Page:On Faraday's Lines of Force.pdf/50

204 Let us now consider the conditions of the conduction of the electric currents within the medium during changes in the electro-tonic state. The method which we shall adopt is an application of that given by Helmholtz in his memoir on- the Conservation of Force.

Let there be some external source of electric currents which would generate in the conducting mass currents whose quantity is measured by $$a_2, b_2 c_2,$$ and their intensity by $$\alpha_2, \beta_2, \gamma_2.$$

Then the amount of work due to this cause in the time $$dt$$ is

$dt\iiint(a_2\alpha_2+b_2\beta_2+c_2\gamma_2)dx dy dz$

in the form of resistance overcome, and

$\frac{dt}{4\pi}\frac{d}{dt}\iiint(a_2\alpha_0+b_2\beta_0+c_2\gamma_0)dx dy dz$

in the form of work done mechanically by the electro-magnetic action of these currents If there be no external cause producing currents, then the quantity representing the whole work done by the external cause must vanish, and we have

$dt\iiint(a_2\alpha_2+b_2\beta_2+c_2\gamma_2)dx dy dz + \frac{dt}{4\pi}\frac{d}{dt}\iiint(a_2\alpha_0+b_2\beta_0+c_2\gamma_0)dx dy dz$

where the integrals are taken through any arbitrary space We must therefore have

$a_2\alpha_2+b_2\beta_2+c_2\gamma_2 = \frac{1}{4\pi}\frac{d}{dt}(a_2\alpha_0+b_2\beta_0+c_2\gamma_0)$

for every point of space; and it must be remembered that the variation of Q is supposed due to variations of $$\alpha_0, \beta_0, \gamma_0$$ and not of $$a_2, b_2 c_2,$$. We must therefore treat $$a_2, b_2 c_2,$$ as constants, and the equation becomes

$a_2\left(\alpha_2+\frac{1}{4\pi}\frac{d\alpha_0}{dt}\right) + b_2\left(\beta_2+\frac{1}{4\pi}\frac{d\beta_0}{dt}\right) + c_2\left(\gamma_2+\frac{1}{4\pi}\frac{d\gamma_0}{dt}\right)=0.$

In order that this equation may be independent of the values of (1,, I)“ (2,, each of these coefﬁcients must =0; and therefore we have the following expressions for the electro-motive forces due to the action of magnets and currents at a distance in terms of the electro-tonic functions,

$\alpha_2 = -\frac{1}{4\pi}\frac{d\alpha_0}{dt}, \ \beta_2 = \frac{1}{4\pi}\frac{d\beta_0}{dt}, \ \gamma_2 = -\frac{1}{4\pi}\frac{d\gamma_0}{dt}.$