# Lorentz Force

_From Wikipedia, the free encyclopedia_

In physics, particularly [electromagnetism](https://en.wikipedia.org/wiki/Electromagnetism), the **Lorentz force** is the combination of electric and magnetic force on a point charge due to electromagnetic fields. If a particle of charge $$ q $$ moves with velocity $$ \mathbf{v} $$ in the presence of an electric field $$ \mathbf{E} $$ and a magnetic field $$ \mathbf{B} $$, then it will experience a force. For any produced force there will be an opposite reactive force. In the case of the magnetic field, the reactive force may be obscure, but it must be accounted for.

$$ \mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right) $$

(in SI units). Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of [Faraday's law of induction](https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction)), and the force on a charged particle which might be traveling near the speed of light (relativistic form of the Lorentz force).

The first derivation of the Lorentz force is commonly attributed to Oliver Heaviside in 1889, although other historians suggest an earlier origin in an 1865 paper by James Clerk Maxwell. Hendrik Lorentz derived it a few years after Heaviside.

## Equation (SI units)

### Charged particle

->![Lorentz force](https://upload.wikimedia.org/wikipedia/commons/7/7c/Lorentz_force_particle.svg =160x)<-

->_Lorentz force $$ \mathbf{F} $$ on a charged particle (of charge $$ q $$) in motion (instantaneous velocity $$ \mathbf{v} $$). The $$ \mathbf{E} $$ field and $$ \mathbf{B} $$ field vary in space and time._<-

The force $$ \mathbf{F} $$ acting on a particle of electric charge $$ q $$ with instantaneous velocity $$ \mathbf{v} $$, due to an external electric field $$ \mathbf{E} $$ and magnetic field $$ \mathbf{B} $$, is given by:

$$ \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) $$

where $$ \times $$ is the vector cross product. All boldface quantities are vectors. More explicitly stated:

$$ \mathbf{F}(\mathbf{r},\mathbf{\dot{r}},t,q) =  q[\mathbf{E}(\mathbf{r},t) + \mathbf{\dot{r}} \times \mathbf{B}(\mathbf{r},t)] $$

in which r is the position vector of the charged particle, t is time, and the overdot is a time derivative.

A positively charged particle will be accelerated in the same linear orientation as the $$ \mathbf{E} $$ field, but will curve perpendicularly to both the instantaneous velocity vector $$ \mathbf{v} $$ and the $$ \mathbf{B} $$ field according to the right-hand rule (in detail, if the thumb of the right hand points along $$ \mathbf{v} $$ and the index finger along $$ \mathbf{B} $$, then the middle finger points along $$ \mathbf{F} $$).

The term $$ q\mathbf{E} $$ is called the **electric force**, while the term $$ q\mathbf{v \times B}$$ is called the **magnetic force**. According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force, with the _total_ electromagnetic force (including the electric force) given some other (nonstandard) name. This article will not follow this nomenclature: In what follows, the term "Lorentz force" will refer only to the expression for the total force.

The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the **Laplace force**.