Page:Elementary Text-book of Physics (Anthony, 1897).djvu/363

§ 296] them. Or, we may consider the tubes of induction as tending to diminish in length and to repel each other, and describe the action on a circuit in terms of the tensions due to the tubes of induction. These various modes of description necessarily yield similar results.

296. Action of a Current on a Magnet Pole.—We will now show that the force between a magnet pole and a circuit carrying a current may be considered as the resultant of forces which act between the pole and the elements of the circuit, and that this action follows the law deduced by Biot directly from his experiments (§ 389). On the view we have taken, this representation of the action is an artificial one, the real action being due to the magnetic field associated with the circuit.

Let $$AB$$ (Fig. 85) represent a circuit carrying the current $$i,$$ placed in a magnetic field in which the permeability is unity; let $$l$$ represent the length of an element of the circuit, and $$H$$ the strength of the magnetic field near that element. If $$N$$ represent the number of unit tubes of force which pass through the circuit, the energy of the circuit is expressed by $$iN.$$ Suppose the circuit displaced so that all parts of it move through the same small distance $$s.$$ The number of tubes of force which pass through it after its displacement is represented by $$N';$$ the energy lost by the displacement is $$i(N - N').$$ This energy is equal to the work done upon the circuit by the forces of the field.

If we consider the closed surface bounded by the planes of the circuit in its two positions and by the cylinder traced by the circuit during its displacement, and remember that there is no free magnetism within this surface, the flux of force over it is zero (§ 56). And since the change in the number of unit tubes passing through the circuit measures the change in the flux of force through the circuit, it is evident that the change in the flux of force through