Electrostatics and Charged Particle Tracing Equations
Under static conditions, Gauss’ law can be written as a variant of Poisson’s equation
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: m/s) can be expressed in terms of the electric potential V (SI unit: V) using the relationship
The equation for the magnetic vector potential A (SI unit: Wb/m) may be written as
where σ (SI unit: S/m) is the electric conductivity, H (SI unit: A/m) is the magnetic field, B (SI unit: T) is the magnetic flux density, and Je (SI unit: A/m2) is the external current density. The constitutive relationship between B and H can be written
where μ0 (SI unit: H/m) is the permeability of free space and M (SI unit: A/m) is the magnetization.
The equation of motion for a charged particle in an electromagnetic field can be written as
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.
For relativistic particles, the particle mass is expressed in terms of the rest mass mr (SI unit: kg):
where c = 2.99792458 × 108 m/s is the speed of light in a vacuum. For nonrelativistic particles, the particle mass can be replaced by the rest mass and the self-imposed magnetic force of the beam is often negligibly small compared to the electric force, allowing The Particle–Field Interaction, Nonrelativistic Interface to be used effectively.
The calculation of the space charge density term and the resulting effect on the electric force is explained in Space Charge Density Calculation in Theory for the Particle–Field Interaction, Nonrelativistic Interface. The contribution of particle motion to the magnetic force is considered in the following section.