Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/250

mutual energy of two current-elements containing charges e, e′ respectively of each kind of electricity, is

${\frac {ee^{\prime }c^{2}}{2r^{3}}}[-\{(\mathbf {r.v^{\prime }} )-(\mathbf {r.v} )\}^{2}+\{(\mathbf {r.v^{\prime }} )+(\mathbf {r.v} )\}^{2}+\{-(\mathbf {r.v^{\prime }} )-(\mathbf {r.v} )\}^{2}-\{(\mathbf {-r.v^{\prime }} )+(\mathbf {r.v} )\}^{2}]$,

or ${\frac {4ee^{\prime }(\mathbf {r.v^{\prime }} )(\mathbf {r.v} )}{r^{3}}}$. If ds, ds′ denote the lengths of the elements, and i, i′ the currents in them, we have

$i\mathbf {ds} =2e\mathbf {v} ,i^{\prime }\mathbf {ds} ^{\prime }=2e^{\prime }\mathbf {v} ^{\prime }$;

so the mutual energy of two current-elements is

${\frac {ii^{\prime }}{r^{3}}}(\mathbf {r.ds^{\prime }} ).(\mathbf {r.ds} )$.

The mutual energy of ids with all the other currents is therefore

$i(\mathbf {ds.a} )$,

where a denotes a vector-potential

$\int _{s^{\prime }}i^{\prime }{\frac {(\mathbf {r.ds^{\prime }} )\mathbf {.r} }{r^{3}}}$.

By reasoning similar to Neumann's, it may be shown that the electromotive force induced in ds by any alteration in the rest of the field is

$-(\mathbf {ds.{\dot {a}}} )$.

and thus a complete theory of induced currents may be constructed.

The necessity for induced currents may be inferred by general reasoning from the first principles of Weber's theory. When a circuit's moves in the field due to currents, the velocity of the vitreous charges in s is, owing to the motion of s, not equal and opposite to that of the resinous charges: this gives rise to a difference in the forces acting on the vitreous and resinous charges in s; and hence the charges of opposite sign separate from each other and move in opposite directions.

The assumption that positive and negative charges move with equal and opposite velocities relative to the matter of