# Power required to hold mutually-repelling particles in a region

Consider some arbitrary type of particle which repels other particles of the same kind (such as an electron). If we uniformly disperse a large number of these particles in a region, they will obviously attempt to spread out infinitely if left undisturbed. How would one find how much power would be needed to apply enough force to redirect the velocity of any escaping particles such that they stay within the region?

Assuming all particles are identical apart from kinetic energy, we can write the force any one particle exerts on any other particle as $$F=\frac{C}{r^{2}}$$ for some arbitrary C with units of $$\text{kg}\cdot \text{m}^{3}\cdot \text{s}^{-2}$$.

However, past this point I am not sure how to proceed. I figure that the power required will be equal to the flux of the kinetic energy of the particles over the surface area of the region, but I do not know how to get there. Ideally, I would like to know how to solve this for any region, but if this is prohibitively difficult, a solution for an arbitrary-sized spherical region and some understanding of how increasing the surface area would affect the answer is good enough.

There is no power required to confine particles to a region. Just put them in a suitably shaped container that makes it impossible for them to escape and they will stay in this region. The walls of the container just have to be able to withstand the pressure force exerted by the particles without any substantial deformation.

• Where does the energy come from to change/stop the momentum of the particle? The kinetic energy must go somewhere, right? Jul 11, 2022 at 20:14
• You don't change the energy of the particles - they undergo elastic collisions with the walls. They change their momentum $\vec p \to -\vec p$ which however does not initiate any measurable move of the container since it has walls from all sides of your charged gas and the average to approximately zero during larger timescales. The kinetic energy $p^2/(2m)$ of the particles remain intact during the elastic colision. Jul 11, 2022 at 20:51

This is more of a question about, "do I need to supply energy to create a force".

[In the steady state]

The kinetic energy isn't kinetic energy. If you confine particles to a region, the field isn't doing any work on it. Zero field energy is being converted into kinetic energy, so you aren't changing the momentum of anything, nothing is moving.

No work is being done on anything so zero energy is being transfered

The existence of a force does not mean energy is being used.

This oftcourse differs from our daily experience where we physically have to use energy to create forces, but in this case, I don't think we are actually creating forces with energy, I don't fully understand this logic applied to the body, however I would assume that our body creates forces, and does WORK on our muscles to push things.

• Re, "daily experience where we physically have to use energy to create force..." With our muscles, yes. But, if I put an object in a clamp—that's pretty close to a daily experience for me—the clamp will continue indefinitely to exert force on the object after I let go of it, with no source of power required. Vertebrate skeletal muscles are different: When a muscle exerts "continuous" force, what's really happening is that thousands of individual muscle fibers are continually "firing" and then relaxing, over and over again, at random intervals. That cycle consumes energy. Jul 11, 2022 at 21:07

I will answer this:

Power required to hold mutually-repelling particles in a region

with an example for electrons.

A beam of electrons in a storage ring is guided with magnetic fields and can be kept in a steady state circle by feeding energy to keep the beam going. To see the effect of the repulsive force between them, go to the center of mass of a volume, dV of the beam . There the electrons are at rest and the repulsive force takes over. To keep them at rest in the center of mass, so that the beam in the ring does not disperse will take corrective action that will take extra power from the ring system, it is part of the complexity of holding the electrons in a ring.

In this chapter we give an introduction to the transverse dynamics of the particles in a synchrotron or storage ring. The emphasis is more on qualitative understanding rather than on mathematical correctness, and a number of simulations are used to demonstrate the physical behavior of the particles. S