The Lorentz Force Law:
F = q(E + v × B)
gives the force F on a particle of charge q in the presence of electric and magnetic fields. In SI units, F is in newtons, q in coulombs, E in volts per meter, B in teslas, and v, which is the velocity of the particle relative to the magnetic field, in meters per second.
Thus, in a pure electric-field system, the force is determined simply by the charge on the particle and the electric field; F = qE
The force acts in the direction of the electric field and is independent of any particle motion.
In pure magnetic-field systems, the situation is somewhat more complex. Here the force
F = q(v × B)
is determined by the magnitude of the charge on the particle and the magnitude of the B field as well as the velocity of the particle. In fact, the direction of the force is always perpendicular to the direction of both the particle motion and that of the magnetic field.
Mathematically, this is indicated by the vector cross product v × B. The magnitude of this cross product is equal to the product of the magnitudes of v and B and the sine of the angle between them; its direction can be found from the fight-hand rule, which states that when the thumb of the fight hand points in the direction of v and the index finger points in the direction of B, the force, which is perpendicular to the directions of both B and v, points in the direction normal to the palm of the hand.
For situations where large numbers of charged particles are in motion, it is convenient
to rewrite the equation in terms of the charge density p (measured in units of coulombs per cubic meter) as
Fv = p(E + v × B) (3.4)
where the subscript v indicates that Fv is a force density (force per unit volume) which in SI units is measured in newtons per cubic meter.
The product p v is known as the current density; J = pv , which has the units of amperes per square meter. The magnetic-system force density corresponding correspondingly can then be written as
Fv = J x B.
For currents flowing in conducting media, that equation can be used to find the force density acting on the material itself. Note that a considerable amount of physics is hidden in this seemingly simple statement, since the mechanism by which the force is transferred from the moving charges to the conducting medium is a complex one.