ArticlePDF Available
Resolution of the ellipsoid paradox in thermodynamics
Theodore J. Yoder and Gregory S. Adkins
a)
Franklin & Marshall College, Lancaster, Pennsylvania 17604
(Received 5 February 2011; accepted 12 May 2011)
We discuss a challenge to the second law of thermodynamics in an optical setting, in which two
black bodies at strategically chosen points inside a perfectly reﬂecting cavity of appropriate shape
apparently transfer energy asymmetrically so that one body experiences a net gain of energy at the
other’s expense. We show how the ﬁnite sizes of the black bodies lead to a resolution of the
apparent paradox. We describe a simulation that allows us to follow the paths of individual rays
and show numerically that the second law requirement of energy balance is satisﬁed. We also
demonstrate that the energy balance condition is satisﬁed in the more general situation where the
cavity and black bodies are of arbitrary shape.
V
C
2011 American Association of Physics Teachers.
[DOI: 10.1119/1.3596430]
I. INTRODUCTION
Imagine a perfectly reﬂecting cavity composed of a sec-
tion of an ellipsoid E
1
[see Fig. 1(a)] with foci at A and B,a
section of a sphere S centered at B, and a section of a larger
ellipsoid E
2
also with foci A and B. At the focal points A and
B place two identical small black bodies at equal tempera-
tures. If we construct the cavity so that no ray from A reaches
the sphere directly, then all of the thermal energy emitted
from A ends at B (in the ray optics approximation) and is
absorbed there, while some of the energy emitted from B is
reﬂected back to B by the spherical section. There is thus a
net ﬂow of energy from A to B, which could be harnessed to
perform useful work. If the energy is not extracted, body B
will become hotter than A. The equilibrium temperatures
must satisfy R
2
A
T
4
A
¼ R
2
B
½1 # X
s
=ð4pÞ&T
4
B
, where R
A
and R
B
are the radii of sources A and B (assumed to be equal, for
now), and X
S
is the solid angle subtended by the spherical
section S as viewed from B. We conclude that there is a vio-
lation of the second law of thermodynamics and a perpetual
motion machine of the second kind.
Variants of the two-ellipsoid conﬁguration we have
described have been proposed and discussed over the
years.
117
The most common variant involves a single small
ellipsoid E
1
and a sphere S as in Fig. 1(b). In this case it is
not as easy to ﬁnd the fraction of energy from source A that
eventually reaches B and the fraction that returns to A, but it
is clear that there is a net ﬂow from A to B. The early discus-
sions of the ellipsoid paradox that we found until that of
Palmer
14
all refer to the version of Fig. 1(b).
The earliest reference to the ellipsoid paradox known to us
was by Fallows in 1959,
1
suggesting a solution. Osborne
2
stated that B will “emit a
correspondingly greater amount of radiant energy at each
wavelength” than A to maintain energy balance. (Osborne’s
solution seems to imply violation of the second law.) Dry-
burgh
3
stated that the explanation “involves only the princi-
ple of rad iation traveling with equal ease in either direction
along a given path.” Bisacre
4
pointed out that “the geometry
of the problem breaks down” if A and B are ﬁnite bodies
context by Greenleaf,
5
and generated much discussion.
68
Guggenheim
6
and then Landsberg
8
summarized the corre-
spondence and focused on a resolution based on the ﬁnite
size of the sources. In an appendix to Ref. 8, Davies pre-
sented an exact calculation showing that for equal source
radii in the small source limit (and for a restricted class of
geometries), the effective solid angle in Fig. 1(b) at source A
for emission of rays absorbed by B equals the solid angle at
B for emission of rays absorbed by A. Landsberg also gave a
by Clausius, Kelvin, Rankine, and Gibbs. Landsberg noted
that a colleague referred to the conﬁguration of Fig. 1(b) as
the “Chinese furnace,” but could not track down the origin
of the name. The ellipsoid paradox was posed as a “brain
teaser” by Helsdo n,
9
but the solution given was too brief to
be clear. An expanded response was provided by Higbie
10
with some related versions of the paradox with different geo-
metries. Boley and Scully
11
presented a detailed discussion
of the paradox of Fig. 1(b), taking into account not only the
internal states of atoms at A and B (assumed to have two
energy states each), but also the radiation in the cavity. They
found that A and B approach a common temperature at large
times. Welford and Winston
12
commented that the extensive
statistical mechanics treatment is unnecessary and presented
Landsberg
13
followed up with a brief note containing refer-
ences to prior discussions. Palmer
14
posted a clear discussion
of the paradox in the form of Fig. 1(a). This document is the
ﬁrst one we have found to use the two-ellipsoid geometry.
Unfortunately, there are no references. Palmer’s version of
the paradox was post ed and discussed in Ref. 15. Muta-
lik
16,17
recently posted the Chinese furnace paradox and a re-
solution in the New York Times.
As mentioned, there have been several proposed resolu-
tions to the ellipsoid paradox. One concerns the ﬁnite size of
the sources. Bodies A and B must have a ﬁnite size to have a
ﬁnite heat capacity, without which the paradox would
become meaningless. Another resolution relaxes the ideali-
zation of a perfectly reﬂecting and non-absorbing shell. A
physical shell would absorb to some extent and, due to its
nonzero temperature, would also emit thermal radiation into
the cavity. After waiting long enough, there would be equi-
librium between sources and the shell at a common tempera-
ture. Other suggestions involve the ray optics approximation
for energy propagation without considering diffraction, the
neglect of quantum effects, and the neglect of ﬁeld modes
inside the cavity.
We consider the paradox in the context of ray optics inside
of a perfectly reﬂecting shell and focus on the effects of ﬁ-
nite size sources. We will show that their ﬁnite size is cru-
cial, and that the paradox can be resolved simply by means
811 Am. J. Phys. 79 (8), August 2011 http://aapt.org/ajp
V
C
2011 American Association of Physics Teachers 811
of geometry and ray optics, as is the consensus of many of
the previous discussions of the paradox. The main point is
illustrated in Fig. 2, which shows the loss of focus for rays
propagating from the focal point A. These rays fail to focus
precisely at point B. Depending on the size of body B, sev-
eral rays miss body B and continue to make (possibly many)
reﬂections until they reach body A or B. The paradox is
resolved by the lack of focus for ﬁnite sized sources, even in
the limit that the sources are small compared to the size of
the cavity.
Our contribution to the resolution of the ellipsoid paradox
is to verify that the second law is satisﬁed quantitatively so
that the transfer of energy from A to B equals that from B to
A for a variety of source radii and cavity conﬁgurations. Our
simulation follows the propagation of rays from A to B and B
to A to show that the energy transfer is the same in each
direction. We can follow the path of an individual ray
through sometimes hundreds of reﬂections to see where it is
ﬁnally absorbed. We also give an analytical analysis showing
exact energy ﬂow balance in a more general geometry.
II. RAY OPTICS SIMULATION
We wrote a computer program to follow the path of an ar-
bitrary ray emitted from either one of the sources. We used
the program to verify in quantitative detail the suggestion
that the resolution of the paradox resides in the ﬁnite sizes of
the sources by showing that the energy emitted by source A
and absorbed by B equals the energy emitted by B and
absorbed by A. We found that the energy ﬂows are equal only
when the angular distribution of the emitted energy follows
Lambert’s emission law, which states that the angular distri-
bution of radiated energy is proportional to cos h dX where h
is the angle between the direction of emission and the surface
normal and dX is the element of solid angle.
18
The necessity
of including the cos h factor is not surprising given that Lam-
bert’s law is a part of the complete black body emission law,
which follows from the second law of thermodynamics.
We considered the paradox in both two- and three-dimen-
sional forms. The two-dimensional variant is simpler to set up
and the behavior of rays in the two-dimensional situation is
signiﬁcantly easier to illustrate. The three-dimensional variant
is more physical, being ﬁnite in extent, and is the version of
the paradox that was originally proposed in the literature. In
two dimensions, diagrams such as those of Fig. 1 are cross-
sections, with inﬁnite extent in the direction (z) transverse to
the page. The sources are right circular cylinders, and the shell
is formed from elliptical cylinders and another right circular
cylinder. Rays are emitted in all directions from every source
point, but it sufﬁces to consider the projection of the path of
the rays on the x-y plane. The z motion is a trivial trip along
the z-axis. In three dimensions, the shell is the surface of revo-
lution of the outline shown in Fig. 1 and the sources are
spheres. The three-dimensional simulation is not signiﬁcantly
more difﬁcult to handle than the two-dimensional case. We
describe the three-dimensional simulation here.
The geometry of Fig. 1(a) can be deﬁned by specifying
just three values: the semi-major axis of the small ellipse a
1
,
the eccentricity of the small ellipse !
1
, and the angle a
between the symmetry axis and a point on the spherical sec-
tion S nearest to A (see the Appendix). An alternative choice
Fig. 2. A spread of 11 rays leaves body A from a point on its surface. These
rays do not come to a focus at the center of body B, and several of the rays
do not hit body B at all. The dots show where each ray hits after one reﬂec-
tion. Of the four that miss B after one reﬂection, three are eventually
absorbed by A and one by B. Rays that reﬂect off E
2
must intersect body B
(which in this example has the same radius as A) because the distance from
the point of reﬂection to B is less than the distance from A to the point of
reﬂection.
Fig. 1. Geometric conﬁgurations for two variants of the ellipsoid paradox, shown in cross-section. Sources A and B are located at the foci of an ellipsoid sec-
tion E
1
. Attached to E
1
is a section of a sphere S with its center at B. In the two-ellipsoid geometry of (a) there is a second and larger ellipsoid section, E
2
, with
the same foci as E
1
. The spherical section in (a) is positioned and constrained so that no rays coming directly from A can hit anywhere on S. The “Chinese
furnace” geometry of (b) is formed entirely of the ellipsoid E
1
and sphere S. The actual cavities are surfaces of revolution obtained from these cross-sections
by revolution on the line containing A and B.
812 Am. J. Phys., Vol. 79, No. 8, August 2011 T. J. Yoder and G. S. Adkins 812
of the third parameter is n = R/a
1
, where R is the radius of the
spherical section of the shell. For the physical situation with
non-zero source radii, the ﬁve parameters a
1
, !
1
, a (or n), and
the radii of the sources sufﬁce to determine all other geomet-
ric quantities in the system.
At the heart of our simulation is an iterative step that takes
a ray from an initial location
~
r
i
¼ðx
i
; y
i
; z
i
Þ inside (or on the
inside surface of) the cavity to a ﬁnal location
~
r
f
¼ðx
f
; y
f
; z
f
Þ on one of the surfaces. The initial direction
of the ray is speciﬁed by a direction unit vector
^
u
i
¼ðu
xi
; u
yi
; u
zi
Þ. After one step the ray will either be
absorbed by A or B if it intercepts one of them, or it will be
reﬂected by the inner surface of E
1
, E
2
, or S. It will have a
new position, and if reﬂected, a new direction unit vector
determined using the law of reﬂection. For each initial set
~
r
i
,
^
u
i
the path of the ray is given parametrically by
~
rðkÞ¼
~
r
i
þ k
^
u
i
: (1)
In three dimensions, let the foci A and B deﬁne the x-axis.
The equation for an ellipsoid centered at
~
r
0
¼ðx
0
; y
0
; z
0
Þ
with circular x cross-section is
ðx # x
0
Þ
2
a
2
þ
ðy # y
0
Þ
2
b
2
þ
ðz # z
0
Þ
2
b
2
¼ 1: (2)
For the surfaces E
1
, E
2
, and S, the values of a and b depend
only on the three parameters a
1
, !
1
, and a (see the Appendix).
We have a = b = R
A
for source A and a = b = R
B
for B. The so-
lution for the intersections of the ray with any one of these
particular surfaces proceeds as follows. Replacing x, y, and z
in Eq. (2) by x(k), y(k), and z(k) given by Eq. (1) results in
an equation quadratic in k. Solving this equation yields val-
ues for k at which the ray (1) intersects the ellipsoid (2). We
next need to remove solutions that represent the intersection
of the ray with any parts of the ellipsoid that are not involved
in the construction of the cavity. Also, we are interested only
in positive solutions, because, by deﬁnition, rays move for-
ward along
^
u
i
, so any negative values for k are deleted.
We used this method to assemble a list of positive values
for k fo r the ﬁrst intersection of the ray with each of the
surfaces: A, B, E
1
, E
2
, and S. If a particular ray does not
intersect one of the surfaces, we added k
max
to the list of can-
max
is greater than the largest dimen-
sion of the cavity. Thus, we have a set of positive k values,
one for each surface involved in constructing the cavity and
two for the sources, from which we choose the minimum
k
min
and advance the ray to the new position
~
r
f
¼
~
r
i
þ k
min
^
u
i
. If
~
r
f
is on A or B the process s tops. Other-
wise, the new direction unit vector is
^
u
f
¼
^
u
i
# 2
^
nð
^
n (
^
u
i
Þ; (3)
where
^
n is the outward-pointing unit normal to the reﬂecting
surface at
~
r
f
. A convenient way to ﬁnd the normal is to use
the fact that it is proportional to
~
rU, where A = 0 is the
equation of the surface of interest. We iterate the basic step
until the ray intersects A or B, at which point it is absorbed.
Some trajectories are shown in Fig. 3. Although we have
shown some interesting paths, most paths have just one
reﬂection like that of Fig. 3(a). The next most common has
two reﬂections. For purposes of calculating means and stand-
ard deviations of quantities of interest, we computed 12 sam-
ples of 200,000 two-dimensional paths starting at A and
ending at B, and the same number from B to A, for each ge-
ometry. The initial angles h (the position of the emitted ray
Fig. 3. Ray diagrams for typical two-dimensional paths, shown in order of increasing numbers of reﬂections. These paths can be thought of as two-dimensional
projections of paths in the cylindrical geometry, or as special cases of the three-dimensional geometry that happen to lie in a plane. The ray shown in (a) is typ-
ical of a single-reﬂection path from A to B. Ray (b) bounces 8 times before returning to A. Ray (c) from B to A has 12 reﬂections. Ray (d) from A to A retraces
its path after a reﬂection at nearly a right angle. Ray (e) from A to B traverses the perimeter of the cavity in a precessing triangle. Ray (f) from B to A enters a
ﬁve-sided nearly resonant situation with 140 reﬂections. Rays (g) from B to A and (h) from A to B form some of the striking patterns that often emerge with
high numbers of reﬂections (here 194 and 352). It is possible to ﬁnd paths that nearly ﬁll the cavity and have many thousands of reﬂections.
813 Am. J. Phys., Vol. 79, No. 8, August 2011 T. J. Yoder and G. S. Adkins 813
on the source) and / (the angle of the emitted ray with
respect to the normal) were chosen randomly subject to the
Lambert emission law. Only a small percentage of paths
have more than 10 or 12 reﬂections. Occasionally an
extremely long path occurs, and, as a result, the mean num-
ber of reﬂections is unexpectedly large. A detailed balance
principle holds, and the numbers for paths from A to B are
the same (within uncertainties) as for paths from B to A. For
example, for the two-ellipsoid geometry of Fig. 1(a) with
a
1
= 2.5, !
1
¼ 0:8, a = p/4, and with source radii
R
A
= R
B
= 0.1, the fractions of paths from A to B having 0,
1,…, 12 reﬂections are 0.0134(1), 0.6700(4), 0.2896(3), 0, 0,
0.0032(1), 0.0054(1), 0, 0.0007(1), 0.0079(1), 0, 0.0022(1),
and 0.0034(1), respectively. (The 1r uncertainties in the last
digit are shown in parentheses.) The average number of
reﬂections is 9.86(17). The numbers for B to A are the same
within the uncertainty.
From the examples shown in Fig. 3 we see that a ray path
can be extremely sensitive to numerical inaccuracy, intro-
duced, for example, by round-off errors. For instance, a
slight variation in the path could change the 352 reﬂection
path of Fig. 3(h) to a much shorter one, and lead to absorp-
tion by one source instead of the other. This sensitivity can
affect even short paths, but it is much more likely for long
ones. Fortunately, long paths are rare. We found noticeable
divergences between the paths generated by different imple-
mentations of the ray tracing simulation after a few hundred
reﬂections, which leads us to distrust the details of the very
long paths. Because most paths are short, our results for
numerical integrals are trustworthy within the given
uncertainty.
The paths of rays leaving the sources obtained using our
two-dimensional simulations are illustrated in Figs. 4 and 5.
These ﬁgures show the ﬁnal destinations of rays emitted
from source A in parts (a) and (b) and from B in parts (d) and
(e) as functions of the angles h and / of the emitted ray. Part
(a) encodes the number of reﬂections undergone by any par-
ticular ray emitted from A and returning to A, as does part
(b) for rays from A to B. Parts (d) and (e) give analogous in-
formation for rays emitted from B. The rays of particular in-
terest are the ones from A to B (shown in colors in part (b)
and from B to A [in color in part (d)]. The red lines in (b) and
(d) represent rays aimed straight at the other source that
reach it without reﬂection.
III. ENERGY FLOW RESULTS
We used our ray optics simulat ion to determine the energy
ﬂow rate between the sources for various geometries. The
power emitted from the area element dA in the frequency
range d" and into the solid angle dX = sin hdhd/ by a black
body source at temperature T is
Fig. 4. (Color) Examples of the fate and number of reﬂections for rays trav-
eling between the sources in the two-dimensional ellipsoid geometry with
a
1
= 2.5, !
1
¼ 0:9, n = 0.9, and R
A
= R
B
= 0.2. Parts (a) and (b) show the desti-
nations of rays given off by source A. The colors in (a) show the number of
reﬂections undergone by rays from A that end on A (with black for rays end-
ing on B) and (b) carries the same information for rays from A to B (with
black for rays ending on A). The angles identify which ray is being fol-
lowed: h describes the position on the source of the emitted ray and / gives
its angle with respect to the normal. The angles are deﬁned so that h = 0 is
the position on one source closest to the other source, and both h and /
increase in the counterclockwise direction. Rays with p )h )0 duplicate
those with 0 )h )p, and so are not shown. Most rays from A to B undergo
one or two reﬂections—a few suffer many reﬂections. A sampling of the col-
ors used to represent numbers of reﬂections is shown on a continuous loga-
rithmic scale in (c). Parts (d) and (e) give the same information as (a) and
(b), only for rays from B to A in (d) and B to B in (e).
Fig. 5. (Color) The same information as in Fig. 4, except for the Chinese
furnace geometry with a
1
= 2.5, !
1
¼ 0:8, a = p/4, R
A
= 0.1, and R
B
= 0.2.
Most of the rays from A to B and from B to A suffer only zero or one reﬂec-
tions while a few undergo many reﬂections. Many of the rays from A to A
have complicated paths.
814 Am. J. Phys., Vol. 79, No. 8, August 2011 T. J. Yoder and G. S. Adkins 814
dP ¼ Ið"; TÞd"dA cos hdX; (4)
where
Ið"; TÞ¼
2h"
3
c
2
1
e
h"=kT
# 1
(5)
is the Planck distribution. Here h is the angle between the
normal to the surface and the direction of the radiation. The
factor cos h is present because black bodies satisfy the Lam-
bert emission law.
18
(A consequence of the presence of cos h
is the fact that the radiance of black bodies is independent of
viewing angle.) The power transfer from A to B in frequency
range d" can be expressed as
dP
A!B
¼ d
~
P
A
pðA ! BÞ; (6)
where the total power radiated from A in frequency range d"
is d
~
P
A
¼ Ið"; TÞd"ðS
A
ÞðpÞ, with S
A
the area of source A,
p ¼
R
dX cos h ¼
R
2p
0
d/
R
p=2
0
dh sin h cos h, and p(A ! B) is
the probability that a ray emitted with the cos h dX distribu-
tion is eventually absorbed by B. The quantities that we
actually calculated were the probabilities p(A ! B) and p(B
! A). We found these to be equal (to within numerical
uncertainties) for sources of equal radii. If the energy trans-
fer balances for each frequency (dP
A!B
= dP
B!A
), we would
expect to ﬁnd
S
A
pðA ! BÞ¼S
B
pðB ! AÞ (7)
or
r *
R
A
R
B
!"
d#1
pðA ! BÞ
pðB ! AÞ
¼ 1; (8)
for the two-dimensional (d = 2) and three-dimensional (d = 3)
cases. We ran 2
5
simulations using the two-ellipsoid or Chi-
nese furnace conﬁgurations, and the two- or three-dimen-
sional geometries. The parameter sets are a
1
= 2.5, !
1
¼ 0:8,
and a = p/4, or a
1
= 2.5, !
1
¼ 0:9, and n = 0.9; 0.1 or 0.2 for
R
A
; and 0.1 or 0.2 for R
B
. Our numerical results for r are con-
sistent with unity to within our uncertainties of a few parts
per thousand. (For each case we did 12 runs of 400,000 rays
emerging from each of the two sources.) We conclude that
the rates of energy ﬂow from A to B and from B to A are
equal, and that there is no paradox.
IV. ANALYTIC ANALYSIS
In this section we give an analytic demonstration for the
energy balance between black bodies in reﬂecting cavities.
The analytic argument is not based on speciﬁc geometric fea-
tures such as spherical sources or specially designed cavities,
and is a signiﬁcant generalization of the previous results.
Consider a perfectly reﬂecting cavity of any shape contain-
ing two black bodies A and B. Rays connecting A and B
undergo some number n = 0,1,2,… of reﬂections, and rays
with a given value of n can be grouped so that all of the rays
within a given group can be continuously deformed into one
another. For example, there are two groups of one-reﬂection
rays between A and B in Fig. 1(a): those hitting E
1
and those
hitting E
2
. The n-reﬂection paths from A to B can be parti-
tioned into inﬁnitesimal elements all of which have propaga-
tion angles that are within Oð! Þ (where ! is an inﬁnitesimal
angle) of each other. A convenient way to perfo rm this parti-
tion is to represent all of the n-reﬂection rays leaving one of
the black bodies on a four-dimensional phase diagram (two
space and two direction coordinates) and partition the n-
reﬂection region into differential elements of Oð! LÞ in each
spatial directi on, where L is a length characteristic of the
sources. For example, a one-reﬂection bundle of rays from
S
1
on A to S
3
on B is shown in Fig. 6.
We now prove the central result that the power P
1
leaving
S
1
and arriving at S
3
is the same as the power P
3
leaving S
3
and arriving at S
1
as lon g as the temperatures of bodies A
and B are the same. The power emitted by S
1
in frequency
range dv equals the radiance factor R
1
times the e´tendue,
19
P
1
¼ R
1
Z
S
1
d
2
r
1
Z
dX
1
cos h
1
; (9)
where S
1
is an inﬁnitesimal element of area on body A and
R
1
: I(",T
1
)d" as in Eq. (4). The points of S
1
are labeled by
the vector
~
r
1
, and d
2
r
1
is the element of area on S
1
. At each
point
~
r
1
on S
1
rays are emitted within a small range of direc-
tions described by dX
1
. The exact directions and total solid
angle of rays emitted from point
~
r
1
are not the same for all
points in S
1
even though S
1
is an inﬁnitesimal element, and
hence the need for the integrals in Eq. (9). The area element
S
1
is small enough that the normal vector
^
n
1
and the cosh
1
factor for all rays in the bundle can be taken to be constant
over S
1.
The integral over the solid angle can be done, giving
the total solid angle DXð
~
r
1
Þ for all rays in the bundle that are
emitted from the point
~
r
1
. (Note that DXð
~
r
1
Þ is of Oð!
2
Þ, and
the e´tendue is of Oð!
4
Þ).
Our ﬁrst task is to show that the e´tendue of the emitted
rays from S
1
is the same as that of the received rays at S
2
.
That is, we need to show that
Z
S
1
d
2
r
1
DXð
~
r
1
Þcos h
1
¼
Z
S
2
d
2
r
2
DXð
~
r
2
Þcos h
2
: (10)
Consider all of the rays leaving point
~
r
1
on S
1
in the inﬁnites-
imal bundle of rays under consideration. These rays ﬁll an
Fig. 6. A bundle of one-reﬂection rays from S
1
on A to S
3
on B by way of S
2
on the reﬂecting envelope. The sizes of S
1
, S
2
, S
3
and, consequently, the
sizes of the solid angles are exaggerated for clarity. The regions S
1
, S
2
, and
S
3
are inﬁnitesimal, so all rays from S
1
to S
2
are approximately parallel, as
are all rays from S
2
to S
3
. The normals
^
n
1
and
^
n
2
to S
1
and S
2
are shown
along with some representative rays.
815 Am. J. Phys., Vol. 79, No. 8, August 2011 T. J. Yoder and G. S. Adkins 815
area element DA
2
on S
2
. The solid angle DXð
~
r
1
Þ can be
expressed as the transverse projection of DA
2
divided by r
2
12
,
where r
12
is the distance between S
1
and S
2
,
DXð
~
r
1
Þ¼
cos h
2
DA
2
r
2
12
¼
cos h
2
r
2
12
Z
S
2
d
2
r
2
Hð
~
r
1
;
~
r
2
Þ: (11)
We have introduced the H function that vanishes unless the
points
~
r
1
and
~
r
2
are connected by a ray in the bundle,
Hð
~
r
1
;
~
r
2
Þ¼
1 if a ray from
~
r
1
reaches
~
r
2
0 otherwise
#
: (12)
Similarly, the solid angle subtended by rays reaching a
generic point
~
r
2
in S
2
can be written as
DXð
~
r
2
Þ¼
cos h
1
DA
1
r
2
12
¼
cos h
1
r
2
12
Z
S
1
d
2
r
1
Hð
~
r
1
;
~
r
2
Þ: (13)
Thus the e´tendue for rays arriving at S
2
is
Z
S
2
d
2
r
2
DXð
~
r
2
Þcos h
2
¼
cos h
1
cos h
2
r
2
12
Z
S
2
d
2
r
2
+
Z
S
1
d
2
r
1
Hð
~
r
1
;
~
r
2
Þ; (14)
which, after reversing the order of integration, is the same as
the e´tendue for rays leaving S
1
. The e´tendue for the reﬂected
2
is the same as for the radiation arriving
there, because the angles (for example, h
2
in Fig. 6) and solid
angles (for example, DXð
~
r
2
Þ in Fig. 6) are both preserved
under reﬂection. An analogous argument shows that the e´ten-
due of the rays arriving at S
3
equals that of rays leaving S
2
,
and thus is the same as the e´tendue of the rays leaving S
1
.
Due to e´tendue conservation and the assumption that no
power is lost upon reﬂection, we can express the power inci-
dent on S
3
as
P
3;in
¼ R
1
Z
S
3
d
2
r
3
DXð
~
r
3
Þcos h
3
: (15)
All is absorbed because B is an ideal absorber.
We now consider the time-reversed bundle of rays from S
3
back to S
1
. Every ray in the original bundle corresponds to a
ray in the reversed bundle having the same path but traced
out in the opposite direction. The power radiated from S
3
along the reversed path is
P
¼ R
3
Z
S
3
d
2
r
3
DXð
~
r
3
Þcos h
3
; (16)
where R
3
= I(",T
3
)d" is the radiance factor of body B. The
incident and radiated powers are balanced in every frequency
range if and only if R
1
¼R
3
, which holds exactly when the
temperatures of A and B are identical.
The argument given for one-reﬂection paths generalizes to
paths with any number of reﬂections. Thus, after integration
of all the inﬁnitesimal ray bundles, we obtain the ﬁnal result
that the energy ﬂow from A to B equals that from B to A as
long as the temperatures of A and B are equal, no matter
what the shapes of the sources or the cavity. Lambert ’s emis-
sion law is an essential element of this delicate balance.
Without the cos h
1
factor in the emitted power in Eq. (9) the
proof would fall apart. Speciﬁcally, because cos h appears
naturally in the solid angle calculations (11) and (13), it has
to be in Eqs. (9) and (10) as well. The detailed balance that
we noticed in the simulations—that the energy ﬂow balances
for each number of reﬂections separately—is a consequence
of the general demonstration. In addition, the general demon-
stration implies energy ﬂow balance on the ﬁner level of in-
ﬁnitesimal bundles of rays.
ACKNOWLEDGMENTS
The authors would like to thank Nina Byers, Amy Lytle,
Etienne Gagnon, and Calvin Stubbins for useful discussions
We acknowledge the support of Franklin & Marshall College
through the Hackman Scholars program.
APPENDIX: GEOMETRY OF THE PAR ADOX
We demonstrate an explicit construction of the geometry
of the paradox. The construction of the geometry of Fig. 1(b)
should be evident from the construction of the more compli-
cated Fig. 1(a), which we now discuss in detail. Three pa-
rameters are necessary to specify the geometry of the shell.
One, which can be taken to be the semi-major axis a
1
of the
Fig. 7. (a) An ellipse with semi-major axis a and semi-minor axis b. The
foci and center are at A, B, and O, and c is half the distance between the
foci. A generic point P is on the ellipse. The ellipse has eccentricity ! ¼ 0:7.
(b) The construction of the ellipsoid paradox geometry using a
1
, !
1
, and a as
parameters. The continuations of ellipsoid cross-section E
1
and sphere cross-
section S are shown as dashed lines.
816 Am. J. Phys., Vol. 79, No. 8, August 2011 T. J. Yoder and G. S. Adkins 816
small ellipse E
1
, serves to set the overall scale and can have
any positive value. For the second parameter we choose the
eccentricity !
1
of the small ellipse. There are two choices of
the third parameter: the angle a shown in Fig. 7(b) or the ra-
tio n = R/a
1
, where R is the radius of the spherical section.
We show the construction in terms of a ﬁrst.
As a quick review, an ellipse of semi-major axis a,
semi-minor axis b, and eccent ricity ! is shown in Fig. 7(a).
The sum of the distances from the foci to any point P on the
perimeter is given by
r
A
þ r
B
¼ 2a: (A1)
Half the distance between the foci is given by c ¼ !a, and
because a
2
= b
2
+ c
2
, it follows that b ¼ a
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 # !
2
p
. The polar
ellipse equation takes either of the forms
r
A
¼
b
2
a # c cos h
A
; (A2a)
r
B
¼
b
2
a þ c cos h
B
: (A2b)
For our construction [see Fig. 7(b)] we choose a
1
> 0 and pa-
rameters !
1
and a such that 0 <!
1
< 1 and 0 < a < p/2, sub-
ject to a constraint to be speciﬁed. Let A and B be the two
foci and O the center of the small ellipse E
1
. Points P
1
and
P
2
delimiting the spherical segment S are on a line from A at
angle a to the centerline given parametrically by
~
rðkÞ¼ðxðkÞ; yðkÞÞ ¼ ð#c; 0Þþkðcos a; sin aÞ; (A3)
with the origin at O and c ¼ !
1
a
1
half the distance between
the foci. Points P
1
and P
2
also lie on the circle
ðx # c Þ
2
þ y
2
¼ R
2
(A4)
of radius R with center at B. The simultaneous solutions to
Eqs. (A3) and (A4) are
k ¼ 2c cos a6
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
R
2
# 4c
2
sin
2
a
p
: (A5)
The distance D between P
2
and P
1
is
D ¼ r
2
# r
1
¼ 2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
R
2
# 4c
2
sin
2
a
p
; (A6)
where r
1
and r
2
are the lengths of the line segments AP
1
and
AP
2
, respectively. We can ﬁnd the radius R of the sphere by
use of the equation,
2a
1
¼ r
1
þ r
0
1
¼ r
1
þ R; (A7)
where r
0
1
¼ R is the length of BP
1
, and the parametric equa-
tion (A2a)
r
1
¼
b
2
1
a
1
# c cos a
¼
ð1 # !
2
1
Þa
1
1 # !
1
cos a
; (A8)
so that
R ¼ 2a
1
# r
1
¼
1 # 2!
1
cos a þ !
2
1
1 # !
1
cos a
!"
a
1
: (A9)
The semi-major axis of the large ellipse E
2
can be found
from the equation,
2a
2
¼ r
2
þ r
0
2
¼ r
1
þ 2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
R
2
# 4c
2
sin
2
a
p
þ R
¼ 2a
1
þ 2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
R
2
# 4c
2
sin
2
a
p
; (A10)
where r
0
2
¼ R is the length of BP
2
, so that
a
2
¼ a
1
þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
R
2
# 4c
2
sin
2
a
p
: (A11)
The large ellipse has the same foci as the small one, and thus
the value of c is the same for both. The eccentricity of E
2
is
!
2
¼
c
a
2
¼ !
1
a
1
a
2
: (A12)
We also need the angles h
1
and h
2
, which are found by
inverting the parametric equations in Eq. (A2),
h
1
¼ cos
#1
1
!
1
ð1 # !
2
1
Þ
a
1
R
# 1
%&
'(
; (A13a)
h
2
¼ cos
#1
1
!
2
ð1 # !
2
2
Þ
a
2
R
# 1
%&
'(
: (A13b)
The condition on the choice of !
1
and a comes from the
requirement that the point Q, where the perpendicular from
B meets the half-line AP
1
))!
, lies outside of the small ellipse.
Point Q lies outside of E
1
because Q is the midpoint of the
chord
P
1
P
2
, which lies outside of E
1
. Because the angle AQB
is a right angle, the angle h is greater than p/2. We see that
the possible positions of P
1
on E
1
are constrained so that
h > p/2. It is a general property of ellipses that
!
1
¼
sin h
sin a þ sin b
; (A14)
which can be shown using the law of sines on triangle ABP
1
along with the ellipse condition r
1
þ r
0
1
¼ 2a
1
. Correspond-
ing to the extreme value h ¼p/2 there are extreme values of
a, either a maximum (with P
1
closer to A) or a minimum
(with P
1
closer to B), which we denote by
!
a. In either case
we have b ¼ p=2 #
!
a and Eq. (A14) becomes
!
1
¼
1
cos
!
a þ sin
!
a
¼
1
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 þ sinð2
!
aÞ
p
: (A15)
For a given value of !
1
an acceptable value of a must lie
between the two solutions
!
a to Eq. (A15) in the range (0, p/2).
(These solutions must exist for the construction to work, and
hence we need !
1
> 1=
ﬃﬃ
2
p
.) Another expression of this condi-
tion is that, for a given value of a in the range (0, p/2), the
eccentricity must satisfy
!
1
>
1
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 þ sinð2aÞ
p
: (A16)
An alternate choice of parameters
14
takes a
1
, !
1
, and n ¼R/a
1
as fundamental. We can invert Eq. (A9) to ﬁnd a in terms of
these parameters !
1
and n,
a ¼ arccos
1 # n þ !
2
1
!
1
ð2 # nÞ
!"
; (A17)
817 Am. J. Phys., Vol. 79, No. 8, August 2011 T. J. Yoder and G. S. Adkins 817
and then the construction of the spherical section and the
large ellipse proceeds as before. The allowable range of n is
given by 2!
1
sin
!
a
1
< n < 2!
1
cos
!
a
1
, where
!
a
1
is the smallest
positive solution of Eq. (A15).
a)
1
J. C. Fallows, “Heat transfer paradox,” The New Scientist 5, 1156 (1959).
2
R. V. Osborne, “Heat transfer paradox,” The New Scientist 5, 1310
(1959).
3
P. M. Dryburgh, “Heat transfer paradox,” The New Scientist 6, 113
(1959).
4
G. H. Bisacre, “Heat transfer paradox,” The New Scientist 6, 113 (1959).
5
J. Greenleaf, “A fallacy in the second law of thermodynamics?,” Bull.
Inst. Phys. Phys. Soc. London 17, 253 (1966).
6
E. A. Guggenheim, “A fallacy in the second law of thermodynamics?,”
Bull. Inst. Phys. Phys. Soc. London 17, 332 (1966).
7
The Editor, “A fallacy in the second law of thermodynamics?,” Bull. Inst.
Phys. Phys. Soc. London 18, 22 (1967).
8
P. T. Landsberg, “A fallacy in the second law of thermodynamics?,” Bull.
Inst. Phys. Phys. Soc. London 18, 228–231 (1967).
9
R. M. Helsdon, “Brain teaser,” Phys. Educ. 7, 414, 446 (1972).
10
J. Higbie, “Chinese furnace,” Phys. Educ. 9, 14–15 (1974).
11
C. D. Boley and M. O. Scully, “The statistical mechanical resolution of a
thermodynamic ‘paradox’,” J. Stat. Phys. 24, 159–174 (1981).
12
W. T. Welford and R. Winston, “The ellipsoid paradox in
thermodynamics,” J. Stat. Phys. 28, 603–606 (1982).
13
P. T. Landsberg, “The ellipsoid paradox of thermodynamics,” J. Stat.
Phys. 34, 357 (1984).
14
L. H. Palmer, “An optical perpetual motion machine of the second kind,”
<www.lhup.edu/~dsimanek/museum/sucker.pdf>. Also see the discussion
15
W. Wu, “Ellipsoid power generation,” <www.ocf.berkeley.edu/~wwu/rid-
dles/hard.shtml#ellipsoidPowerGeneration>.
16
P. Mutalik, “Monday puzzle: Getting something for nothing,” New York
Times, <tierneylab.blogs.nytimes.com/2010/2002/22/monday-puzzle-get-
ting-something-from-nothing/>.
17
P. Mutalik, “The second law strikes back, New York Times, <tierneylab.
blogs.nytimes.com/2010/03/05/the-second-law-strikes-back/>.
18
G. Brooker, Modern Classical Optics (Oxford University Press, Oxford,
2003), p. 249.
19
The e´tendue of an inﬁnitesimal pencil of rays emerging from a source area
dS at angle h with respect to the normal and with all rays conﬁned to a
common solid angle dX is n
2
dS cos h dX, where n is the index of refrac-
tion of the medium. For our use, n ¼1.
818 Am. J. Phys., Vol. 79, No. 8, August 2011 T. J. Yoder and G. S. Adkins 818