Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/242

If the primary current is maintained constant, and the form or relative position of the circuits altered so that M becomes M', the integral secondary current is y₂, where

$R_{2}y_{2}=(M-M'){\dot {y}}_{1}.$

In the case of two circuits placed side by side and in the same direction M diminishes as the distance between the circuits in creases. Hence, the induced current is positive when this distance is increased and negative when it is diminished.

These are the elementary cases of induced currents described in Art. 530.

On Mechanical Action between the Two Circuits.

583.] Let x be any one of the geometrical variables on which the form and relative position of the circuits depend, the electromagnetic force tending to increase x is

$X={\frac {1}{2}}{\dot {y}}_{1}^{2}{\frac {dL}{dx}}+{\dot {y}}_{1}{\dot {y}}_{2}{\frac {dM}{dx}}+{\frac {1}{2}}{\dot {y}}_{2}^{2}{\frac {dN}{dx}}.$

If the motion of the system corresponding to the variation of x is such that each circuit moves as a rigid body, L and N will be independent of x, and the equation will be reduced to the form

$X={\dot {y}}_{1}{\dot {y}}_{2}{\frac {dM}{dx}}.$

Hence, if the primary and secondary currents are of the same sign, the force X, which acts between the circuits, will tend to move them so as to increase M.

If the circuits are placed side by side, and the currents flow in the same direction, M will be increased by their being brought nearer together. Hence the force X is in this case an attraction.

584.] The whole of the phenomena of the mutual action of two circuits, whether the induction of currents or the mechanical force between them, depend on the quantity M, which we have called the coefficient of mutual induction. The method of calculating this quantity from the geometrical relations of the circuits is given in Art. 524, but in the investigations of the next chapter we shall not assume a knowledge of the mathematical form of this quantity. We shall consider it as deduced from experiments on induction, as, for instance, by observing the integral current when the secondary circuit is suddenly moved from a given position to an infinite distance, or to any position in which we know that M = 0.