Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/74

${\begin{aligned}\alpha &=-{\frac {d\omega }{d\xi }}={\frac {dH}{d\eta }}-{\frac {dG}{d\zeta }},\\\beta &=-{\frac {d\omega }{d\eta }}={\frac {dF}{d\zeta }}-{\frac {dH}{d\xi }}\\\gamma &=-{\frac {d\omega }{d\zeta }}={\frac {dG}{d\xi }}-{\frac {dF}{d\eta }}.\end{aligned}}$

(11)

The quantities F, G, H are the components of the vector-potential of the magnetic shell whose strength is unity, and whose edge is the curve s. They are not, like the scalar potential ω, functions having a series of values, but are perfectly determinate for every point in space.

The vector-potential at a point P due to a magnetic shell bounded by a closed curve may be found by the following geometrical construction:

Let a point Q travel round the closed curve with a velocity numerically equal to its distance from P, and let a second point R start from A and travel with a velocity the direction of which is always parallel to that of Q, but whose magnitude is unity. When Q has travelled once round the closed curve join AR, then the line AR represents in direction and in numerical magnitude the vector-potential due to the closed curve at P.

Potential Energy of a Magnetic Shell placed in a Magnetic Field.

423.] We have already shewn, in Art. 410, that the potential energy of a shell of strength φ placed in a magnetic field whose potential is V, is

$M=\phi \iint {\left(l{\frac {dV}{dx}}+m{\frac {dV}{dy}}+n{\frac {dV}{dz}}\right)dS},$

(12)

where l, m, n are the direction-cosines of the normal to the shell drawn from the positive side, and the surface-integral is extended over the shell.

Now this surface-integral may be transformed into a line-integral by means of the vector-potential of the magnetic field, and we may write

$M=-\phi \int {\left(F{\frac {ds}{ds}}+G{\frac {dy}{ds}}+H{\frac {dx}{ds}}\right)ds},$

(13)

where the integration is extended once round the closed curve s which forms the edge of the magnetic shell, the direction of ds being opposite to that of the hands of a watch when viewed from the positive side of the shell.

If we now suppose that the magnetic field is that due to a