Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/164

Rh dimensions of which are small compared with their distance from $$P$$, and with the radii of curvature of the surface.

Round each of these elements conceive a current of strength $$i$$ to flow, the direction of circulation being the same in all the elements as it is in the original circuit.

Along every line forming the division between two contiguous elements two equal currents of strength $$i$$ flow in opposite directions.

The effect of two equal and opposite currents in the same place is absolutely zero, in whatever aspect we consider the currents. Hence their magnetic effect is zero. The only portions of the elementary circuits which are not neutralized in this way are those which coincide with the original circuit. The total effect of the elementary circuits is therefore equivalent to that of the original circuit.

484.] Now since each of the elementary circuits may be considered as a small plane circuit whose distance from $$P$$ is great compared with its dimensions, we may substitute for it an elementary magnetic shell of strength $$i$$ whose bounding edge coincides with the elementary circuit. The magnetic effect of the elementary shell on $$P$$ is equivalent to that of the elementary circuit. The whole of the elementary shells constitute a magnetic shell of strength $$i$$, coinciding with the surface $$S$$ and bounded by the original circuit, and the magnetic action of the whole shell on $$P$$ is equivalent to that of the circuit.

It is manifest that the action of the circuit is independent of the form of the surface $$S$$, which was drawn in a perfectly arbitrary manner so as to fill it up. We see from this that the action of a magnetic shell depends only on the form of its edge and not on the form of the shell itself. This result we obtained before, at Art. 410, but it is instructive to see how it may be deduced from electromagnetic considerations.

The magnetic force due to the circuit at any point is therefore identical in magnitude and direction with that due to a magnetic shell bounded by the circuit and not passing through the point, the strength of the shell being numerically equal to that of the current. The direction of the current in the circuit is related to the direction of magnetization of the shell, so that if a man were to stand with his feet on that side of the shell which we call the positive side, and which tends to point to the north, the current in front of him would be from right to left.