PhD and MPhil Thesis Classes - Université Libre de Bruxelles

3.3 Charged Particle Motionthe charged particles.In order to get insight into the problems of charged particle motion, we brieflyconsider the simplest cases.3.3.2.1 B = 0First, we assume that there is no magnetic field and that the electric field is uniformand directed along x axis. The equation of motion (3.18) is reduced to˙v x = q m E x (3.19)In this trivially simple case, the solution represents a uniformly accelerated motionof the particle, parallel or antiparallel to E x field according to the sign of q. Thesolution isx(t) = x 0 + v x0 t + 1 q2 m E xt 2 (3.20)where x 0 and v x0 are the initial position and velocity. The trajectory of the particle is,in general, a parabola.3.3.2.2 E = 0In this case, a charged particle has a simple cyclotron gyration. Following eq. (3.18),the equation of motion is˙v = q m v × B (3.21)where for simplicity we used the notation dx/dt = v. Taking ẑ to be the direction ofB (B = Bẑ), we have˙v x = q m Bv y ˙v y = q m Bv x ˙v z = 0 (3.22)¨v x = qB m ˙v y = −( qB m )2 v x¨v y = qB m ˙v x = −( qB m )2 v y (3.23)31