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

,

in teslas.

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

and

,

respectively.

Reference

[1] J. D. Jackson, Classical Electrodynamics, 3rd ed., New York: Wiley, 1999.