Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/209

electric arrangement, acts. The current in the circuit at any instant is $I$. He supposes that a magnet is in motion in the neighbourhood of the circuit, and that its potential with respect to the conductor is $V$, so that, during any small interval of time $dt$, the energy communicated to the magnet by the electromagnetic action is $I{\frac {dV}{dt}}dt$.

The work done in generating heat in the circuit is, by Joule's law, Art. 242, $I^{2}Rdt$, and the work spent by the electromotive force $A$, in maintaining the current $I$ during the time $dt$, is $AIdt$. Hence, since the total work done must be equal to the work spent,

whence we find the intensity of the current

Now the value of $A$ may be what we please. Let, therefore, $A=0$, and then

or, there will be a current due to the motion of the magnet, equal to that due to an electromotive force $-{\frac {dV}{dt}}$. The whole induced current during the motion of the magnet from a place where its potential is $V_{1}$ to a place where its potential is $V_{2}$, is

and therefore the total current is independent of the velocity or the path of the magnet, and depends only on its initial and final positions.

In Helmholtz's original investigation he adopted a system of units founded on the measurement of the heat generated in the conductor by the current. Considering the unit of current as arbitrary, the unit of resistance is that of a conductor in which this unit current generates unit of heat in unit of time. The unit of electromotive force in this system is that required to produce the unit of current in the conductor of unit resistance. The adoption of this system of units necessitates the introduction into the equations of a quantity $a$, which is the mechanical equivalent of the unit of heat. As we invariably adopt either the electrostatic or