Electrostatics and Charged Particle Tracing Equations
Under static conditions, Gauss’ law can be written as a variant of Poisson’s equation
(6-1)
where
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ε0 = 8.854187817 × 10-12 F/m is the permittivity of vacuum,
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V (SI unit: V) is the electric potential,
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P (SI unit: C/m2) is the polarization, and
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ρ (SI unit: C/m3) is the charge density.
The electric field E (SI unit: V/m) can be expressed in terms of the electric potential V (SI unit: V) using the relationship
The equation of motion for a nonrelativistic charged particle in an electromagnetic field can be written as
(6-2)
where
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mp (SI unit: kg) is the particle mass,
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v (SI unit: m/s) is the particle velocity,
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e = 1.602176634 × 10-19 C is the elementary charge,
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Z (dimensionless) is the particle charge number, and
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B (SI unit: T) is the magnetic flux density.
In addition to boundary conditions (such as terminals, grounds, and insulated surfaces), the electric potential can also be affected by the presence of charged particles in the simulation domain. The volumetric charge density of the particles can be added to the charge density ρ in Equation 6-1. To avoid double-counting the electrostatic repulsion between charged particles, the Coulomb force is omitted from Equation 6-2 and the Particle-Particle Interaction should not be added to the model.
Just as the charge density of the particles affects the electric potential, it is possible for the current density of moving particles to affect the magnetic field calculation, ultimately perturbing the value of B. However, The self-induced magnetic force on a charged particle beam is significantly weaker than the self-induced electric force if the particles are nonrelativistic. The bidirectional coupling to the magnetic field is included in The Particle–Field Interaction, Relativistic Interface