The Lorentz force is the force that acts on a charged particle due to its presence in electric and magnetic fields. Named after the physicist Hendrik Lorentz, the force F on a particle with charge q travelling with a velocity v is:
where E is the electric field and B is the magnetic field. The first term represents the force due to an electric field, the second due to a magnetic field. The × symbol represents the cross product, which means that the force due to a magnetic field is perpendicular to both the velocity of the particle and the magnetic field. This has the important consequence that a magnetic force cannot do any work on a particle. The magnitude of the magnetic force is FM = Bv sin θ where θ is the angle between the magnetic field and velocity vectors. Its direction can be found using the right hand rule. If the velocity and magnetic field are parallel, then there is no magnetic force produced. This is in contrast with electric fields where the force is always in the same direction of the field and is only zero if there is no electric field.
As a particle in a magnetic field experiences a force perpendicular to its motion, it will undergo circular motion. In general, charged particles follow helical paths around magnetic field lines, with a radius of r = mv/qB for a particle with mass m. This is used in particle accelerators such as cyclotrons. A magnetic field is used to keep charged particles such as electrons or protons following a circular path while an electric field is used to accelerate them at specific points as they go round.