Graphics Reference
In-Depth Information
The other notable feature in the graph is that the location of the peak radiation
intensity moves to the left as the temperature increases. At about 900
◦
K, there is
enough radiation in the visible portion of the spectrum for the eye to detect it. As
a first peek at color, we mention one thing: Radiation with a wavelength between
about 400 nanometers and 700 nanometers is visible to the human eye; radiation
at the 700-nanometer end of the spectrum looks red, and the appearance transi-
tions through yellow, green, and cyan as the wavelength shortens (i.e., the energy
increases); and radiation near the 400-nanometer end of the visible spectrum
looks blue. Since at 900
◦
K the radiated energy at the low-frequency (i.e., long-
wavelength) end of the visible spectrum is larger than that at the high-frequency
end, we see such an object glowing a dull red. As we heat it further, it becomes a
great deal brighter because of the exponent of 4 in the Stefan-Boltzmann law. But
higher frequencies begin to mix in, and we see a combination of red and green
(i.e., an orange and then a yellow color) and eventually a combination of red,
green, and blue, which we perceive as white. By the time an object is glowing
white, it's emitting energy at an amazing rate; at 5000
◦
K, it's radiating at about 35
megawatts per square meter. Clearly this radiation dominates whatever light the
surface might reflect from the ordinary illumination in a room, for instance.
By the way, lamps used in filmmaking and photography are often described
using temperatures; that's shorthand for saying, “The spectrum of light emitted by
this lamp is quite similar to that of black-body radiation of that temperature.” This
can be useful in adjusting a scene to appear illuminated by ordinary incandescent
lamps or by sunlight.
Max Planck developed an expression for the shape of the curve in the graph
above, later supported by theoretical analysis based on quantum theory; he
observed that
1
λ
1
I
(
λ
,
T
)
∝
;
5
hc
λ
e
−
1
kT
where
h
is Planck's constant and
k
is
Boltzmann's constant
(about
10
−
23
JK
−
1
). The precise values are not important to us, but the shape of
the curve is. Because
e
x
=
1
+
x
+
1.38
×
...
, the denominator of the second factor is, for
large
λ
, roughly proportional to 1
/λ
,so
I
(
λ
,
T
)
is proportional to
λ
−
4
; for small
λ
,
the exponential dominates and the curve heads to zero. Note that
I
(
λ
,
T
)Δ
λ
is the
amount of energy at wavelengths between
;
to find the total energy in some range of wavelengths, you have to integrate with
respect to
λ
and
λ
+Δ
λ
, for small values of
Δ
λ
over that range. The corresponding expression, in terms of frequency,
which is the more common descriptor used for light in physics, is
λ
f
3
e
hf
/
kT
1
,
in which frequency appears to the third power, while wavelength appeared to the
fifth power; this is because integration with respect to
f
involves a change of vari-
ables from
R
(
f
,
T
)=
−
f
2
df
.
λ
to
f
, namely,
λ
=
c
/
f
,
d
λ
=
−
c
/
As mentioned earlier, light is a kind of electromagnetic radiation. (Indeed, “light”
is a general term for this, with “visible light” being the radiation that the human
eye can detect. We'll generally follow common usage and mean “visible light”
