Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/458

CHAPTER XXIII.

THEORIES OF ACTION AT A DISTANCE.

On the Explanation of Ampère's Formula given by Gauss and Weber.

846.] The attraction between the elements $ds$ and $ds'$ of two circuits, carrying electric currents of intensity $i$ and $i'$ is, by, Ampère's formula,

$-{\frac {ii'\,ds\,ds'}{r^{2}}}\left(2\cos \epsilon +3{\frac {dr}{ds}}{\frac {dr}{ds'}}\right);$

(1)

or

$-{\frac {ii'\,ds\,ds'}{r^{2}}}\left(2\,r{\frac {d^{2}r}{ds\,ds'}}-3{\frac {dr}{ds}}{\frac {dr}{ds'}}\right);$

(2)

the currents being estimated in electromagnetic units. See Art. 526.

The quantities, whose meaning as they appear in these expressions we have now to interpret, are

$\cos \epsilon ,\;{\frac {dr}{ds}}{\frac {dr}{ds'}},\;{\text{and}}\;{\frac {d^{2}r}{ds\,ds'}};$

and the most obvious phenomenon in which to seek for an interpretation founded on a direct relation between the currents is the relative velocity of the electricity in the two elements.

847.] Let us therefore consider the relative motion of two particles, moving with constant velocities $v$ and $v'$ along the elements $ds$ and $ds'$ respectively. The square of the relative velocity of these particles is

$u^{2}=v^{2}-2vv'\cos \epsilon +v'^{2};$

(3)

and if we denote by $r$ the distance between the particles,

${\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}=v{\frac {dr}{ds}}+v'{\frac {dr}{ds'}},$

(4)

$\left({\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}\right)^{2}=v^{2}\left({\frac {dr}{ds}}\right)^{2}+2vv'{\frac {dr}{ds}}{\frac {dr}{ds'}}+v'^{2}\left({\frac {dr}{ds'}}\right)^{2},$

(5)

${\frac {{\mathfrak {d}}^{2}r}{{\mathfrak {d}}t^{2}}}=v^{2}{\frac {d^{2}r}{ds^{2}}}+2vv'{\frac {d^{2}r}{ds\,ds'}}+v'^{2}{\frac {d^{2}r}{ds'^{2}}},$

(6)