Welcome to the Forum Jager.
The force you want is called the
Lorentz force. I can't easily write equations using the posting software, so go to the Wikipedia article on electrodynamics, at
http://en.wikipedia.org/wiki/Electrodynamics and look at the first equation given.
To calculate the force you will need to know the velocity vectors of the charged particles, the electric field vector, and the magnetic field vector at every point in your plasma device. I would guess you can probably approximate the electric field to be zero. To calculate the magnetic field generated by your magnet and the currents flowing in the plasma will be very difficult. For the B field of the magnet you may be able to simplify the Maxwell equations to a form using the
magnetic scalar potential. If all you are interested in is a theoretical model of the magnet then you can probably find this scalar potential by assuming a very simple, symmetric current distribution in the electromagnets and then using the methods of partial differential calculus to obtain a solution by exploiting the symmetries. On the other hand, for a realistic electromagnet, numerical finite-element analysis using a computer will be needed. For the B field of the plasma you will need to solve simultaneously for the dynamics of the particles using either newtonian or relativistic mechanics (I don’t know which) and the E and B fields using Maxwell’s equations. This type of problem can be tackled in a somewhat simpler way using the
magnetohydrodynamic approximation (see
http://en.wikipedia.org/wiki/Magnetohydrodynamics ). However, even this simpler formulation is
extremely difficult to use, so difficult in fact that the only way to determine the behavior of real plasma reactors is to build them! Even the most sophisticated computer solutions will not be able to predict the complete qualitative behavior of systems such as fusion reactors or MHD electricity generators.
If you want to actually build a device for “compressing a stream of plasma with magnets”, you will face many practical difficulties beyond the problem of generating a theoretical model of the system as described above. You will need a very “hard” vacuum in your reaction vessel; this is far from easy to achieve even in an essentially passive system like a
cryostat for maintaining an experiment at very low temperatures, but a plasma system inherently will “poison” its own vaccum unless you can efficiently pump out plasma ions that escape the experimental beam. You will need a powerful electromagnet. These are hard to build. If they operate at room temperature they will use a lot of electricity, and generate a lot of heat that must be removed. If they are supercooled a large cryostat will be needed. They are potentially
hazardous because of 1) the strong forces on the windings and support structures, 2) the large amount of energy stored in the magnetic field, and 3) the risk of ferromagnetic objects such as tools being violently pulled onto the magnets. You will need a bunch of expensive test equipment.
Etc, etc, etc…
In short: I’d guess building a plasma beam system would be too big a project for a science fair
. Doing a theoretical model would be
very difficult, but not impossible. A mentor familiar with plasma physics would be extremely helpful if you decide to go this route — learning magnetohydrodynamics on your own might prove to be too much.