Lorentz force

In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of F = q ( E + v × B ) {displaystyle mathbf {F} =q(mathbf {E} +mathbf {v} imes mathbf {B} )} f = ρ E + J × B {displaystyle mathbf {f} = ho mathbf {E} +mathbf {J} imes mathbf {B} ,!} F = q [ − ∇ ϕ − ∂ A ∂ t + ∇ ( v ⋅ A ) − ( v ⋅ ∇ ) A ] {displaystyle mathbf {F} =qleft} F = q [ − ∇ x ( ϕ − x ˙ ⋅ A ) + d d t ∇ x ˙ ( ϕ − x ˙ ⋅ A ) ] {displaystyle mathbf {F} =qleft} The total potential energy is then:The equations of motion derived by extremizing the action (see matrix calculus for the notation): d p α d τ = q F α β U β {displaystyle {frac {mathrm {d} p^{alpha }}{mathrm {d} au }}=qF^{alpha eta }U_{eta }} F = q F ⋅ v {displaystyle F=q{mathcal {F}}cdot v} In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of (in SI units). Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the force on a charged particle which might be traveling near the speed of light (relativistic form of the Lorentz force). Historians suggest that the law is implicit in a paper by James Clerk Maxwell, published in 1865. Hendrik Lorentz arrived in a complete derivation in 1895, identifying the contribution of the electric force a few years after Oliver Heaviside correctly identified the contribution of the magnetic force. The force F acting on a particle of electric charge q with instantaneous velocity v, due to an external electric field E and magnetic field B, is given by (in SI units): where × is the vector cross product (all boldface quantities are vectors). In terms of cartesian components, we have: F x = q ( E x + v y B z − v z B y ) , {displaystyle F_{x}=q(E_{x}+v_{y}B_{z}-v_{z}B_{y}),} F y = q ( E y + v z B x − v x B z ) , {displaystyle F_{y}=q(E_{y}+v_{z}B_{x}-v_{x}B_{z}),} F z = q ( E z + v x B y − v y B x ) . {displaystyle F_{z}=q(E_{z}+v_{x}B_{y}-v_{y}B_{x}).}