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

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

$p_{1}=L_{1}{\dot {y}}_{1}+M_{12}{\dot {y}}_{2}+\And c.$

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,

$Y'={\frac {dp}{dt}}-{\frac {dT}{dy}}$

but since $T$ does not involve $y$, the last term disappears.

Hence, the equation of electromotive force is

$E-R{\dot {y}}=Y'={\frac {dp}{dt}},$

or

$E=R{\dot {y}}+{\frac {dp}{dt}}.$

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