Coefficients of Induction for Two Circuits

Heat produced by the Current

(32) On the other side of the equation we have, first,

(9) which represents the work done in overcoming the resistance of the circuits in unit of time. This is converted into heat. The remaining terms represent work not converted into heat. They may be written

Intrinsic Energy of the Currents

(33) If L {\displaystyle L}, M {\displaystyle M}, N {\displaystyle N} are constant, the whole work of the electromotive forces which is not spent against resistance will be devoted to the development of the currents. The whole intrinsic energy of the currents is therefore

(10) This energy exists in a form imperceptible to our senses, probably as actual motion, the seat of this motion being not merely the conducting circuits, but the space surrounding them.

Mechanical Action between Conductors.

(34) The remaining terms,

(11)

represent the work done in unit of time arising from the variations of L {\displaystyle L}, M {\displaystyle M}, and N {\displaystyle N}, or, what is the same thing, alterations in the form and position of the conducting circuits A {\displaystyle A} and B {\displaystyle B}.

Now if work is done when a body is moved, it must arise from ordinary mechanical force acting on the body while it is moved. Hence this part of the expression shows that there is a mechanical force urging every part of the conductors themselves in that direction in which L {\displaystyle L}, M {\displaystyle M}, and N {\displaystyle N} will be most increased.

The existence of the electromagnetic force between conductors carrying currents is therefore a direct consequence of the joint and independent action of each current on the electromagnetic field. If A {\displaystyle A} and B {\displaystyle B} are allowed to approach a distance d s {\displaystyle ds}, so as to increase M {\displaystyle M} from M {\displaystyle M} to M ′ {\displaystyle M’} while the currents are x {\displaystyle x} and y {\displaystyle y}, then the work done will be

and the force in the direction of

{\displaystyle ds} will be

(12)

and this will be an attraction if x {\displaystyle x} and y {\displaystyle y} are of the same sign, and if M {\displaystyle M} is increased as A {\displaystyle A} and B {\displaystyle B} approach.

It appears, therefore, that if we admit that the unresisted part of electromotive force goes on as long as it acts, generating a self-persistent state of the current, which we may call (from mechanical analogy) its electromagnetic momentum, and that this momentum depends on circumstances external to the conductor, then both induction of currents and electromagnetic attractions may be proved by mechanical reasoning.

What I have called electromagnetic momentum is the same quantity which is called by Faraday[4] the electrotonic state of the circuit, every change of which involves the action of an electromotive force, just as change of momentum involves the action of mechanical force.

If, therefore, the phenomena described by Faraday in the ninth Series of his Experimental Researches were the only known facts about electric currents, the laws of Ampère relating to the attraction of conductors carrying currents, as well as those of Faraday about the mutual induction of currents, might be deduced by mechanical reasoning.

In order to bring these results within the range of experimental verification, I shall next investigate the case of a single current, of two currents, and of the six currents in the electric balance, so as to enable the experimenter to determine the values of L {\displaystyle L}, M {\displaystyle M}, N {\displaystyle N}.

Case of a single Circuit.

(35) The equation of the current x {\displaystyle x} in a circuit whose resistance is R {\displaystyle R}, and whose coefficient of self-induction is L {\displaystyle L}, acted on by an external electromotive force ξ {\displaystyle \xi }, is

(13) When ξ {\displaystyle \xi } is constant, the solution is of the form

where a {\displaystyle a} is the value of the current at the commencement, and b {\displaystyle b} is its final value.

The total quantity of electricity which passes in time t {\displaystyle t}, where t {\displaystyle t} is great, is

(14) The value of the integral of x 2 {\displaystyle x^{2}} with respect to the time is

(15)

The actual current changes gradually from the initial value a {\displaystyle a} to the final value b {\displaystyle b}, but the values of the integrals of x {\displaystyle x} and x 2 {\displaystyle x^{2}} are the same as if a steady current of intensity

{\displaystyle {\tfrac {1}{2}}(a+b)} were to flow for a time

{\displaystyle 2{\tfrac {L}{R}}}, and were then succeeded by the steady current b {\displaystyle b}. The time

{\displaystyle 2{\tfrac {L}{R}}} is generally so minute a fraction of a second, that the effects on the galvonometer and dynamometer may be calculated as if the impulse were instantaneous.

If the circuit consists of a battery and a coil, then, when the circuit is first complete, the effects are the same as if the current had only half its final strength during the time

{\displaystyle 2{\tfrac {L}{R}}}. This diminution of the current, due to induction, is sometimes called the counter-current.

(36) If an additional resistance r {\displaystyle r} is suddenly thrown into the circuit, as by breaking contact, so as to force the current to pass through a thin wire of resistance r {\displaystyle r}, then the original current is

{\displaystyle a={\tfrac {\xi }{R}}}, and the final current is

{\displaystyle b={\tfrac {\xi }{R+r}}}.

The current of induction is then

{\displaystyle {\tfrac {1}{2}}\xi {\tfrac {2R+r}{R(R+r)}}}, and continues for a time 2

{\displaystyle {\tfrac {L}{(R+r)}}}. The current is greater than that which the battery can maintain in the two wires R {\displaystyle R} and r {\displaystyle r}, and may be sufficient to ignite the thin wire r {\displaystyle r}.

When contact is broken by separating the wires in air, this additional resistance is given by the interposed air, and since the electromotive force across the new resistance is very great, a spark will be formed across.

If the electromotive force is of the form E sin ⁡ {\displaystyle E\sin {pt}}, as in the case of a coil revolving in a magnetic field, then

where

and tan ⁡