Lorentz force

For four-momentum $p^{\mu }$ and four-velocity $u^{\mu }$, the covariant Lorentz-force law isIts spatial and temporal components are

For smooth stationary fields on ${\mathbb{R}}^3$, set ${\Gamma }^i{}_{tt}=-E^i$ and ${\Gamma }^i{}_{tj}={\Gamma }^i{}_{jt}={ϵ}^i{}_{jk}{B}^k$, with all other Christoffel symbols zero. This torsion-free connection makes the equation $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$ a geodesic equation with time as affine parameter. Arbitrary smooth fields suffice; a metric realization imposes additional potential conditions.

In uniform perpendicular electric and magnetic fields, the guiding centre of a charged particle drifts with ${\mathbf{v}}_D=\mathbf{E}\times \mathbf{B}/|\mathbf{B}{|}^2$. This velocity is independent of the charge and mass; cyclotron motion is superposed on it.

The covariant force on a charged particle is its charge times the electromagnetic field-strength tensor contracted with its four-velocity. Antisymmetry keeps the four-force orthogonal to that velocity and preserves the rest mass.

A wire carrying current $I$ in a magnetic field experiencesThis follows by summing the magnetic part of the Lorentz force over the moving charge carriers.

In a constant magnetic field $B\hat{z}$ with no electric field, a charged relativistic particle moves in a circle in the perpendicular plane. Its angular frequency with respect to proper time iswhereas its angular frequency in an inertial coordinate time is ${\omega }_t=qB/(\gamma m)$. The orbit radius at speed $v$ is

For $qB>0$, choose $A_y=Bx$ and parametrize one clockwise orbit by $x=R\cos ({\omega }_{\tau }\tau )$ and $y=-R\sin ({\omega }_{\tau }\tau )$. Over $\Delta \tau =2\pi /{\omega }_{\tau }$, the relativistic charged-particle action isThe same formula with $|{\omega }_{\tau }|$ applies for either orbit orientation.

The Lorentz force per unit volume is ${\mathbf{f}}_{\mathrm{L}}={\rho }_e\mathbf{E}+\mathbf{j}\times \mathbf{B}/c$ in Gaussian units.

In uniform perpendicular electric and magnetic fields, a charged particle's cyclotron orbit has guiding-centre velocityindependent of its charge and mass.

For $\mathcal{E}=\gamma m{c}^2$, the temporal component of the covariant Lorentz-force equation givesThe magnetic field does no work because $\mathbf{v}\cdot (\mathbf{v}\times \mathbf{B})=0$.

Combining $d(\gamma m\mathbf{v})/dt=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$ with $m{c}^{2}d\gamma /dt=q\mathbf{E}\cdot \mathbf{v}$ gives