Chapter 4

∆A
∆Ω ≡ (4.2.6)
r
Solid angles are dimensionless quantities measured in steradians (sr). Since the surface
rea of the sphere is S
2
a 4π r , the total solid angle subtended by the sphere is
1
1
2
1
2
1
1
2
1
4π
r
Ω= = 4π
(4.2.7)
r
The concept of solid angle in three dimensions is analogous to the ordinary angle in two
dimensions. As illustrated in Figure 4.2.5, an angle
∆ ϕ is the ratio of the length of the
arc to the radius r of a circle:
Figure 4.2.5 The arc
s
ϕ
r
∆
∆ = (4.2.8)
∆ s subtends an angle ∆ ϕ .
Since the total length of the arc is s= 2π
r,
the total angle subtended by the circle is
2π
r
ϕ = = 2π
(4.2.9)
r
In Figure 4.2.4, the area element ∆A2 makes an angle θ with the radial unit vector
en the solid angle subtended by A ∆
th is
2
∆A ⋅rˆ∆A cosθ
∆A
∆Ω = = =
2 2 2n
2
r2 2
r2 2
r2
ˆr ,
(4.2.10)
where 2n 2 cos
A A θ
∆ =∆ is the area of the radial projection of 2 A ∆ onto a s econd sphere S2 of radius r 2 , concentric with S 1 .
As shown in Figure 4.2.4, the solid angle subtended is the same for both ∆A and ∆ A :
1 2n
5