Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/69

due to a magnetic shell is equal to the solid angle subtended by the edge of the shell multiplied by the strength of the shell. As we shall have occasion to refer to solid angles in the theory of electric currents, we shall now explain how they may be measured.

Definition. The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. This area is to be reckoned positive or negative according as it lies on the left or the right-hand of the path of the radius vector as seen from the given point.

Let (ξ, η, ζ) be the given point, and let (x, y, z) be a point on the closed curve. The coordinates x, y, z are functions of s, the length of the curve reckoned from a given point. They are periodic functions of s, recurring whenever s is increased by the whole length of the closed curve.

We may calculate the solid angle ω directly from the definition thus. Using spherical coordinates with centre at (ξ, η, ζ), and putting

$x-\xi =r\sin \theta \cos \phi ,\quad y-\eta =r\sin \theta \sin \phi ,\quad z-\zeta =r\cos \theta ,$

we find the area of any curve on the sphere by integrating

$\omega =\int {(1-\cos \theta )d\phi },$

or, using the rectangular coordinates,

$\omega =\int {\phi }-\int _{0}^{s}{{\frac {z-\zeta }{r^{3}}}\left[(x-\xi ){\frac {dy}{ds}}-(y-\eta ){\frac {dx}{ds}}\right]ds},$

the integration being extended round the curve s.

If the axis of z passes once through the closed curve the first term is 2π. If the axis of z does not pass through it this term is zero.

418.] This method of calculating a solid angle involves a choice of axes which is to some extent arbitrary, and it does not depend solely on the closed curve. Hence the following method, in which no surface is supposed to be constructed, may be stated for the sake of geometrical propriety.

As the radius vector from the given point traces out the closed curve, let a plane passing through the given point roll on the closed curve so as to be a tangent plane at each point of the curve in succession. Let a line of unit-length be drawn from the given point perpendicular to this plane. As the plane rolls round the