Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/293

It follows from this that if $\phi _{1}$ and $\psi _{1}^{\prime }$ are conjugate functions (Art. 183) of $\phi$ and $\psi ^{\prime }$, the curves $\phi _{1}$ may be stream-lines in the sheet for which the curves $\psi _{1}^{\prime }$ are the corresponding equipotential lines. One case, of course, is that in which $\phi _{1}=\psi ^{\prime }$ and $\psi _{1}^{\prime }=-\phi$. In this case the equipotential lines become current-lines, and the current-lines equipotential lines^[1].

If we have obtained the solution of the distribution of electric currents in a uniform sheet of any form for any particular case, we may deduce the distribution in any other case by a proper transformation of the conjugate functions, according to the method given in Art. 190.

652.] We have next to determine the magnetic action of a current-sheet in which the current is entirely confined to the sheet, there being no electrodes to convey the current to or from the sheet.

In this case the current-function $\phi$ has a determinate value at every point, and the stream-lines are closed curves which do not intersect each other, though any one stream-line may intersect itself.

Consider the annular portion of the sheet between the stream-lines $\phi$ and $\phi +\delta \phi$. This part of the sheet is a conducting circuit in which a current of strength $\delta \phi$ circulates in the positive direction round that part of the sheet for which $\phi$ is greater than the given value. The magnetic effect of this circuit is the same as that of a magnetic shell of strength 5 $ at any point not included in the substance of the shell. Let us suppose that the shell coincides with that part of the current-sheet for which $\phi$ has a greater value than it has at the given stream-line.

By drawing all the successive stream-lines, beginning with that for which $\phi$ has the greatest value, and ending with that for which its value is least, we shall divide the current-sheet into a series of circuits. Substituting for each circuit its corresponding magnetic shell, we find that the magnetic effect of the current-sheet at any point not included in the thickness of the sheet is the same as that of a complex magnetic shell, whose strength at any point is $C+\phi$, where $C$ is a constant.

If the current-sheet is bounded, then we must make $C+\phi =0$ at the bounding curve. If the sheet forms a closed or an infinite surface, there is nothing to determine the value of the constant $C$.

1. See Thomson, Camb. and Dub. Math. Journ., vol.iii. p. 286.