Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/239

 Rh the dynamical theory, we shall call $$L_1$$ the coefficient of self-induction of the circuit $$A_1$$ and $$M_{12}$$ the coefficient of mutual induction of the circuits $$A_1$$ and $$A_2$$. $$M_{12}$$ is also called the potential of the circuit $$A_1$$ with respect to $$A_2$$. These quantities depend only on the form and relative position of the circuits. We shall find that in the electromagnetic system of measurement they are quantities of the dimension of a line. See Art. 627.

By differentiating $$T$$ with respect to $$\dot{y}_1$$ we obtain the quantity $$p_1$$ which, in the dynamical theory, may be called the momentum corresponding to $$y_1$$. In the electric theory we shall call $$p_1$$ the electrokinetic momentum of the circuit $$A_1$$. Its value is The electrokinetic momentum of the circuit $$A_1$$ is therefore made up of the product of its own current into its coefficient of self-induction, together with the sum of the products of the currents in the other circuits, each into the coefficient of mutual induction of $$A_1$$ and that other circuit.

On Electromotive Force. 579.] Let $$E$$ be the impressed electromotive force in the circuit $$A$$, arising from some cause, such as a voltaic or thermoelectric battery, which would produce a current independently of magneto-electric induction.

Let $$R$$ be the resistance of the circuit, then, by Ohm's law, an electromotive force $$R\dot{y}$$ is required to overcome the resistance, leaving an electromotive force $$E - R\dot{y}$$ available for changing the momentum of the circuit. Calling this force $$Y'$$, we have, by the general equations, but since $$T$$ does not involve $$y$$, the last term disappears.

Hence, the equation of electromotive force is

The impressed electromotive force $$E$$ is therefore the sum of two parts. The first, $$R\dot{y}$$, is required to maintain the current $$\dot{y}$$ against the resistance $$R$$. The second part is required to increase the electromagnetic momentum $$p$$. This is the electromotive force which must be supplied from sources independent of magneto-electric