises from the interaction between the charged particle and the oscillating electromagnetic field. According to the conservation of momentum, the momentum lost or gained by the particle must be balanced by a corresponding change in the momentum of the electromagnetic field. This ensures that the total momentum of the system is conserved.

## What is the mathematical expression for ponderomotive force?

The ponderomotive force $$F_p$$ on a charged particle with charge $$q$$ and mass $$m$$ in an electric field $$E$$ is given by $$F_p = -\frac{q^2}{4m\omega^2} \nabla E^2$$, where $$\omega$$ is the angular frequency of the oscillating field. This expression shows that the force is proportional to the gradient of the square of the electric field intensity.

## Can ponderomotive force be observed in both static and dynamic fields?

No, ponderomotive force is specifically a result of oscillating or dynamic fields. It does not occur in static fields because it relies on the time-averaged effect of the oscillating field on the charged particle.

## What are some practical applications of ponderomotive force?

Ponderomotive force has several practical applications, particularly in the field of laser physics. It is used in laser cooling and trapping of particles, plasma confinement, and in the acceleration of particles in laser-driven accelerators. It also plays a role in the manipulation of micro- and nanoparticles in optical tweezers.

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