The Lorentz Force

Charged Particle

     The force 'F' acting on a charged particle 'q' with velocity 'v' with external electric field 'E' and magnetic field 'B' is shown by the equation:

Note that the bold quantities are vectors.

     A positively charged particle will accelerate in the same direction as the electric field, but will move orthogonally to both the velocity vector and the the magnetic field (according to the right hand rule: RHR). The term qE is the electric force on the particle while qv x B is the magnetic force on the particle.

Charge Distribution

     For a charge distribution in motion, the force equation becomes:

where dF is the force on a section of the charge distribution of charge dq. If both sides are divided by the volume of a section:

where f is the force density and greek rho is the charge density. Including the current density, we get:

Thus, the total force is the volume integral over the charge distribution:

Force on a Current Carrying Wire

     When a current-carrying wire is placed within a magnetic field, each of the moving charges experiences the Lorentz Force, which in turn can create another force (known as the Laplace Force). By incorporating the Lorentz Force Law to a straight, non-moving wire we get the equation:

where l is the vector for the length and direction of the wire. If the wire is not perfectly straight, the force can be computed using each infinitesimal section of the wire 'dl', and then adding the forces using integration, yielding: