# Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/165

486.]
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MAGNETIC POTENTIAL DUE TO A CIRCUIT.

485.] The magnetic potential of the circuit, however, differs from that of the magnetic shell for those points which are in the substance of the magnetic shell.

If ${\displaystyle \omega }$ is the solid angle subtended at the point ${\displaystyle P}$ by the magnetic shell, reckoned positive when the positive or austral side of the shell is next to ${\displaystyle P}$, then the magnetic potential at any point not in the shell itself is ${\displaystyle \omega \phi }$, where ${\displaystyle \phi }$ is the strength of the shell. At any point in the substance of the shell itself we may suppose the shell divided into two parts whose strengths are ${\displaystyle \phi _{1}}$ and ${\displaystyle \phi _{2}}$, where ${\displaystyle \phi _{1}+\phi _{2}=\phi }$ such that the point is on the positive side of ${\displaystyle \phi _{1}}$ and on the negative side of ${\displaystyle \phi _{2}}$. The potential at this point is

${\displaystyle \omega (\phi _{1}+\phi _{2})-4\pi \phi _{2}}$.

On the negative side of the shell the potential becomes ${\displaystyle \phi (\omega -4\pi )}$ In this case therefore the potential is continuous, and at every point has a single determinate value. In the case of the electric circuit, on the other hand, the magnetic potential at every point not in the conducting wire itself is equal to ${\displaystyle i\omega }$, where ${\displaystyle i}$ is the strength of the current, and ${\displaystyle \omega }$ is the solid angle subtended by the circuit at the point, and is reckoned positive when the current, as seen from ${\displaystyle P}$, circulates in the direction opposite to that of the hands of a watch.

The quantity ${\displaystyle i\omega }$ to is a function having an infinite series of values whose common difference is ${\displaystyle 4\pi i}$. The differential coefficients of ${\displaystyle i\omega }$ with respect to the coordinates have, however, single and determinate values for every point of space.

486.] If a long thin flexible solenoidal magnet were placed in the neighbourhood of an electric circuit, the north and south ends of the solenoid would tend to move in opposite directions round the wire, and if they were free to obey the magnetic force the magnet would finally become wound round the wire in a close coil. If it were possible to obtain a magnet having only one pole, or poles of unequal strength, such a magnet would be moved round and round the wire continually in one direction, but since the poles of every magnet are equal and opposite, this result can never occur. Faraday, however, has shewn how to produce the continuous rotation of one pole of a magnet round an electric current by making it possible for one pole to go round and round the current while the other pole does not. That this process may be repeated in definitely, the body of the magnet must be transferred from one side of the current to the other once in each revolution. To do this without interrupting the flow of electricity, the current is split