Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/287

element of the body, with the forces arising from the magnetization and electric currents.

642.] The nature of the stress of which these are the components may be easily fouud, by making the axis of $x$ bisect the angle between the directions of the magnetic force and the magnetic induction, and taking the axis of $y$ in the plane of these directions, and measured towards the side of the magnetic force.

If we put ${\mathfrak {H}}$ for the numerical value of the magnetic force, ${\mathfrak {B}}$ for that of the magnetic induction, and $2\epsilon$ for the angle between their directions,

$\alpha ={\mathfrak {H}}\cos \epsilon$,	⁠	$\beta ={}$	${\mathfrak {H}}\sin \epsilon$,	⁠	$\gamma =0$,
$a={\mathfrak {B}}\cos \epsilon$,		$b=-{}$	${\mathfrak {B}}\sin \epsilon$,		$c=0$;

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$P_{xx}={\frac {1}{4\pi }}($	${\mathfrak {BH}}\cos ^{2}\epsilon -{\frac {1}{2}}{\mathfrak {H}}^{2})$,
$P_{yy}={\frac {1}{4\pi }}(-{}$	${\mathfrak {BH}}\sin ^{2}\epsilon -{\frac {1}{2}}{\mathfrak {H}}^{2})$,
$P_{zz}={\frac {1}{4\pi }}(-{\frac {1}{2}}{\mathfrak {H}}^{2})$,
$P_{yz}=P_{zx}=P_{zy}=P_{xz}=0$,
$P_{xy}={\frac {1}{4\pi }}{\mathfrak {BH}}\cos \epsilon \sin \epsilon$,
$P_{yx}=-{\frac {1}{4\pi }}{\mathfrak {BH}}\cos \epsilon \sin \epsilon$

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Hence, the state of stress may be considered as compounded of—

(1) A pressure equal in all directions ${}={\frac {1}{8\pi }}{\mathfrak {H}}^{2}$.

(2) A tension along the line bisecting the angle between the directions of the magnetic force and the magnetic induction

(3) A pressure along the line bisecting the exterior angle between these directions ${}={\frac {1}{4\pi }}{\mathfrak {BH}}\sin ^{2}\epsilon$.

(4) A couple tending to turn every element of the substance in the plane of the two directions from the direction of magnetic induction to the direction of magnetic force ${}={\frac {1}{4\pi }}{\mathfrak {BH}}\sin 2\epsilon$.

When the magnetic induction is in the same direction as the magnetic force, as it always is in fluids and non-magnetized solids, then $\epsilon =0$, and making the axis of $x$ coincide with the direction of the magnetic force,