Consider 2D Coulomb gas described by potential V(r) given by e^2 log(r/L), where L is arbitrary length. What is dimension of magnetic force F =(e/m) v x H, where v is velocity in 2D plane and v is perpendicular to the plane.
It is classical system and coupling parameter is defined by gamma = e^2/KB T , where KB is Boltzmann constant and T is temperature. This tells us that e^2 has dimension of energy in 2D i.e. (e V(r) )
what is dimension of 'B' and charge 'e' ??
The right-hand side expression e/m \vec{v} x \vec{B} is a formula for acceleration caused by the Lorentz force acting on the particle which possesses mass m [kg] and charge e [A.s], while moving with velocity v [m/s] perpedicularly to a magnetic field with flux density B [ T ], where the unit Tesla equals T = kg A−1 s−2 . For more systematic list one can refere to any table of physical units, e.g.
https://en.wikipedia.org/wiki/List_of_physical_quantities
and for the Lorentz force a good source of info is given at
https://en.wikipedia.org/wiki/Lorentz_force
Dr Domsta has given a good summary of basic force formulae. But I am wondering just what you are trying to do here considering a magnetic force (which is 3D in nature) interacting with a 2D gas. This cannot happen in any bulk physical system, and if you are considering the magnetic interaction with particles constrained to 2D sheets then I think you will need the services of quantum mechanics, not just Lorentz equations.
Yes, the restriction to 2D problem seems to be not sufficiently explained. But there is a possibility that the author of the question meant a 3D, say ioniside gas in a tube with length very large if compared to the diameter, put, additionaly, between two poles of a strong magnet longwise the magnetic field. Then the problem is well approxiamted by the one described in the question if the particles (ions, charged drops etc.) move perpendicularly to the tube (e.g. rotating arround the axis).
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