Electromagnetism

Lorentz Force

Total electromagnetic force on a charge: electric pulls along E, magnetic perpendicular to v and B

The Lorentz force is the total electromagnetic force on a point charge: F = q(E + v × B). The electric component qE acts along the field; the magnetic component qv×B acts perpendicular to both velocity and B. Magnetic force does no work (always perpendicular to v) — only the electric component changes kinetic energy. Named after Hendrik Lorentz (1895), but the magnetic part predates by Ampère and Faraday. Charges in pure B follow circular orbits with cyclotron frequency ω_c = qB/m and radius r = mv/(qB). Crossed E and B fields give helical drift v_drift = E×B/B² (independent of charge sign). Applications: cyclotrons, mass spectrometers (separate ions by m/q), velocity selectors (E×B), CRT and electron microscopes, plasma confinement, the Hall effect, and the auroras (charged particles spiraling along Earth's field lines).

  • Total forceF = q(E + v × B)
  • Magnetic ⊥ vDoes no work, conserves |v|
  • Cyclotron freqω_c = qB/m
  • Cyclotron radiusr = mv/(qB)
  • E×B driftv = E × B / B² (charge-independent)
  • AuthorLorentz 1895

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Why Lorentz force matters

  • Cyclotrons and synchrotrons. The cyclotron frequency ω_c = qB/m is independent of energy — Lawrence's 1932 cyclotron exploited this to accelerate protons by a small RF voltage applied at every half-orbit. Modern synchrotrons compensate for relativistic mass increase by ramping B in step with energy.
  • Mass spectrometry. Routine identification of isotopes, peptides, drugs, and pollutants relies on bending ion paths in a uniform B and reading r = mv/(qB). High-resolution instruments resolve m/q to 1 part in 10⁶.
  • Plasma physics and fusion. Tokamaks confine 100-million-K plasma with toroidal B; ions and electrons spiral along field lines instead of streaming to the walls. Magnetic mirrors, stellarators, and Z-pinches all rely on Lorentz force confinement geometry.
  • Auroras and the magnetosphere. The Lorentz force funnels solar-wind charged particles down Earth's magnetic field lines toward the poles, where they collide with the upper atmosphere and excite the auroral colors. Same force shields the planet's surface from cosmic rays.
  • Particle physics. Bubble chambers, drift chambers, and tracking detectors at LHC and DESY rely on F = qv × B to curve charged-particle trajectories — the radius reveals momentum, the curvature direction reveals charge sign.
  • CRT displays and electron microscopes. Electron beams are steered by Lorentz force from deflection coils. Even though CRTs have largely retired, every transmission electron microscope, scanning electron microscope, and electron-beam lithography tool still uses qv × B for column optics.
  • Hall effect sensors. Drive a current through a thin slab in a perpendicular B; the Lorentz force pushes carriers sideways until an electric field develops to balance it. Measuring this Hall voltage gives B with a sub-dollar IC; modern automotive sensing relies on it.

Common misconceptions

  • "Magnetic force can accelerate." Magnetic force changes direction but not speed — kinetic energy is invariant under pure qv × B because v·F = 0. To increase a charged particle's energy you need an electric field somewhere along its path (cyclotron RF gap, accelerator cavity, or motional EMF).
  • "Trajectories in B are always circular." Pure circular motion requires uniform B perpendicular to v. With v components both parallel and perpendicular to B, the trajectory becomes a helix; with non-uniform B, it becomes a drifting helix; with E and B both present, an even more general drift curve.
  • "v × B doesn't work in relativity." The non-relativistic Lorentz force is the low-velocity limit of a covariant tensor expression F^μ = qF^μν u_ν, where F^μν is the electromagnetic field tensor and u_ν is the four-velocity. In a different frame, what looks like pure B becomes a mix of E and B, but the proper four-force on the particle is the same.
  • "Magnetic force on a current-carrying wire is something different." F = IL × B for a wire is the integrated qv × B over all the carriers — same physics, just bookkeeping for a continuous flow rather than individual particles.
  • "E and B are alternatives." Almost no real situation uses pure E or pure B alone. Antennas radiate both; capacitors with current have both; even a moving magnet creates an apparent E in the lab frame. F = q(E + v × B) is the actual operative law.
  • "Cyclotron frequency depends on speed." Non-relativistically, ω_c = qB/m has no speed in it — exactly what makes the cyclotron design work. Relativistically, the effective mass is γm, so ω_c = qB/(γm) drops with energy; synchrotrons compensate by raising B as γ rises.

Canonical Lorentz-force motion

  • Pure B, v ⊥ B. Circular orbit, radius r = mv/(qB), period T = 2πm/(qB).
  • Pure B, v with parallel and perpendicular components. Helix: circular gyration around field line plus uniform drift along it.
  • Crossed uniform E and B (E ⊥ B). Cycloidal motion in the lab frame, equivalent to circular gyration plus drift v_drift = E × B / B² in a co-moving frame.
  • Velocity selector. Particles with v = E/B pass straight through unchanged; mismatched speeds are deflected.
  • Magnetic mirror. Region of converging B; the conserved quantity μ = mv⊥²/(2B) (magnetic moment) reflects particles whose v_∥ is small enough.
  • Cyclotron. Particle alternates between two semicircular dees; an RF field across the gap accelerates it each half-orbit. Maximum energy limited by relativistic mass rise breaking ω_c synchronization.

Frequently asked questions

Why does magnetic force do no work?

Power delivered by any force is P = F·v — the dot product of force and velocity. The magnetic component of the Lorentz force is F_mag = qv × B, which by definition of the cross product is always perpendicular to v. The dot product of perpendicular vectors is zero, so F_mag·v = 0 and the magnetic force delivers no power. Kinetic energy ½mv² therefore stays constant — speed |v| is conserved, only direction changes. Only the electric force qE can change a charged particle's kinetic energy. This explains why cyclotron orbits are circles at constant speed and why a particle in a uniform B field never speeds up or slows down on its own.

What's the cyclotron frequency and radius?

A charge q with mass m moving with speed v perpendicular to a uniform magnetic field B traces a circle. Setting magnetic force equal to centripetal force: qvB = mv²/r gives the cyclotron radius (gyroradius) r = mv/(qB). The orbital period is T = 2πm/(qB), so the cyclotron frequency is ω_c = qB/m (or f_c = qB/(2πm)). Notably, ω_c is independent of speed and radius — slower particles trace smaller circles in the same time. This independence is what made the cyclotron particle accelerator possible: drive an oscillating voltage at the fixed cyclotron frequency, and particles accelerate every half-orbit regardless of how fast they're already going (until relativistic mass increase breaks the synchronization).

How does a mass spectrometer use Lorentz force?

A magnetic-sector mass spectrometer ionizes the sample, accelerates ions through a known potential V (giving each ion the same kinetic energy qV), and sends them through a uniform B field. Each ion follows a circular arc with radius r = mv/(qB). Using ½mv² = qV, the radius becomes r = √(2mV/q)/B — proportional to √(m/q), the square root of mass-to-charge ratio. Ions of different m/q land at different positions on the detector plane, separating the sample into a mass spectrum. Modern instruments resolve mass differences as small as 1 part in 10⁶, identifying isotopes, peptides, and trace contaminants. Time-of-flight, quadrupole, and Orbitrap variants use related E and B field combinations.

What is E×B drift?

Place a charged particle in crossed uniform E and B fields. The particle still gyrates around B, but its guiding center drifts with constant velocity v_drift = E × B / B². Remarkable property: this drift velocity is independent of the particle's charge, mass, and speed — positive and negative charges drift the same direction at the same rate. The reason: E accelerates positive charges along itself and negatives the opposite way, but the resulting v × B forces curl them onto a common drift trajectory. E×B drift is fundamental to plasma physics: it explains how plasma flows in tokamaks, the dynamics of the solar wind near planets, and the operation of magnetrons in microwave ovens.

How do auroras form via Lorentz force?

Solar wind plasma — mostly protons and electrons — streams from the Sun and gets captured by Earth's magnetic field. The qv × B force makes captured particles spiral along magnetic field lines like beads on a wire. Earth's dipole field funnels these spirals down to the magnetic poles, where the field lines plunge into the upper atmosphere. There, the spiraling charged particles collide with O₂ and N₂ molecules at altitudes 100–400 km, exciting electronic transitions that release the auroral colors: green from atomic oxygen at 558 nm, red from oxygen at 630 nm at higher altitudes, blue and purple from nitrogen. Without the Lorentz force funneling charges along field lines, solar particles would arrive uniformly over the planet's surface and we'd see no aurora — or, equivalently, no protective magnetosphere shielding the surface.

How does a velocity selector work?

A velocity selector sets up uniform E and B fields perpendicular to each other and perpendicular to the beam direction. A charged particle entering this region experiences electric force qE in one direction and magnetic force qv × B in the opposite. The two forces cancel exactly when v = E/B, regardless of charge or mass. Particles at this matched velocity pass straight through; faster particles are deflected one way, slower the other. By tuning E and B, you select a single velocity from a thermal beam. Velocity selectors precede the magnetic sector in mass spectrometers (so r = mv/(qB) tracks m alone, not m/v); they also clean up beams in fundamental physics experiments where energy uniformity matters.