As an initial case, consider a domain in space for which В = 0 with only a constant electric field affecting the particle trajectory. Orienting the Cartesian coordinate system so that the z-axis points in the direction of E, Fig. 5.1, and imposing an arbitrary initial particle velocity at t = 0, referenced to the origin of the coordinate system, we simply write for Eq.(5.1)

d

m—=q £,k,

where к is the unit vector in the z-direction. This vector representation for the motion of a single charged particle in an electric field can be decomposed into its constituent components

d

dt dt

and these can be solved by inspection:

m

Integrating again with respect to time gives the position of the charged particle at any time t:

The corresponding trajectory for a positive ion is suggested in Fig. 5.2 with the particle located at the origin of the coordinate system at t = 0. This description therefore constitutes a parametric representation of a curve in Cartesian (x, y,z) space and corresponds to the trajectory of the charged particle under the action of an electric field only and for the initial condition specified.

The important feature to note is that the components of motion for this individual charged particle perpendicular to the E-field, that is vx and vy, do not change with time; however, the velocity component in the direction of the E — field, vz(t), is seen from Eq.(5.4) to linearly vary with time. The particle is accelerated in the direction of E for a positive charge and in the opposite

Individual Charge Trajectories direction for a negative charge.