Lorentz force

In physics, the Lorentz force is the force on an electrically charged particle that moves through a magnetic field and possibly also through an electric field. In the absence of an electric field, the strength of the Lorentz force is proportional to the charge q of the particle, its velocity v (a vector), and the strength of the magnetic field. 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.

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 a vector product, which obeys the right hand rule mentioned above. The electric field E is in full generality given by

$\mathbf {E} =-{\boldsymbol {\nabla }}V-k{\frac {d\mathbf {A} }{dt}}$

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 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.