Aequatio Lorentziana

Aequatio Lorentziana (nomen apud gloriam Henrici Antonici Lorentz) describit, quomodo campus electromagneticus vim in particulis quae onus habent pariat, et in aequationibus Maxwellianis basem physicae electromagneticae fundat.

Aequatio Lorentziana forma vectorali (unitatibus MKSA) modo scriptae est sic

${\displaystyle {\vec {\mathbf {F} }}=q({\vec {\mathbf {E} }}+{\vec {\mathbf {v} }}\times {\vec {\mathbf {B} }}),}$

ubi

${\displaystyle {\vec {\mathbf {F} }}}$ est vis electromagnetica (in Newtoniis)

${\displaystyle {\vec {\mathbf {E} }}}$ est campus electricus (in Voltiis per metrum)

${\displaystyle {\vec {\mathbf {B} }}}$ est campus magneticus (in Weberiis per metrum quadratum, aut equivalenter Teslis)

${\displaystyle q}$ est onus electricum particulae (in Coulombiis)

${\displaystyle {\vec {\mathbf {v} }}}$ est velocitas momentanea particulae (in metris per secundum), et

${\displaystyle \times }$ est productum vectorialis sive productum crucis.

Aequatio Lorentziana forma vectorali (unitatibus Gaussiana CGSF) modo scriptae est sic

${\displaystyle {\vec {\mathbf {F} }}=q({\vec {\mathbf {E} }}+{\vec {\frac {\mathbf {v} }{c}}}\times {\vec {\mathbf {B} }})}$

ubi

${\displaystyle {\vec {\mathbf {B} }}}$ est campus magneticus in Gaussibus vel dyniis per Franklin,

${\displaystyle {\vec {\mathbf {E} }}}$ campus electricus in Gaussibus vel dyniis per Franklin,

${\displaystyle q}$ est onus electricum particulae (in Franklinibus)

${\displaystyle {\vec {\mathbf {v} }}}$ est velocitas momentanea particulae (in centimetris per secundum), et

${\displaystyle \times }$ est productum vectorialis sive productum crucis.

Aequatio viris Lorentzianae scribere possumus in forma tensorali covariante (unitatibus MKSA) sic:

${\displaystyle {\frac {dp^{\alpha }}{d\tau }}=qu_{\beta }F^{\alpha \beta }}$

ubi

${\displaystyle \tau }$ est tempus proprium particulae,

q est onus electricum particulae (in Coulombibus),

u est 4-velocitas particulae (in metris per secundum), definita sicut:

${\displaystyle u_{\beta }=\left(u_{0},u_{1},u_{2},u_{3}\right)=\gamma \left(c,v_{x},v_{y},v_{z}\right)\,}$ et

F est tensor campi electromagnetici (in Teslis) definitus sicut:

${\displaystyle F^{\alpha \beta }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}}$.

Pars ${\displaystyle \mu =1}$ viris electromagnetica est:

${\displaystyle \gamma {\frac {dp^{1}}{dt}}={\frac {dp^{1}}{d\tau }}=qu_{\beta }F^{1\beta }=q\left(-u^{0}F^{10}+u^{1}F^{11}+u^{2}F^{12}+u^{3}F^{13}\right).\,}$

ubi ${\displaystyle \tau }$ est tempus propium particulae. Elementa tensoris F electromagnetici substituta obtinemus:

${\displaystyle \gamma {\frac {dp^{1}}{dt}}=q\left(-u^{0}\left({\frac {-E_{x}}{c}}\right)+u^{2}(B_{z})+u^{3}(-B_{y})\right)\,}$

Et si expressim introducimus partes quattuor-velocitatis, deinde obtinemus

${\displaystyle \gamma {\frac {dp^{1}}{dt}}=q\gamma \left(c\left({\frac {E_{x}}{c}}\right)+v_{y}B_{z}-v_{z}B_{y}\right)\,}$

${\displaystyle \gamma {\frac {dp^{1}}{dt}}=q\gamma \left(E_{x}+\left(\mathbf {v} \times \mathbf {B} \right)_{x}\right).\,}$

Calculatio partium ${\displaystyle \mu =2}$ et ${\displaystyle \mu =3}$ est similis, postquam obtinemus aequationem Lorentzianam :

${\displaystyle \gamma {\frac {d\mathbf {p} }{dt}}={\frac {d\mathbf {p} }{d\tau }}=q\gamma \left(\mathbf {E} +(\mathbf {v} \times \mathbf {B} )\right)\,}$.

• Paradoxum geminorum