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

This page has been validated.

579.]

ELECTROKINETIC MOMENTUM.

207

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} + & 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{d p}{d t} - \frac{d T}{d y}

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

Hence, the equation of electromotive force is

E - R \dot{y} = Y^{'} = \frac{d p}{d t},

or

E = R \dot{y} + \frac{d p}{d t} .

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