Lorentz force

In physics the Lorentz force is the force on an electrically charged particle that moves through a magnetic field B and possibly also through an electric field E.

In the absence of an electric field (E = 0), the strength 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 field, and the sine of the angle between the vectors v and B. The direction of the Lorentz force is given by the right hand rule: put your right hand along v with the open palm toward the magnetic field B (a vector). Stretch the thumb of your right hand, the Lorentz force is along it, pointing from your wrist to the tip of your thumb.

If an electric field E is also present then the force q E must be added vectorially to the magnetic component of the Lorentz force (the force on the particle when E = 0).

The force is named after the Dutch physicist Hendrik Antoon Lorentz, who gave its description 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 (sometimes referred to as the magnetic field). The product of v and B is the vector product (a vector with the direction given by the right hand rule mentioned above and of magnitude vBsin α). The electric field E is in full generality given by

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

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 factor k has the same meaning as before. The operator ∇ acting on V gives the gradient of V, while ∇ × A is the curl of A.

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

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

It is possible that the electric field E is absent (zero) and that B is static and non-zero, then the Lorentz force is given by,

$\mathbf {F} =k\,q\,\mathbf {v} \times \mathbf {B} ,$

where k = 1 for SI units and 1/c for Gaussian units. In this form the Lorentz force was given by Oliver Heaviside in 1889, three years before Lorentz.