Page:On Faraday's Lines of Force.pdf/70

224 XI. Spherical electromagnetic Coil Machine.

We have now obtained the electro-tonic function which deﬁnes the action of the one coil on the other. The action of each coil on itself is found by putting $$n^2$$ or $$n'^2$$ for $$nn'$$. Let the ﬁrst coil be connected with an apparatus producing a variable electro-motive force $$F$$ Let us ﬁnd the effects on both wires, supposing their total resistances to be $$R$$ and $$R'$$, and the quantity of the currents $$I$$ and $$I'$$.

Let N stand for $$\frac{2\pi}{3}\frac{a}{3k+k'}$$ then the electro-motive force of the first wire on the second is

$-Nnn'\frac{dI}{dt}$.

That of the second on itself is

$-Nn'^2\frac{dI'}{dt}$.

The equation of the current in the second wire is therefore


 * $$-Nnn'\frac{dI}{dt}-Nn'^2\frac{dI'}{dt}=R'I'.................(1)$$.

The equation of the current in the ﬁrst wire is


 * $$-Nn^2\frac{dI}{dt}-Nnn'\frac{dI'}{dt}+F=RI.................(2)$$.

Eliminating the differential coefﬁcients, we get

$\frac{R}{n}I-\frac{R'}{n'}I'=\frac{F}{n}$,


 * and $$N\left(\frac{n^2}{R}+\frac{n'^2}{R'}\right)\frac{dI}{dt}+I=\frac{F}{R}+N\frac{n'^2}{R'}\frac{dF}{dt}$$..............(3)

from which to ﬁnd $$I$$ and $$I'$$. For this purpose we require to know the value of $$F$$ in terms of $$t$$.

Let us ﬁrst take the case in which $$F$$ is constant and $$I$$ and $$I'$$ initially $$=0$$. This is the case of an electromagnetic coil-machine at the moment when the connexion is made with the galvanic trough.