Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/331

CHAPTER XIV.

CIRCULAR CURRENTS.

Magnetic Potential due to a Circular Current.

694.] The magnetic potential at a given point, due to a circuit carrying a unit current, is numerically equal to the solid angle subtended by the circuit at that point; see Arts. 409, 485.

When the circuit is circular, the solid angle is that of a cone of the second degree, which, when the given point is on the axis of the circle, becomes a right cone. When the point is not on the axis, the cone is an elliptic cone, and its solid angle is numerically equal to the area of the spherical ellipse which it traces on a sphere whose radius is unity.

This area can be expressed in finite terms by means of elliptic integrals of the third kind. We shall find it more convenient to expand it in the form of an infinite series of spherical harmonics, for the facility with which mathematical operations may be performed on the general term of such a series more than counterbalances the trouble of calculating a number of terms sufficient to ensure practical accuracy.

Fig. 46.

For the sake of generality we shall assume the origin at any point on the axis of the circle, that is to say, on the line through the centre perpendicular to the plane of the circle.

Let $O$ (Fig. 46) be the centre of the circle, $C$ the point on the axis which we assume as origin, $H$ a point on the circle.

Describe a sphere with $C$ as centre, and $CH$ as radius. The circle will lie on this sphere, and will form a small circle of the sphere of angular radius $a$.