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

Rh 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,

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 Rh

(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,