Charged particle trajectory in the uniform static electric field. Case: cylindrical. - QuickField simulation example
Engineering question
How to calculate a charged-particle trajectory in a cylindrical electric field?
Answer
Solve the axisymmetric electrostatic field and trace the particle motion from the specified initial position and velocity, neglecting relativistic effects.
Typical applications
cylindrical electrostatic lenses, charged particle beamlines, electrostatic analyzer sections
Download
- Download simulation files (files may be viewed using any QuickField Edition).
Simulation problem
Problem TypeAxisymmetric problem of electrostatics.
Geometry
Given
Relative permittivity of vacuum ε = 1;
Positive potential U+ = 20 V.
Charge (electron) q = -1.602e-19 C
Mass (electron) m = 9.109e-31 kg
Initial velocity vy = 500 000 m/s; vz = vφ = 0 m/s.
Emitter position (0; 0; 0).
Task
Calculate the charged particle trajectory neglecting relativistic effects.
Solution
The analytical solution is a parabola
x(t) = 0,
y(t) = vy * t,
z(t) = 0.5 * Fz / m * t²,
where Fz - is the z-component of the Lorentz force,
t - is time.
Particle trajectory may be calculated using the built-in function in QuickField Electrostatic postprocessor.
Results
Electric field strength Ez = -20 V/m, Er = 0 V/m
Lorentz force Fz = q * Ez = -1.602e-19 * -20 = 3.204e-18 N.
x(t) = 0 m,
y(t) = 500000*t m,
z(t) = 0.5*3.204e-18/9.109e-31 *t² m.
Particle coordinates (x; y; z) m | ||
| time | Theory | QuickField |
| 0 s | (0; 0; 0) | (0; 0; 0) |
| 1e-7 s | (0; 0.050; 0.018) | (0; 0.050; 0.018) |
| 2e-7 s | (0; 0.100; 0.070) | (0; 0.100; 0.070) |
| 3e-7 s | (0; 0.150; 0.158) | (0; 0.150; 0.158) |
| 4e-7 s | (0; 0.200; 0.281) | (0; 0.200; 0.281) |
| 5e-7 s | (0; 0.250; 0.440) | (0; 0.250; 0.440) |
Video
- Charged particle trajectory in the uniform static electric field. Case: cylindrical.. Watch on YouTube
. - View simulation report in PDF