Lorentz force

[3] Historians suggest that the law is implicit in a paper by James Clerk Maxwell, published in 1865.

[6] In many textbook treatments of classical electromagnetism, the Lorentz force law is used as the definition of the electric and magnetic fields E and B.

The interpretation of magnetism by means of a modified Coulomb law was first proposed by Carl Friedrich Gauss.

In 1835, Gauss assumed that each segment of a DC loop contains an equal number of negative and positive point charges that move at different speeds.

[11] If Coulomb's law were completely correct, no force should act between any two short segments of such current loops.

Based on this law, Gauss concluded that the electromagnetic force between two point charges depends not only on the distance but also on the relative velocity.

is the speed of light and ∇· (nabla followed by a middle dot) denotes the divergence of a tensor field.

Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates the energy flux (flow of energy per unit time per unit distance) in the fields to the force exerted on a charge distribution.

The above-mentioned formulae use the conventions for the definition of the electric and magnetic field used with the SI, which is the most common.

In the conventions used with the older CGS-Gaussian units, which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has instead

It was proposed that the force on magnetic poles, by Johann Tobias Mayer and others in 1760,[16] and electrically charged objects, by Henry Cavendish in 1762,[17] obeyed an inverse-square law.

It was not until 1784 when Charles-Augustin de Coulomb, using a torsion balance, was able to definitively show through experiment that this was true.

[21] The modern concept of electric and magnetic fields first arose in the theories of Michael Faraday, particularly his idea of lines of force, later to be given full mathematical description by Lord Kelvin and James Clerk Maxwell.

Oliver Heaviside invented the modern vector notation and applied it to Maxwell's field equations; he also (in 1885 and 1889) had fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object.

Instead, Lorentz made a distinction between matter and the luminiferous aether and sought to apply the Maxwell equations at a microscopic scale.

Using Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangian mechanics (see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.

[27][28] In many cases of practical interest, the motion in a magnetic field of an electrically charged particle (such as an electron or ion in a plasma) can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point.

The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.

Thus, Maxwell's equations do not stand separate from other physical laws, but are coupled to them via the charge and current densities.

In real materials the Lorentz force is inadequate to describe the collective behavior of charged particles, both in principle and as a matter of computation.

By combining the Lorentz force law above with the definition of electric current, the following equation results, in the case of a straight stationary wire in a homogeneous field:[30]

Einstein's special theory of relativity was partially motivated by the desire to better understand this link between the two effects.

From Faraday's law of induction (that is valid for a moving wire, for instance in a motor) and the Maxwell Equations, the Lorentz Force can be deduced.

The reverse is also true, the Lorentz force and the Maxwell Equations can be used to derive the Faraday Law.

In all three cases, Faraday's law of induction then predicts the EMF generated by the change in ΦB.

(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on

The action is the relativistic arclength of the path of the particle in spacetime, minus the potential energy contribution, plus an extra contribution which quantum mechanically is an extra phase a charged particle gets when it is moving along a vector potential.

This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.

The calculation for α = 2, 3 (force components in the y and z directions) yields similar results, so collecting the three equations into one:

The proper form of the Lorentz force law ('invariant' is an inadequate term because no transformation has been defined) is simply