Calculate the Lorentz force F = q(E + v×B) on a charged particle in electric and magnetic fields. Includes cyclotron radius, frequency, and force decomposition.
The Lorentz force is the fundamental electromagnetic force acting on a charged particle moving through electric and magnetic fields. Given by F = q(E + v×B), it combines the electric force (parallel to E, does work) and the magnetic force (perpendicular to v and B, does no work but curves the trajectory). This equation governs everything from cathode ray tubes and mass spectrometers to cyclotron particle accelerators and the aurora borealis.
This Lorentz Force Calculator accepts full 3D vector inputs for velocity, electric field, and magnetic field. It computes the total force vector, decomposes it into electric and magnetic components, and derives key results: cyclotron radius, cyclotron frequency, kinetic energy, and acceleration. Presets cover common scenarios from laboratory electron beams to cyclotron proton orbits.
Understanding the Lorentz force is essential for electromagnetism courses, plasma physics, accelerator design, and space physics. This calculator brings the vector cross product to life with visual force decomposition and a reference table of charged particle properties.
The Lorentz force involves a 3D cross product that is error-prone by hand. This calculator handles the vector math, decomposes forces into electric and magnetic parts, and computes derived quantities (cyclotron radius, frequency, energy) in one step — essential for quickly checking calculations in EM homework, lab work, or accelerator design.
Lorentz Force: F = q(E + v × B) Components: F_electric = qE (parallel to E) F_magnetic = q(v × B) (perpendicular to v and B) Cyclotron Radius: r = mv⊥ / (|q|B) Cyclotron Frequency: ω_c = |q|B / m f_c = ω_c / (2π) Where: q = charge (C) m = mass (kg) v = velocity (m/s) E = electric field (V/m) B = magnetic field (T)
Result: F ≈ 1.76 × 10⁻¹⁵ N, r_cyclotron ≈ 5.69 × 10⁻⁴ m
An electron moving at 10⁶ m/s in a 0.01 T magnetic field and 1000 V/m electric field experiences combined electric and magnetic forces. The cyclotron radius is about 0.57 mm — typical for laboratory-scale electron optics.
The Lorentz force underlies many technologies: CRT displays, mass spectrometers, cyclotron and synchrotron accelerators, magnetic confinement fusion (tokamaks), MRI machines, and Hall-effect sensors. In space physics, it governs charged particle motion in Earth's magnetosphere, creating radiation belts and auroral displays.
In uniform magnetic fields, charged particles follow helical paths (circular perpendicular to B, linear along B). Adding an electric field modifies the orbit: parallel E accelerates along B, perpendicular E causes drift. In non-uniform B fields, gradient and curvature drifts emerge — key to plasma confinement physics.
At relativistic speeds, the equation of motion becomes dp/dt = q(E + v×B) with p = γmv, where γ = 1/√(1−v²/c²). The cyclotron radius increases as r = γmv⊥/(|q|B), and the cyclotron frequency decreases as ω = |q|B/(γm). Synchrotrons compensate by ramping B as particles accelerate.
The magnetic force is always perpendicular to the velocity (F_B = qv×B). Since work = F·ds and ds is parallel to v, the dot product F_B·v = 0 always. The magnetic force changes direction but not speed.
The cyclotron radius (Larmor radius) is the radius of the circular orbit a charged particle follows in a uniform magnetic field. It equals mv⊥/(|q|B), where v⊥ is the velocity component perpendicular to B.
The cyclotron frequency ω_c = |q|B/m is the angular frequency of circular motion in a magnetic field. Remarkably, it is independent of velocity and radius — all particles with the same q/m ratio orbit at the same frequency (non-relativistically).
In many laboratory and astrophysical situations, the magnetic force dominates because particle speeds are high. The electric force matters when E fields are strong or particles are slow (v << E/B).
When E ⊥ B, charged particles drift at velocity v_d = E×B/B² perpendicular to both fields, regardless of charge sign or mass. This is the E×B drift, used in velocity selectors and Hall sensors.
No — it uses non-relativistic mechanics. For particles approaching the speed of light (v > 0.1c), relativistic mass increase changes the cyclotron radius and frequency.