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Charged particle trajectories in QuickField

charged particle trajectories, calculation, trajectory, electrostatic

Charged particle trajectories picture

Version 5.0 introduces the possibility to study trajectories of charged particles movement in the plane-parallel and axisymmetric electrostatic fields. It utilizes our original approaches based on our implementation of Finite-Element technology1 and modern computational algorithms2.

Trajectory calculation uses the following data:

Viewing calculation results, you see:

Calculating trajectories QuickField uses following assumptions:

According to these assumptions, we can describe the trajectory (x(t), y(t), z(t)) of a charged particle in two-dimensional electrostatic field E(x, y) with Newton's system of differential equations:
Charged particle trajectories equation

We reorganize this system of three second degree equations into six first-degree equations and append the following additional equation:
length of the trajectory covered by the particle

Defining the length l(t) of the trajectory covered by the particle in time t. We integrate the resulting system using the Runge-Kutta-Merson method with automatically defined integration step. Numerical integration stops immediately before the finite element's boundary, the step leading outside of the element being excluded. At the last point in the element, we extrapolate the trajectory with cubical segment of its Taylor series relative to time and solve the resulting equation using Tartaglia-Cardano formula and taking into account possible decrease of the equation's degree in homogeneous or zero fields.

PCB design with QuickField