Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/322

the resultant force per unit area along the outward normal is therefore

$-{\tfrac {1}{2}}\mathbf {DE} .t.(1/\rho _{1}+1/\rho _{2})$,

and so we have

$T=-{\tfrac {1}{2}}\mathbf {DE} .t$;

or the pressure at right angles to the lines of force is 1/2DE per unit area—that is, it is numerically equal to the tension along the lines of force.

The principal stresses in the medium being thus determined, it readily follows that the stress across any plane, to which the unit vector N is normal, is

$(\mathbf {D.N} )\mathbf {E} -{\frac {1}{2}}(\mathbf {D.E} )\mathbf {N}$.

Maxwell obtained^[1] a similar formula for the case of magnetic fields; the pouderomotive forces on magnetized matter and on conductors carrying currents may be accounted for by assuming a stress in the medium, the stress across the plane N being represented by the vector

${\frac {1}{4\pi }}(\mathbf {B.N} ).\mathbf {H} -{\frac {1}{8\pi }}(\mathbf {B.H} ).\mathbf {N}$.

This, like the corresponding electrostatic formula, represents a tension across planes perpendicular to the lines of force, and a pressure across planes parallel to them.

It may be remarked that Maxwell made no distinction between stress in the material dielectric and stress in the aether: indeed, so long as it was supposed that material bodies when displaced carry tho contained aether along with them, no distinction was possible. In the modifications of Maxwell's theory which were developed many years afterwards by his followers, stresses corresponding to those introduced by Maxwell were assigned to the aether, as distinct from ponderable matter; and it was assumed that the only stresses set up in material bodies by the electromagnetic field are produced indirectly: they may be calculated by the methods of the theory of elasticity, from & knowledge of the ponderomotive forces exerted on the electric charges connected with the bodies.

1. Maxwell's Treatise in Electricity and Magnetism, § 643.