Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/294

653.] The magnetic potential at any point on either side of the current-sheet is given, as in Art. 415, by the expression

$\Omega =\iint {\frac {1}{r^{2}}}\phi \cos \theta \,dS$,

where $r$ is the distance of the given point from the element of surface $dS$, and $\theta$ is the angle between the direction of $r$, and that of the normal drawn from the positive side of $dS$.

This expression gives the magnetic potential for all points not included in the thickness of the current-sheet, and we know that for points within a conductor carrying a current there is no such thing as a magnetic potential.

The value of $\Omega$ is discontinuous at the current-sheet, for if $\Omega _{1}$ is its value at a point just within the current-sheet, and $\Omega _{2}$ its value at a point close to the first but just outside the current-sheet,

$\Omega _{2}=\Omega -1+4\pi \phi$,

where $\phi$ is the current-function at that point of the sheet.

The value of the component of magnetic force normal to the sheet is continuous, being the same on both sides of the sheet. The component of the magnetic force parallel to the current-lines is also continuous, but the tangential component perpendicular to the current-lines is discontinuous at the sheet. If $s$ is the length of a curve drawn on the sheet, the component of magnetic force in the direction of $ds$ is, for the negative side, ${\frac {d\Omega _{1}}{ds}}$, and for the positive side, ${\frac {d\Omega _{2}}{ds}}={\frac {d\Omega _{1}}{ds}}+4\pi {\frac {d\phi }{ds}}$.

The component of the magnetic force on the positive side therefore exceeds that on the negative side by $4\pi {\frac {d\phi }{ds}}$. At a given point this quantity will be a maximum when $ds$ is perpendicular to the current-lines.

On the Induction of Electric Currents in a Sheet of Infinite Conductivity.

654.] It was shewn in Art. 579 that in any circuit

$E={\frac {dp}{dt}}+Ri$,

where $E$ is the impressed electromotive force, $p$ the electrokinetic momentum of the circuit, $R$ the resistance of the circuit, and $i$ the current round it. If there is no impressed electromotive force and no resistance, then ${\frac {dp}{dt}}=0$, or $p$ is constant.