A general cone is described by a line having one point fixed in space which is free to
rotate. The figure 9-5 illustrates two cones which differ from a right circular cone.
Figure 9-5. Cones generated by moving line about fixed point.
Consider a sphere of radius rand use the origin of the sphere to
construct a cone which intersects the sphere and cuts out an area
S on the surface as illustrated in the accompanying figure. The
area Son the surface of the sphere of radius rwill be proportional
to r^2 since Sis some fraction of the total surface area 4 πr^2. The
ratio S
r^2is therefore a dimensionless ratio and the quantity Ω = S
r^2is called the solid angle subtended at the center of the sphere by the
cone. The solid angle is a measure of how large an object appears to be when viewed
from the origin of the sphere. Solid angles are measured in units called steradians^6
(abbreviated sr) and by definition 1 steradian is the solid angle represented by the
surface area of a sphere equal to the radius of the sphere squared. For example, if
the area Sin the accompanying figure equals r^2 , then the solid angle subtended at
the center of the sphere is said to be 1 steradian. The total solid angle about the
center of the sphere being 4 πsteradians.
For a given oriented surface make the following constructions.
(i) A position vector r from the origin to the point on the oriented surface.
(ii) An element of surface area dS at the terminus of the vector r.
(iii) A unit normal eˆn to the surface at the terminus of the vector r.
(iv) A sphere of radius r=|r |centered at the origin 0.
(^6) The solid angle is really dimensionless and sometimes the terminology of steradians is not used.