Lorentz Force Calculator

About the Lorentz Force Calculator

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.

When This Page Helps

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.

How to Use the Inputs

1. Enter the particle charge and mass (use presets for common particles).
2. Input the velocity vector components (v_x, v_y, v_z) in m/s.
3. Input the electric field (E_x, E_y, E_z) in V/m.
4. Input the magnetic field (B_x, B_y, B_z) in Tesla.
5. Review the total force, electric/magnetic decomposition, and cyclotron parameters.
6. Use the force decomposition chart to visualize the relative contributions.

Example Calculation

Tips & Best Practices

• The magnetic force is always perpendicular to velocity — it curves the path but cannot speed up or slow down the particle.
• The cyclotron frequency is mass-dependent: heavier particles orbit more slowly, which is the basis of mass spectrometry.
• For E×B drift (crossed fields), the drift velocity is independent of charge and mass.
• Check that v << c for non-relativistic validity. At v > 0.1c, use relativistic corrections.
• In practical accelerators, radio-frequency electric fields accelerate particles while magnetic fields curve them into circular paths.

Applications of the Lorentz Force

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.

Particle Motion in Combined Fields

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.

Relativistic Corrections

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.

Frequently Asked Questions

• Why does the magnetic force do no work?

  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.

• What is the cyclotron radius?

  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.

• What is cyclotron frequency?

  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).

• When can I ignore the electric field term?

  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).

• What happens in crossed E and B fields?

  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.

• Is this calculator relativistic?

  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.