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.