In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
(specifically in
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
on a
point charge
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
due to
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
s. A particle of charge moving with a velocity in an
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
and a
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
experiences a force of
(in
SI units[In SI units, is measured in teslas (symbol: T). In Gaussian-cgs units, is measured in gauss (symbol: G). See e.g. )][The -field is measured in ]ampere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elect ...
s per metre (A/m) in SI units, and in oersteds (Oe) in cgs units. ). It says that the electromagnetic force on a charge is a combination of a force in the direction of the electric field proportional to the magnitude of the field and the quantity of charge, and a force at right angles to the magnetic field and the velocity of the charge, proportional to the magnitude of the field, the charge, and the velocity. Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called
Laplace force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an e ...
), the
electromotive force
In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical '' tran ...
in a wire loop moving through a magnetic field (an aspect of
Faraday's law of induction), and the force on a moving charged particle.
Historians suggest that the law is implicit in a paper by
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
, published in 1865.
Hendrik Lorentz
Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorent ...
arrived at a complete derivation in 1895,
identifying the contribution of the electric force a few years after
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
correctly identified the contribution of the magnetic force.
Lorentz force law as the definition of E and B
In many textbook treatments of classical electromagnetism, the Lorentz force law is used as the ''definition'' of the electric and magnetic fields and .
[See, for example, Jackson, pp. 777–8.] To be specific, the Lorentz force is understood to be the following empirical statement:
''The electromagnetic force on a test charge In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be in ...
at a given point and time is a certain function of its charge and velocity , which can be parameterized by exactly two vectors and , in the functional form'':
This is valid, even for particles approaching the speed of light (that is,
magnitude of , ). So the two
vector fields and are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field". The fields are defined everywhere in space and time with respect to what force a test charge would receive regardless of whether a charge is present to experience the force.
As a definition of and , the Lorentz force is only a definition in principle because a real particle (as opposed to the hypothetical "test charge" of infinitesimally-small mass and charge) would generate its own finite and fields, which would alter the electromagnetic force that it experiences. In addition, if the charge experiences acceleration, as if forced into a curved trajectory, it emits radiation that causes it to lose kinetic energy. See for example
Bremsstrahlung
''Bremsstrahlung'' (), from "to brake" and "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typical ...
and
synchrotron light
Synchrotron radiation (also known as magnetobremsstrahlung radiation) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in ...
. These effects occur through both a direct effect (called the
radiation reaction force) and indirectly (by affecting the motion of nearby charges and currents).
Equation
Charged particle
The force acting on a particle of
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
with instantaneous velocity , due to an external electric field and magnetic field , is given by (in
SI units):
where is the vector cross product (all boldface quantities are vectors). In terms of Cartesian components, we have:
In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as: