Averaged Motion of Charged Particles in a Curved Strip

This paper is concerned with the motion of electrically charged particles in \curved" inn-nite strip. The equations of motion are ' = with initial and boundary conditions, where ' is the electric potential, is the particle trajectory, P is the charge distribution, and J is the Jacobian. The lower boundary ? 0 of the strip is grounded. Therefore, once a particle reaches ? 0 ; it loses its charge and becomes immobile. We prove existence and uniqueness of solutions for small time. For nearly axially symmetric initial data, we also show that the solution can be extended until the time when all the particles have migrated to ? 0. A numerical simulation based on our model is implemented. The results indicate that particles tend to accumulate less around the convex parts of ? 0 and more around the concave parts. 1. The Model. We consider a simpliied model of motion of electrically charged spherical particles, with uniform charge and mass, in a \curved" innnite strip in R n : = fx = (x 0 ; x n) 2 R n : g (x 0) < x n < 1g: Let P (x; t) denote the mass (or charge) distribution of the particles at a point x in and time t 0; and ' (x; t) the electric potential. By Maxwell's equation, ' = ?P in : (1.1) A voltage diierence M is maintained between the upper boundary ? 1 = fx n = 1g and the lower boundary