The Lorentz Force

Equation

\[\boxed{\,m\frac{d\mathbf{v}}{dt} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)\,}\]

Physical meaning

The complete law of motion for a charged particle in electromagnetic fields — one half of all plasma physics (Maxwell's equations are the other). The electric term accelerates along \(\mathbf{E}\); the magnetic term bends the velocity without ever changing the speed.

Variables

Symbol	Meaning	SI unit
\(m\)	particle mass	kg
\(q\)	charge (sign matters!)	C
\(\mathbf{v}\)	velocity	m/s
\(\mathbf{E}\)	electric field	V/m
\(\mathbf{B}\)	magnetic flux density	T

Key structural facts

Magnetic force does no work: \(\mathbf{v}\cdot(\mathbf{v}\times\mathbf{B}) = 0\) — kinetic energy responds only to \(\mathbf{E}\). (Any energy change in a magnetic-only simulation is numerical error — the standard integrator diagnostic.)
The cross product couples the velocity components — the source of gyration (cyclotron motion) and, in inhomogeneous fields, of every drift.
As a state-space system \(\mathbf{z} = (\mathbf{x}, \mathbf{v})\), it's the standard ODE benchmark of Chapter 2, with a closed-form leapfrog update.

Applications

Single-particle orbits, drift theory, PIC pushers, mass spectrometers, cyclotrons, aurora physics.

Limitations

Non-relativistic as written (use \(d(\gamma m\mathbf{v})/dt\) for fast particles); point-particle (no radiation reaction); fields are external inputs — self-consistency requires coupling to Maxwell (PIC).

Cyclotron motion — its uniform-field solution
E×B drift, grad-B/curvature drifts — averaged consequences
Leapfrog update — its discrete twin

Quiz

Q1 (computational). An electron moves at \(10^6\) m/s perpendicular to \(B = 0.01\) T. Force magnitude?

Answer

\(F = qvB = 1.6\times10^{-19}\times10^6\times0.01 = 1.6\times10^{-15}\) N — yielding an enormous acceleration \(\sim 1.8\times10^{15}\ \text{m/s}^2\) for so light a particle.

Q2 (conceptual). Why can a magnetic field confine a plasma but not heat it?