Graphics Reference
In-Depth Information
Stefan-Boltzmann law, the second filament must be much smaller than the first.
Which one emits more power in the
invisible
portion of the spectrum?
Exercise 26.8:
A student is given data captured from a digital camera; the
scene viewed by the camera is a spherical frosted incandescent lamp that emits
1.2 W of power in the visible range. The lamp has a radius of 0.0175m. The
camera shutter speed has been adjusted so that the sensor is neither saturated
nor starved; indeed, the values in the image are near the center of the camera's
pixel-value range, and can be assumed to be proportional to radiance. The value
observed for pixels that lie in the lamp is 4000 (on a scale from 0 to 8191). The
student wants to know the constant of proportionality between the radiance of
arriving light and the sensor value. The student says, “The lamp is supposed to
emit uniformly because of its frosting, so except for really tangential angles, we
can figure that the radiance of all outgoing rays is constant. The area of the lamp
is 4
0.000 962m
2
; the hemisphere over which the light is radiated has solid
angle measure approximately 6.28 sr, so the radiance, by division, is the power
divided by the angle and the area, giving
L
r
2
π
≈
≈
/
(
0.000 962m
2
∗
6.28 sr
)
≈
1.2W
/
m
2
sr. So to get the radiance
L
from the sensor value, we multiply by
0.006W
10
−
6
.” Critique the student's approach, and give the correct answer.
Exercise 26.9:
We said that “A 'disk' consisting of all points on the unit sphere
whose spherical distance from a point
P
is less than
r
(where
r
0.006
4000
=
1. 5
×
<π
) has solid
angle measure 2
cos(
r
))
.” But you'd expect, for small
r
, that the solid angle
would contain an
r
2
factor, because the formula for the area of a disk in the plane
contains an
r
2
factor. Reconcile the two by recalling the Taylor series for
cos(
r
)
at
r
=
0.
Exercise 26.10:
Consider a disk of radius
s
in the plane, centered at the origin,
and the point
P
=(
0, 0,
π
(
1
−
h
)
that's distance
h
below the disk. Assume that the disk
is a Lambertian emitter, emitting radiance
L
in every direction from every point,
and is the only surface in the scene.
(a) Write out an integral for the irradiance at
P
.
(b) Evaluate the integral. Switching to polar coordinates on the plane will help.
(c) Show that if
h
<
s
/
−
10, then the irradiance at
P
is essentially the same as it
would be if the disk covered the entire plane (i.e., if
s
were very large). Thus, for
small values of
h
, the irradiance is nearly constant.
(d) Show that for
h
h
)
2
.
Exercise 26.11:
Show that any plane wave
E
traveling along the
x
-axis as in
Equation 26.6 can be expressed as the sum of two plane waves,
E
|
and
E
◦
,the
first being linearly polarized and the second being circularly polarized. Hints: The
axis of the linearly polarized wave
will be
0
>
4
s
, the irradiance is within 1% of
π
L
(
s
/
A
z
T
; the magnitude of the
A
y
circularly polarized wave will be
A
y
+
A
z
.
Exercise 26.12:
Suppose that in Figure 26.11, we draw a circle of radius
r
about the refraction point. On the line tangent to the top of this circle, we place
equispaced points, and from each point draw a ray toward the refraction point.
These rays refract with different angles into the lower material. Each refracted ray
meets a horizontal line tangent to the
bottom
of the circle.
(a) Draw a figure depicting this situation.
(b) Show that the points of intersection on the bottom line are also equispaced.
(c) What is the ratio of the bottom-line spacing to the top-line spacing?
Exercise 26.13:
Use Planck's formula for blackbody radiation
R
(
f
,
T
)
in
terms of frequency to approximate the location of the power peak in
R
(
f
,
T
)
as a
