Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/343

due to ordinary resistance, and the generation or absorption of heat at the junction of two metals. We shall call the first the frictional generation of heat by the current, and, as we have seen, it is proportional to the square of the current, and is the same whether the current be in the positive or the negative direction. The second we may call the Peltier effect, which changes its sign with that of the current.

The total heat generated in a portion of a compound conductor consisting of two metals may be expressed by

$H={\frac {R}{J}}C^{2}t-\Pi Ct,$

where $H$ is the quantity of heat, $J$ the mechanical equivalent of unit of heat, $R$ the resistance of the conductor, $C$ the current, and $t$ the time; $\Pi$ being the coefficient of the Peltier effect, that is, the heat absorbed at the junction due to the passage of unit of current for unit of time.

Now the heat generated is mechanically equivalent to the work done against electrical forces in the conductor, that is, it is equal to the product of the current into the electromotive force producing it. Hence, if $E$ is the external electromotive force which causes the current to flow through the conductor,

$JH=CEt=RC^{2}t-J\Pi Ct,$

whence

$E=RC-J\Pi .$

It appears from this equation that the external electromotive force required to drive the current through the compound conductor is less than that due to its resistance alone by the electromotive force $J\Pi$. Hence $J\Pi$ represents the electromotive contact force at the junction acting in the positive direction.

This application, due to Sir W. Thomson^[1], of the dynamical theory of heat to the determination of a local electromotive force is of great scientific importance, since the ordinary method of connecting two points of the compound conductor with the electrodes of a galvanometer or electroscope by wires would be useless, owing to the contact forces at the junctions of the wires with the materials of the compound conductor. In the thermal method, on the other hand, we know that the only source of energy is the current of electricity, and that no work is done by the current in a certain portion of the circuit except in heating that portion of the conductor. If, therefore, we can measure the amount of the

1. Proc. R. S. Edin., Dec. 15, 1851; and Trans. R. S. Edin., 1854.