Begin2.DVI

( v) A unit sphere about the origin.

(vi) Connect the points on the boundary of dS to the origin and form a cone which

will intersect both the unit sphere and the sphere of radius r.

(vii) Use the dot product given by ˆen·
r =rcos θto find the angle θbetween the unit

normal to the surface and the position vector
r.

An example using the above constructions is illustrated in the figure 9-6. In

the figure 9-6, the cone, constructed using the boundary of the element of surface

area dS , intersects the sphere of radius rto produce an element of surface area dΩ.

The element of surface area dΩcan also be thought of as the projection of dS onto

the sphere of radius r. This projection is given by dΩ = cos θ dS where θis the angle

between the normal to the surface and the position vector
r.

Figure 9-6. Solid angle as surface area on unit sphere.

The solid angle subtended at the origin does not depend upon the size of the sphere

about the origin and so one can write

dω

(1)^2 =

dΩ

r^2 =⇒ dω =

dΩ

r^2

Using the result ˆen·
r =rcos θor cos θ= ˆen·
r

r

one obtains

dΩ = cos θ dS =eˆn·
r

r

dS =
r ·d

S


r

where dS
=ˆendS is a vector element of area.