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
.

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