Lorentz force
The Lorentz force is the electromagnetic force acting on a charged particle moving through electric and magnetic fields, combining the effects of both fields into a single vector expression \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}), where q is the particle's charge, \mathbf{E} is the electric field, \mathbf{v} is the particle's velocity, and \mathbf{B} is the magnetic field.[1] The electric component q\mathbf{E} acts parallel to the electric field and can accelerate or decelerate the particle along its direction, while the magnetic component q\mathbf{v} \times \mathbf{B} is always perpendicular to both \mathbf{v} and \mathbf{B}, resulting in a deflection without changing the particle's kinetic energy since it performs no work.[2] This force was derived in its modern form by Dutch physicist Hendrik Antoon Lorentz in 1895 as part of his theory of electrons and electromagnetic phenomena in moving bodies.[3] The Lorentz force law serves as a fundamental principle in classical electrodynamics, linking Maxwell's equations to the dynamics of individual charged particles and enabling predictions of their trajectories in electromagnetic environments.[4] It underpins key phenomena such as cyclotron motion, where charged particles spiral in uniform magnetic fields, and Hall effects in conductors.[2] Historically, experimental verification came through J.J. Thomson's 1897 cathode ray studies, which measured the electron's charge-to-mass ratio using the force.[2] In applications, the Lorentz force drives technologies including particle accelerators for high-energy physics research, electric motors and generators in power systems, magnetohydrodynamic propulsion in plasmas, and magnetic resonance imaging in medicine by influencing charged ion flows.[5] Its relativistic generalization remains essential in special relativity and quantum electrodynamics, ensuring consistency across scales from subatomic particles to astrophysical plasmas.[4]Classical Definition and Properties
Point Particle in Electromagnetic Fields
The Lorentz force represents the total electromagnetic force acting on a point charge q moving with velocity \mathbf{v} in the presence of an electric field \mathbf{E} and a magnetic field \mathbf{B}, given by the vector equation \mathbf{F} = q \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right). This expression, named after Hendrik Lorentz who formulated it in 1895, combines the contributions from both fields to describe the motion of charged particles in electromagnetic environments.[6] The electric component of the force, \mathbf{F}_E = q \mathbf{E}, arises directly from Coulomb's law generalized to continuous field distributions, as established by Maxwell's equations; it acts equally on stationary or moving charges and points in the direction of \mathbf{E} for positive q.[2] The magnetic component, \mathbf{F}_B = q \mathbf{v} \times \mathbf{B}, emerges from the interaction between moving charges, derived by considering the force on a charge in a current-carrying wire under a magnetic field: the force per unit length on the wire is I \times \mathbf{B}, where I = n q A v (with n the charge density, A the cross-sectional area), leading to the per-charge form q \mathbf{v} \times \mathbf{B} for consistency with experimental observations of deflections in magnetic fields.[2] Adding these yields the full Lorentz force, which governs particle dynamics in combined fields. Key physical properties distinguish the components: the electric force is independent of velocity and can accelerate or decelerate the particle along its direction, potentially doing work; in contrast, the magnetic force is always perpendicular to both \mathbf{v} and \mathbf{B}, resulting in no work done on the particle (\mathbf{F}_B \cdot \mathbf{v} = 0) and thus no change in kinetic energy from the magnetic field alone.[2] In a uniform magnetic field with no electric field, this perpendicularity causes charged particles to follow circular paths, modified to helical trajectories if the initial velocity has a component parallel to \mathbf{B}; the superposition of fields allows complex paths, such as cycloidal motion when \mathbf{E} and \mathbf{B} are perpendicular. The law's fields are defined in a given inertial frame, with the overall expression invariant under Lorentz transformations between frames, ensuring consistency in special relativity. A representative example is the cyclotron motion of a charged particle in a uniform magnetic field \mathbf{B} perpendicular to the initial velocity plane, where the centripetal force balance \frac{m v^2}{r} = q v B yields the orbital radius r = \frac{m v}{q B} and angular frequency \omega = \frac{q B}{m}, independent of speed and enabling particle acceleration in devices like cyclotrons.Continuous Charge and Current Distributions
The Lorentz force law, originally formulated for point charges, extends naturally to continuous distributions of charge and current by considering the force per unit volume, known as the force density. For a charge density \rho and current density \vec{J} in electromagnetic fields \vec{E} and \vec{B}, the force density is given by \vec{f} = \rho \vec{E} + \vec{J} \times \vec{B}. This expression arises from summing the contributions over infinitesimal charge elements dq = \rho \, dV with associated velocities, where the convective current \vec{J} = \rho \vec{v} replaces the discrete velocity \vec{v}.[7] The total force \vec{F} on a finite volume V containing such distributions is then the volume integral \vec{F} = \int_V (\rho \vec{E} + \vec{J} \times \vec{B}) \, dV. This integral form accounts for the net electromagnetic force on macroscopic matter, such as plasmas or conductors, and facilitates analysis of momentum transfer from the fields to the material.[7] The force density \vec{f} connects directly to Maxwell's equations through the electromagnetic energy-momentum conservation laws, particularly via the Poynting theorem extended to momentum. The theorem implies that the rate of change of mechanical momentum density equals the negative of the electromagnetic force density plus the divergence of the momentum flux, with the electromagnetic momentum density \vec{g} = \epsilon_0 \vec{E} \times \vec{B} (in SI units). This \vec{g} represents the field's intrinsic momentum, analogous to the energy density in the original Poynting formulation for energy flow. The full momentum balance is \frac{\partial \vec{g}}{\partial t} + \nabla \cdot \overleftrightarrow{T} = -\vec{f}, where \overleftrightarrow{T} is the Maxwell stress tensor, providing a tensorial description of momentum transfer across surfaces enclosing the volume. This framework resolves paradoxes in field-matter interactions by attributing momentum to the fields themselves.[8][9] In magnetized materials, the Lorentz force can be applied by modeling the magnetization \vec{M} as equivalent bound currents, which contribute to the total \vec{J}. The volume bound current density is \vec{J}_m = \nabla \times \vec{M}, supplemented by surface currents \vec{K}_m = \vec{M} \times \hat{n}, allowing the force on the material to be computed as \int_V (\rho \vec{E} + (\vec{J}_f + \vec{J}_m) \times \vec{B}) \, dV, where \vec{J}_f denotes free currents. This approach treats permanent magnets or ferromagnetic materials as distributions of amperian current loops, enabling calculation of forces like those in magnetic levitation devices. A representative example is the force between two coaxial circular current loops, which illustrates the integral application. The magnetic field \vec{B} due to the first loop carrying current I_1 is computed via the Biot-Savart law at points along the second loop with current I_2; the force on the second loop is then \vec{F} = I_2 \int d\vec{l}_2 \times \vec{B}_1, where the integral is over the loop contour. This calculation generally requires evaluation using complete elliptic integrals and demonstrates attractive or repulsive behavior depending on current directions. This underpins applications in electromagnetic actuators.[10]Formulations in Different Unit Systems
The Lorentz force law in the International System of Units (SI) is expressed as\mathbf{F} = q \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right),
where q is the charge, \mathbf{E} is the electric field in volts per meter (V/m), \mathbf{v} is the velocity of the particle, and \mathbf{B} is the magnetic field in teslas (T).[11] This formulation incorporates the permeability of free space \mu_0 and permittivity of free space \epsilon_0 implicitly through the definitions of the fields in Maxwell's equations, such as Ampère's law in the form \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}.[12] In Gaussian units, a centimeter-gram-second (cgs) system, the Lorentz force is
\mathbf{F} = q \left( \mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B} \right),
where c is the speed of light, \mathbf{E} and \mathbf{B} both have units of statvolts per centimeter (equivalent to gauss for \mathbf{B}), reflecting the unified treatment of electric and magnetic fields in vacuum. Numerically, the Gaussian unit for B is the gauss (G), where 1 T = 10,000 G.[13] This system sets \epsilon_0 = 1/(4\pi) and \mu_0 = 4\pi / c^2, which eliminates explicit constants in Coulomb's law but introduces factors of $4\pi in equations like Ampère's law: \nabla \times \mathbf{B} = \frac{4\pi}{c} \mathbf{J} + \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t}.[12] Heaviside-Lorentz units, a rationalized variant of the Gaussian system, modify the force to
\mathbf{F} = q \left( \mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B} \right),
with field units identical to Gaussian (statvolts/cm for \mathbf{E}, gauss for \mathbf{B}), but setting \epsilon_0 = 1 and \mu_0 = 1/c^2 to remove $4\pi factors from Maxwell's equations for greater symmetry. In this system, Ampère's law simplifies to \nabla \times \mathbf{B} = \mathbf{J}/c + (1/c) \partial \mathbf{E}/\partial t, avoiding the unrationalized $4\pi present in Gaussian units. The Heaviside-Lorentz fields scale relative to Gaussian by a factor of $1/\sqrt{4\pi} for B.[11][12] Conversions between these systems involve rescaling fields and charges. Gaussian and Heaviside-Lorentz units are favored in theoretical physics for their simplicity in relativistic contexts and reduced constant clutter, whereas SI units predominate in experimental and engineering applications due to their alignment with practical measurements like the ampere.[12]