Many experiments in physics call for a beam of charged particles. The stability and “optics” of charged-particle beams are influenced by the electric and magnetic forces that the individual charged particles in the beam exert on one another. Consider a beam of positively charged ions with kinetic energy E0. At t = 0 the beam has a radius R0, a uniform charge density ρ0, and is traveling in the x-direction.
(a) Derive an expression for the radial force on a charged particle in the beam at some initial radial position r0 (note that there will be electric and magnetic forces acting on the particle).
(b) Apply F = ma to determine the radial velocity of the charged particle when its radial position has increased from r0 to 2r0. You’ll need to make one assumption here – the charge density of the particle beam remains uniform (but not constant in time!) as the beam expands.
(1) Fmag = Q(v x B)
(2) Fnet = Q[E + (v x B)]
The Attempt at a Solution
I am not sure how to treat this situation. Do the charges in the center of the beam act like a current-carrying wire of circular cross section? In this case the volume current density would be J = I/πR02 at t=0 and the magnetic field generated by the beam at the position of the particle would be B = μ0I/2πr0. Would the electric field be that of a line charge, E = kρ0/r0? Can I plug these fields into (2) to get the radial force?
Are any of these trains of thought going in the right direction? Thank you.