Lorentz Force Calculator (charge & wire)

Understanding the Lorentz force calculator — concept and principle

This calculator gives the magnetic force on a moving charged particle, F = qvB sin θ, and for a particle moving perpendicular to the field also its circular (cyclotron) radius and frequency. For a current-carrying wire, switch the mode to wire and enter the current in place of the charge and the wire length in place of the velocity to get F = BIL sin θ.

The force always points sideways, so the electron goes round in a circle without speeding up.

How it is calculated

Force: F = q · v · B · sin θ, where θ is the angle between velocity and field.
Cyclotron radius: r = m · v · sin θ / (|q| · B), using only the speed across the field.
Cyclotron frequency: f = |q| · B / (2π · m).

Units: charge in coulombs, speed in m/s, field in tesla, mass in kilograms. The default values describe an electron (charge magnitude 1.602 × 10⁻¹⁹ C, mass 9.109 × 10⁻³¹ kg).

The principle

A magnetic field pushes on a moving charge in a direction perpendicular to both its velocity and the field (F = qv × B). Only the velocity component across the field matters, hence the sin θ. Because the force is always at right angles to the motion, it changes the direction but never the speed, so no work is done. A particle crossing a uniform field therefore circles: setting the magnetic force qvB equal to the centripetal force mv²/r gives r = mv/(qB). The time for one loop, 2πr/v = 2πm/(qB), does not depend on speed, which is the principle behind the cyclotron.

Worked example

Inputs: q = 1.602 × 10⁻¹⁹ C, v = 10⁶ m/s, B = 0.5 T, θ = 90°, m = 9.109 × 10⁻³¹ kg.
Force: 1.602 × 10⁻¹⁹ × 10⁶ × 0.5 × sin 90° = 8.01 × 10⁻¹⁴ N.
Radius: 9.109 × 10⁻³¹ × 10⁶ ÷ (1.602 × 10⁻¹⁹ × 0.5) = 1.1372 × 10⁻⁵ m (about 11.4 µm).
Frequency: 1.602 × 10⁻¹⁹ × 0.5 ÷ (2π × 9.109 × 10⁻³¹) = 1.3995 × 10¹⁰ Hz (about 14.0 GHz).

How to read the result

A single electron feels a tiny force, yet its tiny mass bends it into a circle of only a few micrometres. The electron cyclotron frequency is about 28 GHz per tesla; a proton, about 1,836 times heavier, circles about 1,836 times more slowly.

Common mistakes

Forgetting the sign of the charge: an electron's force points opposite to a positive charge's.
Entering a wire current in charge mode; in wire mode the radius and frequency show — because they have no meaning.
Ignoring relativity at high speed: the classical radius and frequency are off by the Lorentz factor γ (about 15% at half the speed of light).

Keep in mind: The radius and frequency outputs assume motion perpendicular to a uniform magnetic field and ignore electric fields and relativistic effects. At other angles the particle follows a helix, and the radius shown is that of the helix, from v · sin θ.

Frequently asked questions

Why does the magnetic force do no work?

It is always perpendicular to the velocity, so it never adds or removes kinetic energy.

What is the full Lorentz force?

F = q(E + v × B), including the electric force qE. This calculator covers only the magnetic part.

Which way does the force point?

Use the right-hand rule for a positive charge: point the fingers along v, curl them toward B, and the thumb shows F.

Sources: CODATA recommended values of the fundamental physical constants, NIST (electron mass and elementary charge) · D. J. Griffiths, Introduction to Electrodynamics, Cambridge University Press, chapter 5

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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