\Section[Lorentz transformation of electromagnetic force]{75.}{The Lorentz transformation of electromagnetic force} \index{Force!Lorentz transformation of}% \index{Lorentz transformation!for electromagnetic force}% The Lorentz transformation for an observer~$S'$ moving relatively to~$S$ with a velocity~$u$ along the $x$-axis is \[ x_{1}' = q(x_{1} - ux_{4}),\quad x_{2}' = x_{2},\quad x_{3}' = x_{3},\quad x_{4}' = q(x_{4} - ux_{1}), \Tag{(75.1)} \] where \[ q = (1 - u^{2})^{-\frac{1}{2}}. \] \PageSep{180} We use $q$ instead of~$\beta$ in order to avoid confusion with the component~$\beta$ of magnetic force. \index{Force!mechanical force due to}% \index{Mechanical force of electromagnetic field}% We have \[ \frac{\dd x_{1}'}{\dd x_{1}} = \frac{\dd x_{4}'}{\dd x_{4}} = q,\quad \frac{\dd x_{1}'}{\dd x_{4}} = \frac{\dd x_{4}'}{\dd x_{1}} = -qu,\quad \frac{\dd x_{2}'}{\dd x_{2}} = \frac{\dd x_{3}'}{\dd x_{3}} = 1, \Tag{(75.2)} \] and all other derivatives vanish. To calculate the electromagnetic force for~$S'$ in terms of the force for~$S$, we apply the general formulae of transformation~\Eq{(23.21)}. Thus \begin{align*} \gamma' = F'^{12} &= \frac{\dd x_{1}'}{\dd x_{\alpha}}\, \frac{\dd x_{2}'}{\dd x_{\beta}}\, F^{\alpha\beta} \\ &= \frac{\dd x_{1}'}{\dd x_{1}}\, \frac{\dd x_{2}'}{\dd x_{2}}\, F^{12} + \frac{\dd x_{1}'}{\dd x_{4}}\, \frac{\dd x_{2}'}{\dd x_{2}}\, F^{42} \\ &= q\gamma - quY. \end{align*} Working out the other components similarly, the result is \[ \left. \begin{aligned} X' &= X, & Y' &= q(Y - u\gamma), & Z' &= q(Z + u\beta)\Add{,} \\ \alpha' &= \alpha, & \beta' &= q(\beta + uZ), & \gamma' &= q(\gamma - uY), \end{aligned} \right\} \Tag{(75.3)} \] which are the formulae given by Lorentz. The more general formulae when the velocity of the observer~$S'$ is $(u, v, w)$ become very complicated. We shall only consider the approximate results when the square of the velocity is neglected. In that case $q = 1$, and the formulae~\Eq{(75.3)} can be completed by symmetry, viz.\ \[ \left. \begin{alignedat}{3} X' &= X &&- w\beta &&+ v\gamma\Add{,} \\ \alpha' &= \alpha &&+ wY &&- vZ. \end{alignedat} \right\} \Tag{(75.4)} \] \Section{76.}{Mechanical effects of the electromagnetic field} According to the elementary laws, a piece of matter carrying electric charge of density~$\rho$ experiences in an electrostatic field a mechanical force \[ \rho X,\quad \rho Y,\quad \rho Z \] per unit volume. Moving charges constituting electric currents of amount $(\sigma_{x}, \sigma_{y}, \sigma_{z})$ per unit volume are acted on by a magnetic field, so that a mechanical force \[ \gamma\sigma_{y} - \beta\sigma_{z},\quad \alpha\sigma_{z} - \gamma\sigma_{x},\quad \beta\sigma_{x} - \alpha\sigma_{y} \] per unit volume is experienced.