Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/60

the closed curve, and not on the form of the surface which is bounded by it, it must be possible to determine the induction through a closed curve by a process depending only on the nature of that curve, and not involving the construction of a surface forming a diaphragm of the curve.

This may be done by finding a vector ${\mathfrak {A}}$ related to ${\mathfrak {B}}$, the magnetic induction, in such a way that the line-integral of ${\mathfrak {A}}$, extended round the closed curve, is equal to the surface-integral of ${\mathfrak {B}}$, extended over a surface bounded by the closed curve.

If, in Art. 24, we write F, G, H for the components of ${\mathfrak {A}}$, and a, b, c for the components of ${\mathfrak {B}}$, we find for the relation between these components

$a={\frac {dH}{dy}}-{\frac {dG}{dz}},\quad b={\frac {dF}{dz}}-{\frac {dH}{dx}},\quad c={\frac {dG}{dx}}-{\frac {dF}{dy}}$

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The vector ${\mathfrak {A}}$, whose components are F, G, H, is called the vector-potential of magnetic induction. The vector-potential at a given point, due to a magnetized particle placed at the origin, is numerically equal to the magnetic moment of the particle divided by the square of the radius vector and multiplied by the sine of the angle between the axis of magnetization and the radius vector, and the direction of the vector-potential is perpendicular to the plane of the axis of magnetization and the radius vector, and is such that to an eye looking in the positive direction along the axis of magnetization the vector-potential is drawn in the direction of rotation of the hands of a watch.

Hence, for a magnet of any form in which A, B, C are the components of magnetization at the point x y z, the components of the vector-potential at the point ξ η ζ, are

${\begin{matrix}F=\iiint {\left(B{\frac {dp}{dz}}-C{\frac {dp}{dy}}\right)dxdydz},\\G=\iiint {\left(C{\frac {dp}{dx}}-A{\frac {dp}{dz}}\right)dxdydz},\\H=\iiint {\left(A{\frac {dp}{dy}}-B{\frac {dp}{dx}}\right)dxdydz};\end{matrix}}$

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where p is put, for conciseness, for the reciprocal of the distance between the points (ξ η ζ,) and (x, y, z), and the integrations are extended over the space occupied by the magnet.

406.] The scalar, or ordinary, potential of magnetic force, Art. 386, becomes when expressed in the same notation,

$V=\iiint {\left(A{\frac {dp}{dx}}+B{\frac {dp}{dy}}+C{\frac {dp}{dz}}\right)dxdydz}.$

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