The Lorentz force law describes the force experienced by a charged particle moving in electric and magnetic fields. It is fundamental to classical electromagnetism and essential for understanding the motion of charged particles in many physical and astrophysical contexts.
When particle speeds satisfy v≪c, the above vector form accurately describes the force. The particle’s momentum is given by p=mv, and Newton's second law
For particles with speeds comparable to the speed of light, v≲c, relativistic effects become significant. The particle’s momentum is
p=γmv,whereγ=1−v2/c21
is the Lorentz factor.
In this regime, the force and motion are described covariantly by the four-vector equation
dτdpμ=qFμνuν,
where
pμ=muμ is the four-momentum,
uμ is the four-velocity,
τ is the particle's proper time,
Fμν is the electromagnetic field tensor.
This fully relativistic formulation ensures consistency with special relativity and correctly handles time dilation and momentum-energy relations.
¶ Force on a Distribution of Particles (Non-Relativistic Case)
In realistic physical systems such as plasmas, we often deal with continuous distributions of many particles rather than tracking individual particle trajectories. To generalize the Lorentz force law to such systems, we derive the force per unit volume (force density).
Start with the Lorentz force law for a single particle of charge qi and velocity vi:
Fi=qi(E+vi×B).
Now consider a small volume element dV that contains many particles. The total force on all particles within dV is the sum over individual forces:
i∑Fi=i∑qi(E+vi×B).
Factor out the common fields (assumed uniform across dV):
i∑Fi=(i∑qi)E+(i∑qivi)×B.
Define the charge densityρ and current densityJ as
ρ=dV1i∑qi,J=dV1i∑qivi.
Then the force per unit volume, or force density, is:
f=ρE+J×B.
This is the Lorentz force density, and it governs the motion of continuous media such as fluids or plasmas.
¶ Force on a Distribution of Particles (Relativistic Case)
We start with the relativistic Lorentz force law for a single particle of charge q and four-velocity uν:
dτdpμ=qFμνuν,
where pμ=muμ is the four-momentum and τ is the proper time.
For a continuous distribution of particles (e.g. a plasma), we define the four-current density:
Jν(x)=i∑qi∫uiνδ(4)(x−xi(τ))dτ.
This expression sums the contribution of all particles, weighted by their charge and four-velocity, using Dirac delta functions to localize the current at each particle’s position in spacetime.