Introduction to the Charged Particle Tracing Interface Theory
Motion of charged particles in electromagnetic field is best understood by starting with the Lagrangian for a charge in an electromagnetic field:
where
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 mp (SI unit: kg) is the particle mass,
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 c = 2.99792458 × 108 m/s is the speed of light in a vacuum,
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 v (SI unit: m/s) is the particle velocity,
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 Z (dimensionless) is the particle charge,
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 e = 1.602176634 × 10−19 C is the elementary charge,
•
 A (SI unit: Wb/m) is the magnetic vector potential, and
•
 V (SI unit: V) is the electric scalar potential.
For low velocities, after subtracting the rest energy, the Lagrangian becomes:
The equations of motion are given by the Lagrange equation:
where q (SI unit: m) is the particle position vector. The right-hand side of the Lagrange equation is:
Using some elementary vector calculus, this can be rewritten as:
So, the Lagrange equation becomes:
(4-1)
The second term on the left-hand side of Equation 4-1 represents the total differential which can be expressed as:
(4-2)
Inserting Equation 4-2 into Equation 4-1 results in:
(4-3)
Defining the electric field as:
and the magnetic flux density as:
the equation of motion for a charged particle in an electromagnetic field becomes
(4-4)
The term on the right-hand side of Equation 4-4 is called the Lorentz force. So far, only the Electric Force and Magnetic Force have been considered. Additional forces, such those introduced by the Friction Force and Ionization Loss features, can be added to the right-hand side if necessary.