Lorentz Force Calculator

Enter a charge, its velocity, a magnetic field, the angle between them, and an optional electric field to get the magnetic force (F = qvB sinθ), electric force (F = qE), combined force, and the velocity-selector balance speed (v = E/B).

Quick Facts

Lorentz force law
F = q(E + v × B)
The total electromagnetic force on charge q moving with velocity v through electric field E and magnetic field B.
Magnetic force
F_B = qvB sinθ
Zero when v is parallel to B (θ = 0°); maximum when v is perpendicular to B (θ = 90°).
Electric force
F_E = qE
Acts along the field direction and does not depend on the particle's speed.
Velocity selector
v = E / B
The speed at which the electric and magnetic forces exactly cancel, used in mass spectrometers.

Your Results

Calculated
Magnetic Force
-
F = qvB sinθ, in newtons (N)
Electric Force
-
F = qE, in newtons (N)
Combined Force
-
√(F_B² + F_E²), assumes the two forces act perpendicular to each other
Velocity-Selector Balance Speed
-
v = E / B, the speed at which forces cancel exactly

Ready

Enter charge, velocity, magnetic field, angle, and (optionally) an electric field, then press Calculate.

Formula and Method for the Lorentz Force

The Lorentz force is the complete electromagnetic force on a charged particle: F = q(E + v × B), where q is the particle's charge, E is the electric field, v is its velocity, and B is the magnetic field. It has two distinct parts: an electric force qE that pushes the charge along the field direction regardless of motion, and a magnetic force q(v × B) that acts only while the charge is moving and always points perpendicular to both v and B. This calculator computes the magnitude of each part separately, combines them, and reports the classic velocity-selector balance speed.

How the calculation works

The magnetic force magnitude follows directly from the cross product: F_B = qvB sin(θ), where θ is the angle between the velocity and magnetic field vectors. This term is zero when the charge moves parallel to B (θ = 0° or 180°) and largest when it moves perpendicular to B (θ = 90°) — the force never does work on the charge because it is always perpendicular to v, so it changes direction but not speed. The electric force is simpler: F_E = qE, independent of velocity or angle. Because F_B is always perpendicular to v (and, for the common crossed-field case, also perpendicular to F_E), the calculator combines the two magnitudes with the Pythagorean relationship F = √(F_B² + F_E²) to give a single combined-force figure. It also reports v = E/B, the exact speed at which the electric and magnetic forces cancel — the operating principle of a velocity selector.

Common mistakes and practical notes

  • Forgetting the angle: the magnetic force depends on sin(θ), not just the field strength — a charge moving parallel to B feels no magnetic force at all, even in a very strong field.
  • Mixing up force and direction: this calculator gives magnitudes. The direction of the magnetic force is found separately with the right-hand rule: point your fingers along v, curl them toward B, and your thumb points along F for a positive charge (reverse it for a negative charge).
  • Charge sign matters for direction, not magnitude: flipping the sign of q reverses both force directions but leaves F_B and F_E unchanged in size.
  • Unit consistency: keep charge in coulombs, velocity in m/s, magnetic field in tesla, and electric field in V/m (or use the built-in unit selectors) — mixing unit systems is the most common source of wrong answers.

Frequently Asked Questions

What is the Lorentz force?
The Lorentz force is the total electromagnetic force on a charged particle: F = q(E + v × B), where q is the charge, E is the electric field, v is the particle's velocity, and B is the magnetic field. It combines the electric force (qE) with the magnetic force (qv × B) into one vector equation.
How do you calculate the magnetic part of the Lorentz force?
The magnetic force magnitude is F = qvB sin(θ), where θ is the angle between the velocity vector and the magnetic field vector. The force is zero when velocity is parallel to B (θ = 0°) and maximum when velocity is perpendicular to B (θ = 90°). Its direction is perpendicular to both v and B, given by the right-hand rule for a positive charge.
What is a velocity selector and how does it relate to the Lorentz force?
A velocity selector uses perpendicular electric and magnetic fields arranged so the electric force (qE) opposes the magnetic force (qvB). Only particles moving at v = E/B pass through undeflected because the two Lorentz-force components exactly cancel; all other speeds are deflected. This principle is used in mass spectrometers and cathode-ray tubes.
Does the sign of the charge change the direction of the force?
Yes. Reversing the sign of q reverses the direction of both the electric force (qE) and the magnetic force (qv × B) while leaving their magnitudes unchanged. A positive charge and a negative charge moving the same way through the same fields feel equal but opposite forces.

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