ELECTROMAGNETIC INDUCTION

Electromagnetic Momentum of a Current

(22) We may begin by considering the state of the field in the neighbourhood of an electric current. We know that the magnetic forces are excited in the field, their direction and magnitude depending according to known laws upon the form of the conductor carrying the current. When the strength of the current is increased, all the magnetic effects are increased in the same proportion. Now, if the magnetic state of the field depends on motions of the medium, a certain force must be exerted in order to increase or diminish these motions, and when the motions are excited they continue, so that the effect of the connection between the current and the electromagnetic field surrounding it, is to endow the current with a kind of momentum, just as the connection between the driving-point of a machine and a fly-wheel endows the driving-point with an additional momentum, which may be called the momentum of the fly-wheel reduced to the driving-point. The unbalanced force acting on the driving-point increases this momentum, and is measured by the rate of its increase.

In the case of electric currents, the resistance to sudden increase or diminution of strength produces effects exactly like those of momentum, but the amount of this momentum depends on the shape of the conductor and the relative position of its different parts.

Mutual Action of two Currents

(23) If there are two electric currents in the field, the magnetic force at any point is that compounded of the forces due to each current separately, and since the two currents are in connexion with every point of the field, they will be in connexion with each other, so that any increases or diminution of the one will produce a force acting with or contrary to the other.

Dynamical Illustration of Reduced Momentum

(24) As a dynamical illustration, let us suppose a body C so connected with two independent driving-points A and B that its velocity is p times that of A together with q times that of B.

Let u be the velocity of A, v that of B, and w that of C.

Let δx δy δz be their simultaneous displacements, then by the general equation of dynamics[1],

…

where X and Y are the forces acting at A and B.

But

and

Substituting, and remembering that d x {\displaystyle dx} and d y are independent,

(1) We may call

the momentum of

referred to A {\displaystyle A}, and

its momentum referred to B {\displaystyle B}; then we may say that the effect of the force X {\displaystyle X} is to increase the momentum of C {\displaystyle C} referred to A {\displaystyle A}, and that of Y {\displaystyle Y} to increase its momentum referred to B {\displaystyle B}.

If there are many bodies connected with A {\displaystyle A} and B {\displaystyle B} in a similar way but with different values of p {\displaystyle p} and q {\displaystyle q}, we may treat the question in the same way by assuming

where the summation is extended to all the bodies with their proper values of

{\displaystyle a}, and q {\displaystyle q}. Then the momentum of the system referred to A

And referred to B

And we shall have

(2) Where

X {\displaystyle X} and Y {\displaystyle Y} are the external forces acting on A {\displaystyle A} and B {\displaystyle B}.

(25) To make the illustration more complete we have only to suppose that the motion of A {\displaystyle A} is resisted by a force proportional to its velocity, which we may call R u {\displaystyle Ru}, and that of B {\displaystyle B} by a similar force, which we may call S v {\displaystyle Sv}, R {\displaystyle R} and S {\displaystyle S} being coefficients of resistance. Then if ξ {\displaystyle \xi } and η {\displaystyle \eta } are the forces on A {\displaystyle A} and B

(3) If the velocity of

{\displaystyle A} be increased at the rate

{\displaystyle {\tfrac {du}{dt}}}, then in order to prevent B {\displaystyle B} from moving a force,

must be applied to it.

This effect on B {\displaystyle B}, due to an increase of the velocity of A {\displaystyle A}, corresponds to the electromotive force on one circuit arising from an increase in the strength of a neighbouring circuit.

This dynamical illustration is to be considered merely as assisting the reader to understand what is meant in mechanics by Reduced Momentum. The facts of the induction of currents as depending on the variations of the quantity called Electromagnetic Momentum, or Electrotonic State, rest on the experiments of Faraday[2], Felici[3], &c.