Home | Survey | Topics | Index | References | Dictionary | Contributing | Gallery | Community
| Solid angle | Prev topic | Next topic Fig. 1, Fig. 2 |
The solid angle extends the meaning of an angle in a plane (two dimensions) to an angle in space (three dimensions). It is defined as a ratio, ω, of an area, A, of a region of a spherical surface, to the square of the surface radius, R:
| ω = A / R 2 | (1) |
The "unit" of the solid angle is steradian, abbreviated sr. Note that the steradian is a non-dimensional unit and can be omitted. By defintion, the area A subtends the solid angle, ω, at the center of the sphere. A simple case of the solid angle of a spherical cap is shown in Fig. 1. The concept of the solid angle is crucial in radiometry where it is used to quantify the power of radiation propagating within a range of directions in space.
Note that the solid angle, ω, which is subtended by an area A at the surface of a sphere, is also subtended by any three-dimensional surface, S, composed of oriented elements, dA, whose central projection onto the sphere surface has an area A. This leads to a generalized definition of the solid angle as a surface integral over S, of the ratio of the projected surface element dA = ξ · dA, where ξ is the direction from the projection center to the element, to the square of the distance of that element from the projection center, R.
An exact expression for the solid angle suspended by a spherical cap of area, A, can be derived as follows, in reference to Fig. 2:
| ω | = A / R 2 | |
| = [ ∫0φ 2π R 2 sinφ dφ ] / R 2 | ||
| = 2π (1 - cosφ) | (2) |
It follows from Eq. 2 that the solid angles subtended by a hemispherical surface and the entire sphere surface at the sphere center equal 2π and 4π, respectively.
In the first approximation, when the apex half-angle (Fig. 1), φ [rad] << 1, the solid angle subtended by a spherical cap with an area A can be expressed as follows:
| ω | = A / R 2 | |
| = π r 2 / R 2 | ||
| = π φ 2 | (3) |
because r ≈ R φ in this case. At φ = 1° (~17.5 mrad) the relative error of the approximation of the solid angle by Eq. 3 equals ~0.0025%. Even at φ = 10° (~175 mrad) that error equals ~0.25%.
| CITATION: Jonasz M. 2007. Solid angle (www.tpdsci.com/Tpc/RdmSldAng.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 01-Mar-2007 Modified: 23-Feb-2008 Peer-reviewed: PENDING |
| Copyright 2005-2008 MJC Optical Technology. All rights reserved. | Terms of use | Menu |