Page:Philosophical magazine 21 series 4.djvu/195

so that within the conductor

$\alpha =-2{\frac {C}{R^{2}}}y,\ \beta =2{\frac {C}{R^{2}}}x,\ \gamma =0.$

(23)

Beyond the conductor, in the space round it,

$\phi =2C\tan ^{-1}{\frac {y}{x}},$

(24)

$\alpha ={\frac {d\phi }{dx}}=-2C{\frac {y}{x^{2}+y^{2}}},\ \beta ={\frac {d\phi }{dy}}=2C{\frac {x}{x^{2}+y^{2}}},\ \gamma ={\frac {d\phi }{dz}}=0.$

(25)

If $\rho ={\sqrt {x^{2}+y^{2}}}$ is the perpendicular distance of any point from the axis of the conductor, a unit north pole will experience a force $={\frac {2C}{\rho }}$, tending to move it round the conductor in the direction of the hands of a watch, if the observer view it in the direction of the current.

Let us now consider a current running parallel to the axis of $z$ in the plane of $xz$ at a distance $\rho$. Let the quantity of the current be $c'$, and let the length of the part considered be $l$, and its section $s$, so that ${\frac {c'}{s}}$ is its strength per unit of section. Putting this quantity for $\rho$ in equations (12, 13, 14), we find

$X=-\mu \beta {\frac {c'}{s}}$

per unit of volume; and multiplying by $ls$, the volume of the conductor considered, we find

${\begin{array}{ll}X&=-\mu \beta c'l\\\\&=-2\mu {\dfrac {Cc'l}{\rho }}\end{array}}$

(26)

showing that the second conductor will be attracted towards the first with a force inversely as the distance.

We find in this case also that the amount of attraction depends on the value of $\mu$, but that it varies directly instead of inversely as $\mu$; so that the attraction between two conducting wires will be greater in oxygen than in air, and greater in air than in water.

We shall next consider the nature of electric currents and electromotive forces in connexion with the theory of molecular vortices.