Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/284

placed in a field of magnetic force whose components are $\alpha$, $\beta$, $\gamma$, is

Hence, if the force urging the element to move without rotation in the direction of $x$ is $X_{1}\,dx\,dy\,dz$,

and if the moment of the couple tending to turn the element about the axis of $x$ from $y$ towards $z$ is $L\,dx\,dy\,dz$,

The forces and the moments corresponding to the axes of $y$ and $z$ may be written down by making the proper substitutions.

640.] If the magnetized body carries an electric current, of which the components are $u$, $v$, $w$, then, by equations C, Art. 603, there will be an additional electromagnetic force whose components are $X_{2}$, $Y_{2}$, $Z_{2}$, of which $X_{2}$ is

Hence, the total force, $X$, arising from the magnetism of the molecule, as well as the current passing through it, is

The quantities $a$, $b$, $c$ are the components of magnetic induction, and are related to $\alpha$, $\beta$, $\gamma$, the components of magnetic force, by the equations given in Art. 400,

$a=\alpha +4\pi A$, $b=\beta +4\pi B$, $c=\gamma +4\pi C$.

(5)

The components of the current, $u$, $v$, $w$, can be expressed in terms of $\alpha$, $\beta$, $\gamma$ by the equations of Art. 607,

$4\pi u={\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}$, $4\pi v={\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}$, $4\pi w={\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}$.

(6)

Hence

$X$	${}={\frac {1}{4\pi }}\left\{(a-\alpha ){\frac {d\alpha }{dx}}+(b-\beta ){\frac {d\beta }{dx}}+(c-\gamma ){\frac {d\gamma }{dx}}+b\left({\frac {d\alpha }{dy}}-{\frac {d\beta }{dx}}\right)+c\left({\frac {d\alpha }{dx}}-{\frac {d\gamma }{dx}}\right)\right\}$,
	${}={\frac {1}{4\pi }}\left\{a{\frac {d\alpha }{dx}}+b{\frac {d\alpha }{dy}}+c{\frac {d\alpha }{dz}}-{\frac {1}{2}}{\frac {d}{dx}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\right\}$.

(7)