Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/378

If therefore an electromotive force $E$ be introduced, acting in the conductor from $A$ to $B,$ and if this causes the potential at $C$ to exceed that at $D$ by $P,$ then the same electromotive force $E$ introduced into the conductor from $C$ to $D$ will cause the potential at $A$ to exceed that at $B$ by the same quantity $P.$

The electromotive force $E$ may be that of a voltaic battery introduced between the points named, care being taken that the resistance of the conductor is the same before and after the introduction of the battery.

282.] If

$D_{pr}+D_{qs}-D_{ps}-D_{qr}=0,$

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the conductor $A_{p}A_{q}$ is said to be conjugate to $A_{r}A_{s},$ and we have seen that this relation is reciprocal.

An electromotive force in one of two conjugate conductors produces no electromotive force or current along the other. We shall find the practical application of this principle in the case of the electric bridge.

The theory of conjugate conductors has been investigated by Kirchhoff, who has stated the conditions of a linear system in the following manner, in which the consideration of the potential is avoided.

(1) (Condition of 'continuity.') At any point of the system the sum of all the currents which flow towards that point is zero.

(2) In any complete circuit formed by the conductors the sum of the electromotive forces taken round the circuit is equal to the sum of the products of the current in each conductor multiplied by the resistance of that conductor.

We obtain this result by adding equations of the form (1) for the complete circuit, when the potentials necessarily disappear.

Heat Generated in the System.

283.] The mechanical equivalent of the quantity of heat generated in a conductor whose resistance is $R$ by a current $C$ in unit of time is, by Art. 242,

$JH=RC^{2}.$.

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We have therefore to determine the sum of such quantities as $RC^{2}$ for all the conductors of the system.

For the conductor from $A_{p}$ to $A_{q}$ the conductivity is $K_{pq},$ and the resistance $R_{pq},$ where

$K_{pq}\cdot R_{pq}=1.$

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The current in this conductor is, according to Ohm's Law,

$C_{pq}=K_{pq}(P_{p}-P_{q}).$

(16)