Lorentz force

In physics the Lorentz force is the force on an electrically charged particle that moves through a magnetic field and an electric field; hence the Lorentz force has a magnetic and an electric component.

The strength of the magnetic component of the Lorentz force is proportional to the charge q of the particle, the speed v (the size of the velocity v) of the particle, the intensity B of the magnetic induction, and the sine of the angle between the vectors v and B. The direction of the magnetic component is given by the right hand rule: put your right hand along v with fingers pointing in the direction of v and the open palm toward the vector B. Stretch the thumb of your right hand, then the Lorentz force is along it, pointing from your wrist to the tip of your thumb.

The electric component is equal to q E (charge of the particle times the electric field), which must be added vectorially to the magnetic component of the Lorentz force in order to obtain the total force.

The force is named after the Dutch physicist Hendrik Antoon Lorentz, who gave its equation in 1892.

Mathematical description

The Lorentz force F is given by the expression

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

where k is a constant depending on the units. In SI units k = 1; in Gaussian units k = 1/c, where c is the speed of light in vacuum (299 792 458 m s⁻¹ exactly). The quantity q is the electric charge of the particle and v is its velocity. The vector B is the magnetic induction. The product v × B is the vector product (a vector with the direction given by the right-hand rule and of magnitude v B sin α with α the angle between v and B). The electric field E is is the sum of a longitudinal (curl-free) component and a transverse (divergence-free) component (provided the Coulomb gauge ∇ • A = 0 is chosen) ,

$\mathbf {E} =-{\boldsymbol {\nabla }}V-k{\frac {\partial \mathbf {A} }{\partial t}},$

where V is a scalar (electric) potential and the (magnetic) vector potential A is connected to B via

$\mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} .$

The operator ∇ acting on V gives the gradient of V, while ∇ × A is the curl of A. Since ∇ × (∇ V) = 0 and ∇ • A = 0, the components are curl-free and divergence-free, respectively.

Note that the Lorentz force does not depend on the medium; the electric force does not contain the electric permittivity ε and the magnetic force does not the contain magnetic permeability μ.

If B is static (does not depend on time) then A is also static and

$\mathbf {E} =-{\boldsymbol {\nabla }}V.$

Non-relativistically, the electric field E may be absent (zero) while B is static and non-zero; the Lorentz force is then given by,

$\mathbf {F} =k\,q\,\mathbf {v} \times \mathbf {B} ,\quad {\hbox{with}}\quad F=k\,q\,v\,B\sin \alpha ,$

where k = 1 for SI units and 1/c for Gaussian units and α the angle between v and B. The italic quantities are the strenghts (lengths) of the corresponding vectors

$F\equiv |\mathbf {F} |,\quad v\equiv |\mathbf {v} |,\quad B\equiv |\mathbf {B} |.$

The Lorentz force as a vector (cross) product was given by Oliver Heaviside in 1889, three years before Lorentz.

In special relativity the Lorentz force transforms as a four-vector under a Lorentz transformation, giving a linear combination of E and B, because relativistically the fields E and B are components of the same second rank tensor and do not have an independent existence.