Lorentz force

In physics, the Lorentz force is the force on an electrically charged particle that moves through a magnetic plus an electric field.

The Lorentz force has two vector components, one proportional to the magnetic field and one proportional to the electric field. These components must be added vectorially to obtain the total force.

1. The strength (absolute value) of the magnetic component is proportional to four factors: the charge q of the particle, the speed 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.

2. The electric component of the Lorentz force is equal to q•E (charge of the particle times the electric field). It is in the same direction as E for positively charged particles and in the opposite direction of E for negatively charged particles.

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, also referred to as the cross product.

As any vector field, the electric field E appearing in the Lorentz force F is the sum of a longitudinal (curl-free) component and a transverse (divergence-free) component. The following form holds when the Coulomb gauge ∇ • A = 0 is chosen,

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

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

$\mathbf {B} (\mathbf {r} ,t)={\boldsymbol {\nabla }}\times \mathbf {A} (\mathbf {r} ,t).$

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 of E are indeed 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

$\mathbf {E} =-{\boldsymbol {\nabla }}V\quad {\hbox{and}}\quad \mathbf {F} =-q{\boldsymbol {\nabla }}V.$

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, non-bold, quantities are the strengths (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. Because relativistically the fields E and B are components of the same second rank tensor, a Lorentz transformation gives a linear combination of E and B, and hence in relativity theory these two fields do not have an independent existence.