ローレンツ力

${\displaystyle {\boldsymbol {F}}(t)=q{\boldsymbol {E}}(t,{\boldsymbol {r}}(t))+q{\boldsymbol {v}}(t)\times {\boldsymbol {B}}(t,{\boldsymbol {r}}(t))}$

${\displaystyle {\boldsymbol {\mathit {F}}}=q({\boldsymbol {\mathit {v}}}\times {\boldsymbol {\mathit {B}}})}$

${\displaystyle {\begin{aligned}dW&={\boldsymbol {F}}\cdot d{\boldsymbol {r}}\\&=q({\boldsymbol {E}}+{\boldsymbol {v}}\times {\boldsymbol {B}})\cdot d{\boldsymbol {r}}\\\end{aligned}}}$

${\displaystyle {\begin{aligned}dW_{m}&=q({\boldsymbol {v}}\times {\boldsymbol {B}})\cdot d{\boldsymbol {r}}\\&=q{\boldsymbol {v}}\cdot ({\boldsymbol {v}}\times {\boldsymbol {B}})dt=0\\\end{aligned}}}$

${\displaystyle {\begin{aligned}dW_{e}&=q{\boldsymbol {E}}\cdot d{\boldsymbol {r}}\\&=q{\boldsymbol {v}}\cdot {\boldsymbol {E}}dt=w\,dt\\\end{aligned}}}$

${\displaystyle \rho (t,{\boldsymbol {x}})=\sum _{i}q_{i}\delta ({\boldsymbol {x}}-{\boldsymbol {r}}_{i}(t))}$

${\displaystyle {\boldsymbol {j}}(t,{\boldsymbol {x}})=\sum _{i}q_{i}{\boldsymbol {v}}_{i}(t)\delta ({\boldsymbol {x}}-{\boldsymbol {r}}_{i}(t))}$

${\displaystyle {\boldsymbol {F}}(t)=\sum _{i}q_{i}\left({\boldsymbol {E}}(t,{\boldsymbol {r}}_{i}(t))+{\boldsymbol {v}}_{i}(t)\times {\boldsymbol {B}}(t,{\boldsymbol {r}}_{i}(t))\right)}$

${\displaystyle {\boldsymbol {E}}(t,{\boldsymbol {r}}_{i}(t))=\int \!\!d^{3}x\,\delta ({\boldsymbol {x}}-{\boldsymbol {r}}_{i}(t)){\boldsymbol {E}}(t,{\boldsymbol {x}})}$

${\displaystyle {\boldsymbol {B}}(t,{\boldsymbol {r}}_{i}(t))=\int \!\!d^{3}x\,\delta ({\boldsymbol {x}}-{\boldsymbol {r}}_{i}(t)){\boldsymbol {B}}(t,{\boldsymbol {x}})}$

${\displaystyle {\boldsymbol {F}}(t)=\int \!\!d^{3}x\,\left(\rho (t,{\boldsymbol {x}}){\boldsymbol {E}}(t,{\boldsymbol {x}})+{\boldsymbol {j}}(t,{\boldsymbol {x}})\times {\boldsymbol {B}}(t,{\boldsymbol {x}})\right)}$

${\displaystyle {\dot {p}}_{\mu }=q{\dot {z}}^{\nu }F_{\nu \mu }(z)}$

${\displaystyle (F_{01},F_{02},F_{03})=(E_{1}/c,E_{2}/c,E_{3}/c),~(F_{32},F_{13},F_{21})=(B_{1},B_{2},B_{3})}$

${\displaystyle {\dot {z}}^{\mu }=(c{\dot {t}},{\dot {t}}{\boldsymbol {v}})}$

${\displaystyle {\dot {\boldsymbol {p}}}=q{\dot {t}}{\boldsymbol {E}}(t,{\boldsymbol {r}})+q{\dot {t}}{\boldsymbol {v}}\times {\boldsymbol {B}}(t,{\boldsymbol {r}})}$

${\displaystyle {\boldsymbol {f}}={\frac {d{\boldsymbol {p}}}{dt}}={\frac {\dot {\boldsymbol {p}}}{\dot {t}}}=q{\boldsymbol {E}}(t,{\boldsymbol {r}})+q{\boldsymbol {v}}\times {\boldsymbol {B}}(t,{\boldsymbol {r}})}$

${\displaystyle {\boldsymbol {\mathit {F}}}=q({\boldsymbol {\mathit {v}}}\times {\boldsymbol {\mathit {B}}})}$