167, 168] STEADY FLOW IN CONDUCTORS. 335 



This is the equation of Joule's Law*: 



The heat developed in any portion of conductor in unit time is 

 equal to the resistance of that portion of the conductor multiplied 

 by the square of the current traversing it. This law is universally 

 true, whether the flow is steady or not. 



168. Sources of Electromotive Force. Suppose we have 

 a closed circuit of a number of different conductors. We have 

 already seen that if all are of the first class there can be no cur- 

 rent. Suppose that one only is of the second class, and let its 

 suffix be 1. Then for all the others we have 



EM + E M + E n _ lt n -f E nz = 0, 



so that the total electromotive force around the circuit is 



E = E 12 E m + E nl = E 12 -f- E 2n + Em, 



depending only on the conductor of the second class and the two 

 of the first class in contact with it. Such an arrangement is called 

 a galvanic or voltaic cell. 



In a conductor of the second class traversed by a current, 

 chemical actions go on, whose laws were discovered by Faraday 

 and Helmholtz. Such actions belong to the subject of electro- 

 chemistry, which is a branch of thermodynamics, and will be 

 treated by the author elsewhere. For the same reason the theory 

 of thermoelectromotive forces will not be treated here. 



We have so far considered impressed electromotive forces to 

 exist only at certain surfaces, where the potential is discontinuous. 

 If, starting at any equipotential surface in a closed conductor, we 

 plot the potential as an ordinate, on a diagram in which the 

 abscissa is the resistance from the initial to any other equipotential 

 surface, the curve will be composed of portions of parallel straight 

 lines, whose slope is proportional to the current. The total 

 impressed electromotive force will be equal to the sum of sudden 

 rises minus the sum of sudden falls as we pass in the direction of 

 the downward slope. It is evident that the discontinuities may 

 occur at as many points as we please, and that provided the alge- 

 braic sum is the same the current will be unchanged. It is 

 evident, comparing the two figures in which this is the case, and 



* Joule. " On the Heat evolved by Metallic Conductors of Electricity, and in 

 the Cells of a Battery during Electrolysis." Phil. Mag. 19, p. 260, 1841. 



