About Lorentz Force
The Lorentz force is the combined force on a charged particle due to electric and magnetic fields, expressed as F = q(E + v × B), where q is the particle's charge, E is the electric field vector, v is the particle's velocity, and B is the magnetic flux density. The magnetic component (qv × B) is always perpendicular to the velocity, doing no work on the particle but continuously changing its direction — this causes circular motion in a uniform magnetic field, with orbital radius r = mv/(|q|B) (the cyclotron radius or Larmor radius). When both E and B fields are present, the particle can drift perpendicular to both fields at the E × B drift velocity v_drift = E × B / B², a result that is field-strength independent and underlies plasma confinement physics, the Hall effect, and crossed-field electron tubes such as magnetrons.
This simulator integrates the relativistic Lorentz force equations numerically using a Runge-Kutta solver, letting you independently vary the electric field strength and direction, magnetic flux density, particle charge, mass, and initial velocity. Observable trajectories include pure circular orbits (E = 0), helical motion (v has a component parallel to B), cycloid paths (E perpendicular to B), and E × B drift of gyrating particles — all rendered as colour-coded 3D trails.
Frequently Asked Questions
Why does the magnetic force never do work on a moving charge?
Work is defined as the dot product F · ds = F · v dt. The magnetic component of the Lorentz force is F_B = qv × B, which is always perpendicular to v by the definition of the cross product. Therefore F_B · v = 0 for all v, meaning the magnetic force does zero work at every instant. It can only change the direction of motion, not the speed. This is why a charged particle moving in a pure magnetic field maintains constant kinetic energy — its speed never changes, only its direction curves.
What is the cyclotron radius (Larmor radius) and how does it depend on particle properties?
When a charged particle moves perpendicular to a uniform magnetic field B, it follows a circle of radius r_L = mv⊥/(|q|B), where m is the particle's mass, v⊥ is the speed perpendicular to B, and |q| is the charge magnitude. Heavier particles (larger m) and faster particles (larger v⊥) spiral more widely; stronger fields or higher charges produce tighter circles. For a proton in Earth's equatorial magnetic field (B ≈ 3×10⁻⁵ T) moving at 10⁷ m/s, r_L ≈ 3.5 km — this governs the radius of trapped radiation-belt particles around Earth.
What is E × B drift and why is it independent of charge and mass?
When a charged particle gyrates in crossed E and B fields (E ⊥ B), each gyration is slightly stretched on one side (where E accelerates the particle, increasing the gyroradius) and compressed on the other (where E decelerates it). The net effect is a steady sideways drift at velocity v_drift = (E × B) / B², perpendicular to both E and B. Crucially, both the larger gyroradius half-circle and the smaller gyroradius half-circle scale identically with m and q, making v_drift independent of particle mass and charge. This means positive and negative particles drift in the same direction — unlike the curvature drift — and the effect produces a net plasma fluid velocity used in fusion reactor confinement models.
What is a cyclotron and how does it use the Lorentz force?
A cyclotron is a particle accelerator that uses a static magnetic field to curve a proton or ion beam into a spiral path, and alternating electric fields applied at the "D" electrodes (dees) to accelerate the particle each time it crosses the gap. Because the cyclotron period T = 2πm/(|q|B) is independent of particle speed (for non-relativistic particles), the same RF frequency resonates with the particle throughout its acceleration. Early cyclotrons (Lawrence, 1930) reached a few MeV; modern sector-focusing cyclotrons reach 500–600 MeV for proton therapy in cancer treatment, delivering millimetre-precise dose to tumours.
What is helical motion and when does it occur?
If a charged particle's initial velocity has a component v‖ parallel to B in addition to v⊥ perpendicular to B, the magnetic force acts only on v⊥, producing circular motion in the plane perpendicular to B. Meanwhile v‖ is unaffected and carries the particle along B. The combined motion is a helix: radius r_L = mv⊥/(|q|B), pitch = v‖T = 2πmv‖/(|q|B). This is exactly the mechanism by which energetic electrons from the solar wind spiral along Earth's magnetic field lines into the polar regions, producing the aurora borealis and aurora australis.
How does the Hall effect relate to the Lorentz force?
When a current-carrying conductor is placed in a magnetic field perpendicular to the current, the Lorentz force deflects the charge carriers (electrons or holes) to one side of the conductor, building up a transverse electric field (the Hall field) that balances further deflection. In steady state, E_Hall = v_drift × B. Measuring the Hall voltage across the conductor gives the sign of the charge carriers (positive holes vs negative electrons) and their density n: V_H = IB/(nqt) where t is the conductor thickness. Modern Hall sensors (in brushless motors, ABS wheel-speed sensors, joysticks) exploit this effect for contactless position and current sensing.
What is magnetic mirror confinement and how does it relate to Lorentz force?
A magnetic mirror is a region of increasing field strength at the ends of a magnetic bottle. As a helically gyrating particle moves toward the stronger field region, its perpendicular velocity v⊥ increases (magnetic moment conservation: μ = mv⊥²/2B = const) while parallel velocity v‖ decreases to conserve energy. If the particle reaches the mirror point before escaping, v‖ → 0 and the particle is reflected. This principle was used in early fusion experiments (e.g., LLNL's Mirror Fusion Test Facility). The Earth's Van Allen belts are natural magnetic mirrors: particles bounce between north and south magnetic poles, trapped for years.
Can the Lorentz force be relativistic?
Yes — at speeds approaching c, the relativistic form of Newton's second law is d(γmv)/dt = q(E + v × B), where γ = 1/√(1 − v²/c²) is the Lorentz factor. The relativistic cyclotron radius becomes r_L = γmv⊥/(|q|B), so faster particles spiral more widely. Importantly, T = 2πγm/(|q|B) now depends on γ (hence on speed), so the simple cyclotron resonance breaks. This "relativistic detuning" is why synchrotrons (used in CERN's LHC) vary the magnetic field B as the beam accelerates — maintaining resonance requires increasing B in proportion to γm.
What is gradient drift in a non-uniform magnetic field?
In a magnetic field with a spatial gradient ∇B (varying in strength but not direction), a gyrating particle spends more time on the weaker-field side (larger gyroradius) than the stronger-field side (smaller gyroradius). This asymmetry produces a net drift v_∇B = (mv⊥²/2|q|) · (B × ∇B) / B³. Unlike E × B drift, gradient drift is charge-sign dependent: positive and negative particles drift in opposite directions, producing a net current in the plasma. In Earth's magnetosphere, gradient drift causes the ring current — a belt of westward-flowing current at 3–5 Earth radii responsible for the magnetic storm disturbances measured on the surface.
How does the magnetron in a microwave oven use the Lorentz force?
A magnetron is a crossed-field vacuum tube: electrons emitted from a central cathode are simultaneously attracted radially toward an outer anode (electric field E, radially outward) and deflected azimuthally by an axial magnetic field B. The Lorentz force (E × B drift) causes electrons to orbit the cathode. Resonant cavities machined into the anode are tuned to 2.45 GHz; electrons interact with these cavities to transfer kinetic energy to the microwave field, amplifying it. The magnetron converts ~65% of DC input power to microwave power — far more efficient than older thermionic amplifiers — making affordable microwave ovens possible since the 1960s.
What is the adiabatic invariant of cyclotron motion?
The magnetic moment of a gyrating particle μ = mv⊥²/(2B) is an adiabatic invariant: it is conserved when the magnetic field changes slowly compared with the cyclotron period. This invariant governs particle trapping in magnetic mirrors and in planetary radiation belts: as B increases, v⊥ must increase proportionally (since μ = const and E = ½m(v‖² + v⊥²) = const gives a constraint between v‖ and v⊥). Adiabatic invariants are also used in the design of magnetic traps for neutral atoms and in the analysis of charged-particle motion in tokamak magnetic fields.