Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/297

Rh $$x$$ and to $$y$$ at the positive surface, and $$\alpha^\prime$$, $$\beta^\prime$$ those on the negative surface Rh Rh

Within the sheet the components vary continuously from $$\alpha$$ and $$\beta$$ to $$\alpha^\prime$$ and $$\beta^\prime$$.

The equations

which connect the components $$F$$, $$G$$, $$H$$ of the vector-potential due to the current-sheet with the scalar potential $$\Omega$$, are satisfied if we make Rh

We may also obtain these values by direct integration, thus for $$F$$,

Since the integration is to be estimated over the infinite plane sheet, and since the first term vanishes at infinity, the expression is reduced to the second term; and by substituting $\frac{d}{dy}\frac{1}{r}$ for $-\frac{d}{dy^\prime}\frac{1}{r}$, and remembering that $$\phi$$ depends on $$x^\prime$$ and $$y^\prime$$ and not on $$x$$, $$y$$, $$z$$, we obtain

If $$\Omega^\prime$$ is the magnetic potential due to any magnetic or electric system external to the sheet, we may write Rhand we shall then have Rhfor the components of the vector-potential due to this system.