Last update: July 22, 2023

The Lorentz Force: Pillar of Electromagnetism

The Lorentz Force

At the crossroads of electricity and magnetism lies one of the most elegant and powerful forces in physics: the Lorentz force. Named after Dutch physicist Hendrik Lorentz (1853-1928) who formalized it in the late 19th century, this force governs the behavior of charged particles in electromagnetic fields. As the founding principle of all conversion between electricity and motion, it underpins most of our technological infrastructure: from energy production to its use, from medical devices to communication tools. In other words, it is the omnipresent pillar of our technological civilization.

N.B.:
In environments where collisions are rare or absent (such as in particle accelerators or plasma), electrons can reach relativistic speeds. There, mastering the electric and magnetic fields via the Lorentz force is essential to accelerate, focus, and control particles.

The Essence of the Lorentz Force

Whether the particle is an electron, a proton, or an ion, its interaction with electromagnetic fields is entirely described by the Lorentz force: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$
On the left side of the equation is the mechanical force exerted on the particle; on the right is specified the mechanism behind this force: the electric field acts through direct attraction or repulsion, while the magnetic field imposes a transverse deflection as soon as the particle has velocity.

F = Lorentz force (in newtons, N)
q = electric charge of the particle (in coulombs, C)
E = electric field (in volts per meter, V/m)
v = velocity vector of the particle (in meters per second, m/s)
B = magnetic induction field (in teslas, T)

The Electric Component: Acceleration and Braking

The electric part of the Lorentz force is the most intuitive. An electron placed between the plates of a capacitor, for example, will be attracted to the positive plate and repelled by the negative plate. This force is independent of the particle's speed: whether it is stationary or moving, the action of the electric field remains the same.

The Magnetic Component: Perpendicular Deflection

The magnetic part of the Lorentz force has a fascinating particularity: it always acts perpendicular to the direction of motion. This geometric property has profound consequences. Unlike the electric force, which accelerates or decelerates a particle in its direction of propagation, the magnetic force bends the trajectory without changing the speed in magnitude.

An electron entering a uniform magnetic field at a right angle will follow a circular path, kept in orbit by the magnetic force acting as a centripetal force. If the entry is at any angle, the trajectory takes the form of a spiral: the electron circles while gradually advancing in the direction of the magnetic field.

The Northern Lights: A Lorentz Spectacle

Nature offers its own spectacular demonstration of the Lorentz force in the polar auroras. Charged particles emitted by the Sun during solar flares, mainly electrons and protons, are captured by the Earth's magnetic field. Guided by the field lines toward the magnetic poles, these particles follow spiral trajectories along these lines.

When they penetrate the upper atmosphere, they collide with oxygen and nitrogen atoms, exciting them. The return of these atoms to their ground state is accompanied by the emission of photons, creating the green, red, or purple luminous veils that dance in the night sky. This celestial ballet is none other than the visible signature of billions of particles subjected to the Lorentz force.

The Lorentz Force: A Universal Force

The Lorentz force thus stands as a universal pillar of physics. It connects our electric motors to the northern lights, our electronic circuits to the mysteries of the quantum world. More than an elegant equation, it represents the key that transformed our understanding of the universe and made possible the technological civilization in which we live.