The force behind motors, auroras and particle accelerators
A charged particle moving through a magnetic field feels a sideways push. This push turns electric motors, steers electron beams in old television tubes, focuses beams in particle accelerators and guides solar wind particles toward the poles, where they light up the auroras.
The formulas
The magnetic part of the Lorentz force is:
F = |q| × v × B × sin θ
- q is the particle’s charge (1.602 × 10⁻¹⁹ coulomb for a proton or an electron, twice that for an alpha particle).
- v is its speed, B the magnetic field in teslas, and θ the angle between the velocity and the field.
Because the force is always perpendicular to the velocity, the particle moves in a circle of radius:
r = m × v sin θ ÷ (|q| × B)
and goes around at the cyclotron frequency f = |q| × B ÷ (2π × m). Any velocity along the field is unaffected, so the path becomes a helix whose pitch is v cos θ ÷ f.
Example
A proton moving at 1,000 km/s at right angles to a 1-tesla field:
- Force: 1.602 × 10⁻¹⁹ × 10⁶ × 1 = 1.6 × 10⁻¹³ N
- Radius: 1.673 × 10⁻²⁷ × 10⁶ ÷ (1.602 × 10⁻¹⁹ × 1) ≈ 1.04 cm
- Cyclotron frequency: about 15.2 MHz, one turn every 66 nanoseconds
An electron at the same speed would circle 1,836 times faster on a radius 1,836 times smaller, because its mass is that much lower.
Things to keep in mind
- Speed does not change. The magnetic force does no work. Only an electric field can speed a particle up or slow it down.
- Direction depends on the sign of the charge. Protons and electrons curve in opposite directions, which is how detectors tell them apart. The calculator gives magnitudes only.
- Relativity. Above about 10% of the speed of light, the particle’s momentum grows faster than its speed, so the real radius is larger than this calculator shows. Large accelerators use synchrotrons that raise the field as the particles speed up.
- Mass spectrometers use exactly this formula: ions of different mass-to-charge ratio follow circles of different radius and land at different places on a detector.