dA θ d ω Ω n Fig. 2.6: The flux density of radiation carried by a beam in the direction ˆ Ω through a surface element dA is proportional to cos θ = ˆ n · ˆ Ω . 2.7.3 Relationship between Flux and Intensity We previously defined the flux F as the total power incident on a unit area of surface. We then defined intensity in terms of a flux contribution arriving from a very small element of solid angle d ω centered on a given direction of propagation ˆ Ω . It follows that the flux incident on, passing through, or emerging from an arbitrary surface is given by an integral over the relevant range of solid angle of the intensity. Let us start by considering the flux emerging upward from a hor- izontal surface: it must be an integral of the intensity I ( ˆ Ω ) over all possible directions ˆ Ω directed skyward; i.e., into the 2 π steradi- ans of solid angle corresponding to the upper hemisphere. There is one minor complication, however. Recall that intensity is defined in terms of flux per unit solid angle normal to the beam . For our horizon- tal surface, however, only one direction is normal; radiation from all other directions passes through the surface at an oblique angle (Fig.
Flux and Intensity 47 2.6). Thus, we must weight the contributions to the flux by the co- sine of the incidence angle relative to the normal vector ˆ n . For the upward-directed flux F ↑ , we therefore have the following relation- ship: F ↑ = integraldisplay 2 π I ↑ ( ˆ Ω ) ˆ n · ˆ Ω d ω . (2.57) The above expression is generic: it doesn’t depend on one’s choice of coordinate system. In practice, it is convenient to again introduce spherical polar coordinates, with the z -axis normal to the surface: F ↑ = integraldisplay 2 π 0 integraldisplay π /2 0 I ↑ ( θ , φ ) cos θ sin θ d θ d φ , (2.58) where we have used (2.49) to express d ω in terms of θ and φ . For the downward flux, we integrate over the lower hemisphere, so we have F ↓ = − integraldisplay 2 π 0 integraldisplay π π /2 I ↓ ( θ , φ ) cos θ sin θ d θ d φ . (2.59) Since I is always positive, the above definitions always yield posi- tive values for F ↑ and F ↓ . Key fact: For the special case that the intensity is isotropic — that is, I is a constant for all directions in the hemisphere, then the above integrals can be evaluated to yield F = π I . (2.60) Key fact: The net flux is defined as the difference between upward- and downward-directed fluxes: F net ≡ F ↑ − F ↓ , (2.61) which can be expanded as F net = integraldisplay 2 π 0 integraldisplay π 0 I ( θ , φ ) cos θ sin θ d θ d φ = integraldisplay 4 π I ( ˆ Ω ) ˆ n · ˆ Ω d ω . (2.62)
48 2. Properties of Radiation Note, by the way, that the notation used throughout this subsec- tion implies that we are relating a broadband intensity to a broadband flux . Identical relationships hold between the monochromatic inten- sity I λ and the monochromatic fluxes F ↑ λ and F ↓ λ .
You've reached the end of your free preview.
Want to read all 73 pages?