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

Rh 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.

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 $$C$$.