(26) In the electromagnetic field the values of L M N {\displaystyle N} depend on the distribution of the magnetic effects due to the two circuits, and this distribution depends only on the form and relative position of the circuits. Hence L {\displaystyle L}, M {\displaystyle M}, N {\displaystyle N} are quantities depending on the form and relative position of the circuits, and are subject to variation with the motion of the conductors. It will be presently seen that L {\displaystyle L}, M {\displaystyle M}, N {\displaystyle N} are geometrical quantities of the nature of lines, that is, of one dimension in space; L {\displaystyle L} depends on the form of the first conductor, which we shall call A {\displaystyle A}, N {\displaystyle N} on that of the second, which we call B {\displaystyle B}, and M {\displaystyle M} on the relative position of A {\displaystyle A} and B {\displaystyle B}.

(27) Let ξ {\displaystyle \xi } be the electromotive force acting on A {\displaystyle A}, x {\displaystyle x} the strength of the current, and R {\displaystyle R} the resistance, then R x {\displaystyle Rx} will be the resisting force. In steady currents the electromotive force just balances the resisting force, but in variable currents the resultant force ξ

R x {\displaystyle \xi =Rx} is expanded in increasing the “electromagnetic momentum”, using the word momentum merely to express that which is generated by a force acting during a time, that is, a velocity existing in a body.

In the case of electric currents, the force in action is not ordinary mechanical force, at least we are not as yet able to measure it as common force, but we call it electromotive force, and the body moved is not merely the electricity in the conductor, but something outside the conductor, and capable of being affected by other conductors in the neighbourhood carrying currents. In this it resembles rather the reduced momentum of a driving-point of a machine as influenced by its mechanical connections, than that of a simple moving body like a cannon ball, or water in a tube.

Electromagnetic Relations of two Conducting Circuits

(28) In the case of two conducting circuits, A {\displaystyle A} and B {\displaystyle B}, we shall assume that the electromagnetic momentum belonging to A {\displaystyle A} is

L x + M y , {\displaystyle Lx+My,}

and that belonging to B {\displaystyle B},

M x + N y , {\displaystyle Mx+Ny,}

where L {\displaystyle L}, M {\displaystyle M}, N {\displaystyle N} correspond to the same quantities in the dynamical illustration, except that they are supposed to be capable of variation when the conductors A {\displaystyle A} or B {\displaystyle B} are moved.

Then the equation of the current x {\displaystyle x} in A {\displaystyle A} will be

(4) and that of y {\displaystyle y} in B {\displaystyle B}

(5) where ξ {\displaystyle \xi } and η {\displaystyle \eta } are the electromotive forces, x {\displaystyle x} and y {\displaystyle y} the currents, and R {\displaystyle R} and S {\displaystyle S} the resistances in A {\displaystyle A} and B {\displaystyle B} respectively.

Induction of one Current by another.

(29) Case 1st. Let there be no electromotive force on B {\displaystyle B}, except that which arises from the action of A, and let the current of A {\displaystyle A} increase from 0 to the value x {\displaystyle x}, then

whence

(6)

that is, a quantity of electricity Y {\displaystyle Y}, being the total induced current, will flow through B {\displaystyle B} when x {\displaystyle x} rises from 0 to x {\displaystyle x}. This is induction by variation of the current in the primary conductor. When M {\displaystyle M} is positive, the induced current due to increase of the primary current is negative.

Induction of Motion by a Conductor.

(30) Case 2nd. Let x {\displaystyle x} remain constant, and let M {\displaystyle M} change from M {\displaystyle M} to M ′ {\displaystyle M’}, then

(7) so that if M {\displaystyle M} is increased, which it will be by the primary and secondary circuits approaching each other, there will be a negative induced current, the total quantity of electricity passed through B {\displaystyle B} being Y {\displaystyle Y}.

This is induction by the relative motion of the primary and secondary conductors.

Equation of Work and Energy.

(31) To form the equation between work done and energy produced, multiply (1) by x {\displaystyle x} and (2) by y {\displaystyle y}, and add

Here ξ x {\displaystyle \xi x} is the work done in unit of time by the electromotive force ξ {\displaystyle \xi } acting on the current x {\displaystyle x} and maintaining it, and η y {\displaystyle \eta y} is the work done by the electromtoive force η {\displaystyle \eta }. Hence the left-hand side of the equation represents the work done by the electromotive forces in unit of time.