Excerpt from The University of Colorado Studies, Vol. 1

Apply the above to a small closed surface fixed in space sur rounding one and only one of the elementary magnets. The number of lines of force outward through the surface cannot vary, even if the magnet should move completely out Of the enclosure. Hence there must be as many lines of force running toward the magnet as away from it.

The question now arises: If we adopt the electronic theory of the conduction of electricity, can we generalize the second law so as to hold for material bodies, even in case the change is due to motion of the electrons? It is immediately evident that we cannot do so without Special hypotheses, for if we surround an electron the charge of which is e, by a small closed surface, the number Of lines of electric force through the surface outward is 4 7r e, and if the electron moves out of the enclosure the number changes to zero.

We may, however, generalize the second law as follows: The magneto-motive force around any closed curve equals the rate Of in crease Of the number of lines of electric force running through any surface bounded by the curve, provided we take account of that in increase only which is due to the cutting of lines of force across the closed curve.

In the case Of a magneto-motive force produced by a conduction current of electricity, this cutting of lines of electric force across the closed curve is due primarily to the motion of the electrons (at least according to the modern views). Let us consider a constant current Of electrons flowing in the positive direction around a circuit through the line 1 in Fig. 1. AS each electron with its charge flows around the circuit Once, the entire number Of its lines of force 4 r e cut across the line 1. At some instant during its motion, the elec tron passes through the surface S, and, the number of its lines run ning through this surface in the positive direction, decreases by the amount 4 71 e. Hence, on the whole, the change in the number of lines Of force through this surface, due to the motion of this electron, is zero. The current has been assumed constant. Hence, as a whole, the number of lines passing through S, remains the same; which means that as a whole just as many lines Of force are withdrawn from the positive direction through the surface, owing to the passage Of electrons through S, as are thrust through by cutting the edge of the surface. But the magneto-motive force equals the number Of lines of force cutting through lper second, and therefore equals 4 7r times the sum of the electron charges passing through S, per second. This latter we consider to be the current through S, . The same is true of any other surface, S2 bounded by l, and hence the sum of the electrons passing through S, per second, is the same as through S2, or the total sum Of the charges passing outward through the closed surface S, 82 per second is zero. This means that the flow of electrons is similar to that of an incompressible fiuid.

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