Lorentz Force

The Lorentz force is the force exerted on a particle carrying electric charge by an electric field and a magnetic field. It combines an electric contribution, which depends on charge and electric-field strength, with a magnetic contribution that also depends on the particle’s velocity. Together with Maxwell’s equations, the force law provides a central description of classical electromagnetism: the equations determine the fields, while the force law determines how charged matter responds to them. (feynmanlectures.caltech.edu)

Mathematical formulation

In the International System of Units (SI), the law is

where $q$ is the particle’s signed charge, $\mathbf v$ its velocity, and $\mathbf E$ and $\mathbf B$ the fields evaluated at its instantaneous position and time. All quantities refer to the same reference frame. The multiplication symbol $\times$ denotes the vector cross product, rather than ordinary scalar multiplication. Force is measured in newtons, electric field in volts per metre, and magnetic field in teslas. (feynmanlectures.caltech.edu)

The electric contribution is $\mathbf F_E=q\mathbf E$. It points along the electric field for a positive charge and against it for a negative charge. The magnetic contribution is $\mathbf F_B=q\,\mathbf v\times\mathbf B$, with magnitude

where $\theta$ is the angle between velocity and magnetic field. It vanishes for a stationary charge or for motion parallel or antiparallel to the field. Its direction follows the right-hand rule for positive charge and reverses for negative charge; thus an electron and a proton moving with the same velocity are deflected in opposite directions. (feynmanlectures.caltech.edu)

The equation is a local force law, not a statement that fields must be uniform or time-independent. A particle’s trajectory is obtained by combining it with an equation of motion and the specified spatial and temporal field distributions. (damtp.cam.ac.uk)

Work and energy

A distinctive property of the magnetic contribution is that it is perpendicular to the particle’s velocity. Consequently, it does no instantaneous mechanical work on a point charge:

The rate of change of the particle’s kinetic energy is therefore

An electric field can change the particle’s speed and energy, whereas a magnetic field alone changes its direction without changing its kinetic energy in this idealized description. (openstax.org)

This distinction explains why magnetic bending and electric acceleration perform different functions in charged-particle instruments. Magnetic fields steer beams; electric fields supply or remove their kinetic energy. (pressbooks.online.ucf.edu)

Motion in uniform fields

In nonrelativistic mechanics, a particle of mass $m$ obeys

With no electric field and a uniform magnetic field, motion perpendicular to the field is circular: the magnetic force provides the centripetal force. The orbit radius and angular frequency are

where $v_\perp$ is the velocity component perpendicular to the field. The quantity $\omega_c$ is called the cyclotron angular frequency. These expressions assume that other forces and radiation losses are negligible. (openstax.org)

A nonzero velocity component parallel to the field remains unchanged, producing a helical trajectory. Purely parallel motion is straight, while purely perpendicular motion is circular. The sense of rotation depends on the sign of the charge. (openstax.org)

Crossed electric and magnetic fields can also produce zero net force. For a beam directed perpendicular to both fields, with the two force contributions opposed, undeflected motion requires $v=E/B$. A velocity selector uses this condition to transmit particles of a selected speed, independent of their mass and, for nonzero charge, charge magnitude. (pressbooks.online.ucf.edu)

Currents and distributed charge

The force law extends from individual particles to continuous charge distributions. For charge density $\rho$ and current density $\mathbf J$, its force-per-unit-volume form is

The total force is obtained by integrating this density over the relevant volume. This formulation connects individual charged-particle motion with electromagnetic forces on extended matter. (scipp.ucsc.edu)

For a thin current-carrying wire, the magnetic force on a short element is

where $d\boldsymbol{\ell}$ points along conventional current. A straight wire segment in a uniform field experiences $\mathbf F=I\boldsymbol{\ell}\times\mathbf B$. These expressions describe forces on moving charge carriers in an electrical conductor. A closed current loop in a uniform field has zero net magnetic force but can experience a torque—the operating principle underlying many electric motors. (openstax.org)

Relativistic description

The Lorentz force retains its three-vector form at relativistic speeds, but the equation of motion must use relativistic momentum:

with $\gamma=(1-v^2/c^2)^{-1/2}$ and $c$ the speed of light. Replacing $d\mathbf p/dt$ by $m\,d\mathbf v/dt$ is generally inappropriate at these speeds. (feynmanlectures.caltech.edu)

Electric and magnetic fields mix under a Lorentz transformation. A force attributed partly to magnetism in one inertial frame can be described differently in another. Their unified relativistic expression is

where $\tau$ is proper time, $u_\nu$ is four-velocity, and $F^{\mu\nu}$ is the electromagnetic field tensor, with compatible sign conventions. (damtp.cam.ac.uk)

Historical background and applications

The law is named after Hendrik Antoon Lorentz, whose electron theory developed James Clerk Maxwell’s electromagnetic theory into an account of electrical and optical phenomena in matter. Lorentz’s work connected microscopic charged particles with observable electromagnetic behavior. (nobelprize.org)

Applications include magnetic-sector mass spectrometry, which distinguishes ions through their deflection, and particle accelerators, which combine electric acceleration with magnetic steering. In a cyclotron, repeated electric acceleration occurs as a magnetic field bends particles around successive orbits; the approximately energy-independent nonrelativistic cyclotron frequency enables synchronization with an alternating accelerating field. (pressbooks.online.ucf.edu)