Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/167

 Rh their common axis. They are evidently oblate figures, being flattened in the direction of the axis. They meet each other in the line of the circuit at angles of 15°.

The force acting on a magnetic pole placed at any point of an equipotential surface is perpendicular to this surface, and varies inversely as the distance between consecutive surfaces. The closed curves surrounding the section of the wire in Fig. XVIII are the lines of force. They are copied from Sir W. Thomson's Paper on 'Vortex Motion .' See also Art. 702.

Action of an Electric Circuit on any Magnetic System.

488.] We are now able to deduce the action of an electric circuit on any magnetic system in its neighbourhood from the theory of magnetic shells. For if we construct a magnetic shell, whose strength is numerically equal to the strength of the current, and whose edge coincides in position with the circuit, while the shell itself does not pass through any part of the magnetic system, the action of the shell on the magnetic system will be identical with that of the electric circuit.

Reaction of the Magnetic System on the Electric Circuit.

489.] From this, applying the principle that action and reaction are equal and opposite, we conclude that the mechanical action of the magnetic system on the electric circuit is identical with its action on a magnetic shell having the circuit for its edge.

The potential energy of a magnetic shell of strength $$\phi$$ placed in a field of magnetic force of which the potential is $$V$$, is, by Art. 410,Rhwhere $$l$$, $$m$$, $$n$$ are the direction-cosines of the normal drawn from the positive side of the element $$dS$$ of the shell, and the integration is extended over the surface of the shell. Now the surface-integralRhwhere $$a$$, $$b$$, $$c$$ are the components of the magnetic induction, represents the quantity of magnetic induction through the shell, or,