Lorentz Force & Particle Motion Calculator

Calculate trajectory radius, cyclotron frequency, and drift velocity for charged particles in fields.

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Electric and Magnetic Fields Affect Charged Particles Differently

The Lorentz force F=q(E+v×B) combines an electric force that can change speed with a magnetic force that is perpendicular to velocity and therefore does no work by itself. In a uniform magnetic field with velocity perpendicular to B, the magnetic force supplies centripetal force and the particle follows a circle of radius r=mv/(|q|B). The cyclotron angular frequency is ωc=|q|B/m in the nonrelativistic limit.

If velocity has a component parallel to B, that component remains unchanged while the perpendicular component circles, producing a helix. Crossed electric and magnetic fields can produce drifts or velocity selection depending on geometry. At relativistic speeds, momentum rather than mv must be used.

F=q(E+v×B),   r=mv⊥/(|q|B),   ωc=|q|B/m
SymbolMeaningWhy it appears / units
qParticle chargeC; sign determines force direction.
vParticle velocitym/s vector.
BMagnetic fieldT; magnetic force depends on perpendicular velocity.
EElectric fieldN/C or V/m; can do work on the charge.

Positive and negative charges curve in opposite directions in the same magnetic field. Magnetic fields redirect motion without changing kinetic energy when acting alone, while electric fields can speed up or slow down particles.

For perpendicular motion, the orbit radius should grow with momentum. r=mv/(|q|B), so increasing mass or speed increases radius, while increasing charge magnitude or magnetic field decreases it. Reversing the charge reverses the curvature direction but not the radius magnitude.

Worked Examples

Example 1: Proton 1000eV in B=0.5T
r=mp×v/(q×B)
Result: r=0.91 cm (cyclotron radius)
Used in medical proton therapy cyclotrons
Example 2: Cyclotron frequency, proton, B=1T
f_c=1.602e-19×1/(2π×1.673e-27)
Result: f_c=15.24 MHz
Medical cyclotrons operate at this RF frequency
Example 3: Proton radius
v=1×106m/s, B=0.5T, v⊥B
Result: r≈2.09cm
Radius grows with momentum and shrinks with charge magnitude or field strength.
Example 4: Velocity selector
E=2×104V/m, B=0.1T
Result: v=E/B=2×105m/s
Opposing electric and magnetic forces cancel at one selected speed.

Common Mistakes

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Using total speed instead of perpendicular speed for magnetic radius

Only the component perpendicular to B contributes to circular bending.

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Forgetting charge sign when finding direction

Magnitude uses |q|, but the vector force reverses for negative charge.

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Saying magnetic force increases particle kinetic energy

v×B is perpendicular to v, so qv×B does zero instantaneous work.

Frequently Asked Questions

Why is cyclotron frequency mass-dependent but radius speed-dependent?
f_c=qB/2πm: depends only on charge-to-mass ratio — not speed. This is why cyclotrons work: all protons at any speed take the same time per revolution. r=mv/qB: faster particles curve less (larger radius).
Mass spectrometry connection?
Different ions have different m/q ratios → different cyclotron radii in same B field. Mass spectrometers separate ions by their trajectories. Used in chemical analysis, carbon dating, protein identification (proteomics).
Why does a magnetic field not change particle speed?
The magnetic force is perpendicular to velocity, so F·v=0 and no work is done. It changes direction of momentum rather than kinetic-energy magnitude.
What makes a helical path?
A velocity component perpendicular to B produces circular motion while the parallel component remains uniform, combining into a helix.
What is cyclotron frequency?
It is the rotation frequency of a nonrelativistic charged particle in a uniform magnetic field. Its angular form is ωc=|q|B/m and is independent of speed in the ideal nonrelativistic model.
When does relativity matter?
As particle speed becomes a significant fraction of c, momentum is γmv and the simple radius and cyclotron-frequency expressions need relativistic correction.
Why does a magnetic field not change a charged particle’s speed by itself?
The magnetic force qv×B is perpendicular to the instantaneous velocity, so its power F·v is zero. It changes the direction of motion rather than kinetic energy. A perpendicular uniform magnetic field therefore produces circular motion with constant speed in the ideal nonrelativistic model.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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