Charged particle trajectory in the uniform static electric field. Case: plane-parallel.

QuickField simulation example

Problem Type
Plane-parallel problem of electrostatics.

Geometry

Distance between the plates d = 1 m.

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 v_x = v_x = 500 000 m/s; v_z = 0 m/s.
Emitter position (0; 0; 0) m.

Task
Calculate the particle trajectory neglecting relativistic effects.

Solution
The analytical solution gives parabolic trajectory.
x(t) = v_x*t [m],
y(t) = 0.5 * F_y / m * t² [m],
z(t) = 0 [m],
where F_y = q*E_y - is the y-component of the Lorentz force,
            t - is time.

Particle trajectory may be calculated using the built-in function in QuickField Electrostatic postprocessor or by the free tool TrajectoryTracer.

Results
Electric field strength E_x = 0 V/m, E_y = -20 V/m.
Lorentz force F_y = -1.602e-19 * -20 = 3.204e-18 N.
x(t) = 5e5*t m,
y(t) = 0.5*3.204e-18/9.109e-31 * t² m,
z(t) = 0 m.

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