Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/379

We shall suppose, however, that the value of the current is not that given by Ohm's Law, but $X_{pq},$ where

$X_{pq}=C_{pq}+Y_{pq},$

(17)

To determine the heat generated in the system we have to find the sum of all the quantities of the form

$R_{pq}X_{pq}^{2},$

or

$JH=\Sigma {R_{pq}C_{pq}^{2}+2R_{pq}C_{pq}Y_{pq}+R_{pq}Y_{pq}^{2}}.$

(18)

Giving $C_{pq}$ its value, and remembering the relation between $K_{pq}$ and $R_{pq},$ this becomes

$\Sigma (P_{p}-P_{q})(C_{pq}+2Y_{pq})+R_{pq}Y_{pq}^{2}.$

(19)

Now since both $C$ and $X$ must satisfy the condition of continuity at $A_{p},$ we have

$Q_{p}=C_{p1}+C_{p2}+\mathrm {\&c.} +C_{pn},$

(20)

$Q_{p}=X_{p1}+X_{p2}+\mathrm {\&c.} +X_{pn},$

(21)

therefore

$0=Y_{p1}+Y_{p2}+\mathrm {\&c.} +Y_{pn}.$

(22)

Adding together therefore all the terms of (19), we find

$\Sigma (R_{pq}X_{pq}^{2})=\Sigma P_{p}Q_{p}+\Sigma R_{pq}Y_{pq}^{2}$

(23)

Now since $R$ is always positive and $Y^{2}$ is essentially positive, the last term of this equation must be essentially positive. Hence the first term is a minimum when $Y$ is zero in every conductor, that is, when the current in every conductor is that given by Ohm's Law.

Hence the following theorem:

284.] In any system of conductors in which there are no internal electromotive forces the heat generated by currents distributed in accordance with Ohm's Law is less than if the currents had been distributed in any other manner consistent with the actual conditions of supply and outflow of the current.

The heat actually generated when Ohm's Law is fulfilled is mechanically equivalent to $\Sigma P_{p}Q_{q},$ that is, to the sum of the products of the quantities of electricity supplied at the different external electrodes, each multiplied by the potential at which it is supplied.