Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/162

130 But we know that in all magnets Rh Hence or the electromagnetic force due to a straight current of infinite length is perpendicular to the current, and varies inversely as the distance from it.

479.] Since the product $$Tr$$ depends on the strength of the current it may be employed as a measure of the current. This method of measurement is different from that founded upon electrostatic phenomena, and as it depends on the magnetic phenomena produced by electric currents it is called the Electromagnetic system of measurement. In the electromagnetic system if $$i$$ is the current, Rh

480.] If the wire be taken for the axis of $$z$$, then the rectangular components of $$T$$ are Rh

Here $$X\, dx + Y\, dy + Z\, dz$$ is a complete differential, being that of Rh

Hence the magnetic force in the field can be deduced from a potential function, as in several former instances, but the potential is in this case a function having an infinite series of values whose common difference is $$4\pi i$$. The differential coefficients of the potential with respect to the coordinates have, however, definite and single values at every point.

The existence of a potential function in the field near an electric current is not a self-evident result of the principle of the conservation of energy, for in all actual currents there is a continual expenditure of the electric energy of the battery in overcoming the resistance of the wire, so that unless the amount of this expenditure were accurately known, it might be suspected that part of the energy of the battery may be employed in causing work to be done on a magnet moving in a cycle. In fact, if a magnetic pole, $$m$$, moves round a closed curve which embraces the wire, work is actually done to the amount of $$4 \pi m i$$. It is only for closed paths which do not embrace the wire that the line-integral of the force vanishes. We must therefore for the present consider the law of force and the existence of a potential as resting on the evidence of the experiment already described.