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Lorentz Force Calculator

Use the Lorentz force calculator to compute the effect of the magnetic field on charged particles.

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Magnetic force80.1088 fNqvB sin 90° — perpendicular to both v and B, so it changes direction without changing speed
Radius of the circular path11.3713 µmmv⊥ ÷ qB
Cyclotron frequency13.9962 GHzqB ÷ 2πm — independent of the SPEED, which is the entire principle of the cyclotron
Period of one orbit71.4477 ps
Work done by the magnetic force0 J, alwaysthe force is perpendicular to the motion, so a magnetic field can never change a particle's kinetic energy — only its direction
Acceleration in the electric field1.759e+14 m/s²qE ÷ m — 1.793e+13 g
Electric force1.602e-16 Nparallel to the field, and it DOES do work
Energy gained across one metre1.602e-16 J1,000 eV — which is what an electronvolt means: the energy one elementary charge gains across one volt
Energy in electronvolts1,000 eVfor a charge of 1e across 1,000 V — the conversion is exactly the elementary charge in joules
Speed after accelerating through 1 kV from rest18.7554 Mm/s6.256% of the speed of light — above about 10% the non-relativistic formula starts to fail
ParticleElectronmass 9.109e-31 kg, charge -1e
Charge-to-mass ratio175.882 GC/kgthe only combination that appears in the cyclotron frequency, which is why it is measurable so precisely
Speed as a fraction of light0.3336%non-relativistic, so these formulas hold well
Kinetic energy4.555e-19 J2.8428 eV

The formula

F = qvB sin θ; r = mv ÷ qB; f_c = qB ÷ 2πm; a = qE ÷ m

A magnetic field does no work

The magnetic force is always perpendicular to the velocity, so it changes a particle's direction and never its speed. A charged particle in a uniform magnetic field therefore travels in a circle at constant speed, and its kinetic energy is exactly conserved. Electric fields are the opposite: they act along their own direction and do work freely.

The cyclotron frequency qB/2πm is independent of speed. That is a remarkable and useful fact — as a particle gains energy its circle grows, but it completes each orbit in the same time, so a fixed-frequency accelerating voltage stays in step with it. This is what makes a cyclotron possible, and it fails only when the particle becomes relativistic and its effective mass rises.

If the velocity is not perpendicular to the field, the parallel component is untouched and the path becomes a helix. This is why charged particles spiral along the Earth's field lines towards the poles, and why aurorae appear where they do.

Above about 10% of the speed of light the non-relativistic formulas here begin to fail, understating both the orbit radius and the energy. An electron reaches that after accelerating through only about 2.5 kV, so relativistic corrections matter in quite ordinary equipment.