# The Cluster's Lorentz Force

A page about the Lorentz force — the force on a charged particle moving through electromagnetic fields.

## The Lorentz force law

The Lorentz force is the force on a particle of charge q moving with velocity v through electric and magnetic fields:
F = q (E + v x B)

In vector form, the magnetic force is perpendicular to both v and B, with magnitude F_B = q v B sin(theta), where theta is the angle between v and B. The electric force is parallel (or antiparallel) to E.

In the cluster, the edit Lorentz force law gives an edit force on an edit particle.

## The cyclotron motion

In a uniform magnetic field B (E = 0), a charged particle undergoes circular motion in the plane perpendicular to B:
- Cyclotron frequency: omega_c = q B / m
- Cyclotron radius (Larmor radius): r_L = m v_perp / (q B)
- Along B: uniform motion (constant v_parallel)
- Net trajectory: helix

For an electron in B = 1 Tesla: omega_c / (2pi) = 28 GHz, r_L = 3.4 mm for v_perp = 10^6 m/s.

In the cluster, the edit cyclotron motion gives an edit circular motion.

## The drift motions

If E is present (or B is non-uniform), the guiding center drifts:
- **E x B drift**: v_E = E x B / B^2 (independent of charge, mass, or velocity)
- **Gradient-B drift**: v_gB = (m v_perp^2 / (2 q B^3)) (B x nabla B)
- **Curvature drift**: v_c = (m v_parallel^2 / (q B^2 R_c)) (b x n_c) where R_c is the curvature radius

All drifts cause charge separation, which creates polarization currents.

In the cluster, the edit drift motions give an edit guiding center drift.

## Applications

- **Particle accelerators**: Cyclotrons, synchrotrons (magnetic steering, electric acceleration)
- **Plasma physics**: Magnetic confinement (tokamaks, stellarators), plasma waves
- **Aurora**: Charged particles from the Sun spiraling along Earth's magnetic field lines into the atmosphere
- **Mass spectrometry**: Deflecting ions by mass-to-charge ratio
- **Hall effect**: Transverse voltage from magnetic deflection of current carriers
- **MHD (magnetohydrodynamics)**: Force on conducting fluids, solar physics, fusion

In the cluster, edit applications include:
- edit Particle accelerators
- edit Plasma physics
- edit Aurora
- edit Mass spectrometry
- edit Hall effect
- edit MHD

## The relativistic form

In relativistic mechanics: dp/dt = q (E + v x B), where p = gamma m v is the relativistic momentum. The power transferred is dE/dt = q E . v (magnetic force does no work). In covariant form: dp^mu / d tau = q F^{mu nu} u_nu, where F^{mu nu} is the electromagnetic field tensor.

In the cluster, the edit relativistic form gives an edit power transfer.

## This force

This page is about the Lorentz force. F = q (E + v x B). omega_c = qB/m. r_L = m v_perp / (q B). The force is real.