Charged Particle Subjected to Lorentz Force
When subjected to electric and magnetic fields, the Lorentz force determines Newtonian particle motion. This Demonstration describes the effect of a homogeneous magnetic field in the direction combined with a homogeneous electric field in an arbitrary direction on the trajectory of a charged particle, given its charge, mass, initial position and initial velocity.
Consider a particle of charge coulombs and mass kilograms subjected to an electric field
in newtons per coulomb and a magnetic field
The resulting force is given by the Lorentz force:
In Cartesian coordinates, the position vector is
then the velocity is
and the acceleration is
In this case, Newton's second law,
can be written as
Substituting the vectors gives
Doing the cross products and rearranging terms gives
These are coupled second-order ordinary differential equations that can be solved by either analytical or numerical methods. Numerically, as done in this Demonstration, the solution needs initial conditions for the velocity and the position, given by
 J. D. Jackson, Classical Electrodynamics, 3rd ed., New York: Wiley, 1999.