1. Quantum capacitor at a metal–liquid interface
Fredy R. Zypmana)
Department of Physics, Yeshiva University, New York, New York 10033
͑Received 17 March 2000; accepted 4 September 2000͒
͓DOI: 10.1119/1.1328352͔
I. PROBLEM
Consider a metal immersed in an electrolyte. As the metal
electrostatic potential, V, is externally varied, a layer of sol-
ute deposits on the metal surface. The charge, Q, deposited
on the metal surface provides topographic information. Ex-
perimentalists typically report the differential capacitance
CϭdQ/dV as a function of V, C(V). ‘‘Normally’’ C is a
nonconstant function of V. However, for large solute con-
centrations, the C(V) curves show plateaus, that is, voltage
regions in which the capacitance does not change. This
‘‘anomalous’’ behavior is presented and explained in this
problem.
The complete understanding of metal–electrolyte inter-
faces is of great relevance in physics applied to biomedicine.
For example in electrocardiograms and electroencephalo-
grams, the electrodes are in contact with the skin, which
plays the role of an electrolyte ͑dehydrated skin is an insu-
lator͒. Another example is the connection of nerves by me-
tallic wires. The nerve–wire interface is of an electrolyte–
metal kind. In orthopedic implants, metal–metal interfaces
are a concern. An artificial hip must be soft and squishy in
the region close to the femur so as not to abrade the much
softer bone, but hard everywhere else. Thus, a typical artifi-
cial hip is made of ‘‘soft’’ titanium in contact with hard alloy
Co–Cr–Mo. Due to differences in chemical potential, that
metal interface is prone to corrosion.
In the formation of adlayers of mostly organic adsorbates
on metals1
such as Ag, Au, and particularly liquid Hg an
interesting ‘‘cut’’ appears in the differential capacitance Cd
curve as a function of potential. The first such observation
was made by Lorenz, who studied the capacitance of a solu-
tion of nanoic acid near a mercury electrode.2
The effect was
initially attributed to film formation via lateral interactions,
and theoretically studied by Rangarajan and co-workers us-
ing mean field techniques3,4
and a lattice gas model for the
surface. However, more recent experiments by Wandlowski
and De Levie5
show a rather sharp cut in the differential
capacity curve that is very difficult to understand in terms of
a classical model of adlayer formation.
Figure 1 shows a typical plot of differential capacitance
versus voltage for an aqueous solution–metal interface, for
concentrations below a critical value. The height and peak
position changes with solute concentration, but the general
shape remains the same.
However, above the critical concentration, there appear
plateaus in the C–V curve, as shown in Fig. 2.
The problem in understanding this phenomenon is that
independently of the suddenness of the formation of the
monolayer, there is no reasonable classical explanation for
the almost perfectly flat region inside the cut.
Problem. Explain this behavior.
Hint. Postulate the existence of a surface state of trapped
electrons. A very simple model shows that indeed C
ϭconstant by invoking surface trapped electrons in a con-
ducting state, which completely shield the metal charge.
Once this band is filled then the excess charge is screened by
the solution and the C–V curve recovers its normal appear-
ance. The cuts appear when the adlayer undergoes a metal–
insulator transition ͑as, for example, the Cs–Au alloy͒,
which is related to a percolation transition.
II. SOLUTION
We consider the electrons at the interface to be immersed
in the step potential due to a potential difference between the
electrolyte and the electrode, and a thin, confining film, de-
posited on the surface ͑Fig. 3͒.
We define ⌽1(x) as the wave function to the left of the
interface, and ⌽2(x) as that to the right of it ͑Fig. 4͒. The
total wave function is continuous across the interface, that is,
⌽1(0)ϭ⌽2(0)ϵ⌽(0), where xϭ0 is the coordinate defin-
ing the plane of the interface. Since there is a delta function
NEW PROBLEMS
Christopher R. Gould, Editor
Physics Department, Box 8202
North Carolina State University, Raleigh, North Carolina 27695
‘‘New Problems’’ solicits interesting and novel worked problems for use in undergraduate physics
courses beyond the introductory level. We seek problems that convey the excitement and interest of
current developments in physics and that are useful for teaching courses such as Classical Mechanics,
Electricity and Magnetism, Statistical Mechanics and Thermodynamics, ‘‘Modern’’ Physics, and
Quantum Mechanics. We challenge physicists everywhere to create problems that show how contem-
porary research in their various branches of physics uses the central unifying ideas of physics to
advance physical understanding. We want these problems to become an important source of ideas and
information for students of physics and their teachers. All submissions are peer-reviewed prior to
publication. Send manuscripts directly to Christopher R. Gould, Editor.
601 601Am. J. Phys. 69 ͑5͒, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers

2. at xϭ0, the wave function may have a noncontinuous deriva-
tive. By integrating the Schro¨dinger equation in a neighbor-
hood of the origin, the derivative difference is6
ប2
2m
͓⌽2Ј͑0͒Ϫ⌽1Ј͑0͔͒ϩ␣⌽͑0͒ϭ0, ͑1͒
where, as shown in Fig. 4, ␣ represents the strength of the ␦
function at the origin, and is, for a thin layer, the product of
the potential well and its spatial extension.
We are interested in charge accumulation at the interface.
Then, we look for electron bound eigenstates. These eigen-
states ͑in this particular problem there is only one͒ must have
negative total energy E. If they did not, they would be delo-
calized states. As in the case of transmission and reflection
through a potential step, if 0ϽEϽV0 then the electron is
delocalized for xϽ0 and confined within a small region for
xϾ0. If, on the other hand, V0ϽE, then the electron state is
delocalized for all x. In the case of a potential step ͑that is,
our problem with ␣ϭ0͒, there cannot be states completely
confined around xϭ0, since for EϽ0 the only solution to the
Schro¨dinger equation would be ⌽ϵ0. However, nontrivial
solutions can exist when ␣ 0. Thus we search for exponen-
tial solutions decaying away from the origin and with
EϽ0:
⌽1͑x͒ϭAe␯x
, ͑2͒
⌽2͑x͒ϭAeϪ␮x
, ͑3͒
EϭϪ
ប2
␯2
2m
, ͑4͒
EϭVϪ
ប2
␮2
2m
, ͑5͒
where ␮ and ␯ are real constants.
We next define a new parameter, ␤, to simplify the nota-
tion:
␯ϭͱ2mV
ប2 sinh ␤, ͑6͒
Fig. 1. Typical differential interfacial capacitance of mercury in contact
with aqueous solution of guanidinium nitrate for concentrations below
0.3 M.
Fig. 2. Typical differential interfacial capacitance of mercury in contact
with aqueous solution of guanidinium nitrate for concentrations above
0.3 M.
Fig. 3. Diagram of the system show-
ing the metal, the electrolyte, the thin
layer, and the electron wave function.
602 602Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

3. ␮ϭͱ2mV
ប2 cosh ␤. ͑7͒
From these definitions and Eqs. ͑4͒ and ͑5͒, one obtains the
state given by
␤ϭ
1
2
log
2m␣2
ប2
V
. ͑8͒
Let Q1 and Q2 be the net charges to the left and to the
right of the interface, respectively.
Then
Q1ϭe ͵Ϫϱ
0
͉⌽1͑x͉͒2
dxϭ
eA2
2␯
, ͑9͒
where the normalization constant A is obtained by requiring
that
͵Ϫϱ
0
͉⌽1͑x͉͒2
dxϩ ͵0
ϩϱ
͉⌽2͑x͉͒2
dxϭ1:
A2
ϭ2
V
␣
cosh ␤ sinh ␤. ͑10͒
In the previous expression, we have made use of the fact
that Eq. ͑1͒ is equivalent to ␮ϩ␯ϭ2ma/ប2
.
Then
Q1ϭ
e␮
␮ϩ␯
. ͑11͒
And, similarly
Q2ϭ
e␯
␮ϩ␯
. ͑12͒
The charge at the interface capacitor can now be evaluated
QcϭQ1ϪQ2ϭe
␮Ϫ␯
␮ϩ␯
ϭe
ប2
V
2m␣2 . ͑13͒
The capacitance, CϭdQc /dV, is
Cϭ
eប2
2m␣2 , ͑14͒
which is a constant that depends only on the properties of the
layer through the parameter ␣.
Thus by simply adding a small confining layer at the in-
terface, it is possible to explain the flat characteristics of the
total capacitance. As the external voltage in the electrolytic
cell changes, the thin layer may appear or disappear, thus
creating ‘‘normal’’ C–V regions, and flat ones.
ACKNOWLEDGMENTS
I would like to thank Dr. Steven J. Eppell and Dr. Lesser
Blum for useful comments. The National Cancer Institute
through Grant No. CA77796-01 has financially supported
this work.
a͒
Electronic mail: zypman@ymail.yu.edu
1
E. Budevski, G. Staikov, and W. J. Lorenz, Electrochemical Phase For-
mation and Growth ͑Wiley, New York, 1996͒, pp. 200–210.
2
Wolfgang Lorenz, ‘‘The rate of absorption and of two-dimensional asso-
ciation of fatty acids at the mercury-electrolyte interface,’’ Z. Elektro-
chem. 62 ͑1͒, 192–199 ͑1958͒.
3
M. V. Sangaranarayanan and S. K. Rangarajan, ‘‘Adsorption-isotherms for
neutral organic-compounds—A hierarchy in modeling,’’ J. Electroanal.
Chem. Interfacial Electrochem. 176 ͑1-2͒, 45–64 ͑1984͒.
4
M. V. Sangaranarayanan and S. K. Rangarajan, ‘‘Adsorption-isotherms—
microscopic modeling,’’ J. Electroanal. Chem. Interfacial Electrochem.
176 ͑1-2͒, 119–137 ͑1984͒.
5
T. Wandlowski, G. Jameson, and R. De Levie, ‘‘Two-Dimensional Con-
densation of Guadinidium Nitrate at the Mercury-Water Interface,’’ J.
Phys. Chem. 97 ͑39͒, 10119–10126 ͑1993͒.
6
C. Cohen-Tannoudji, Bernard Diu, and Frank Laloe¨, Quantum Mechanics
͑Wiley, New York, Paris, 1977͒, Vol. 1, p. 87.
Fig. 4. Potential function representing
the metal, the interface, and the elec-
trolyte.
603 603Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

4. Electromagnetically induced transparency
Tao Panga)
Department of Physics, University of Nevada, Las Vegas, Nevada 89154-4002
͑Received 6 June 2000; accepted 21 September 2000͒
͓DOI: 10.1119/1.1331303͔
I. SCOPE
Imagine that you are taking a quick walk and you are in
fact traveling faster than light. This is what has been
achieved in a recent experiment by Lene Hau and her
co-workers.1
In that experiment, the group velocity of a
pulsed laser was effectively reduced to about 1 mile per hour
͑0.45 m/s͒ in a cold, laser-dressed sodium atom cloud. In an
earlier experiment, the same group had successfully slowed
the group velocity of light to 38 miles per hour ͑17 m/s͒ in a
similar system.2
Propagation of light in a medium is a well-studied subject
even though there have been some very tough questions,
such as the group velocity exceeding the speed of light in
vacuum.3
Five years ago, it was demonstrated for the first
time that the speed of light can be reduced significantly4
in a
cold atom cloud that is in a state called electromagnetically
induced transparency.5,6
What has made the experimental
work of Hau et al. so unique is that the orders of magnitude
in the reduction of speed of light, which cannot commonly
be accomplished by increasing the index of refraction, is
achieved instead by the extremely rapid variation of the in-
dex with the frequency, and the elimination of light absorp-
tion at resonance frequency—a quantum phenomenon result-
ing from the coupling and interaction between lasers and
electrons at different atomic levels. Here we would like to
highlight some basic understanding of this exciting phenom-
enon.
From the Maxwell equations for a propagating electro-
magnetic wave with angular frequency ␻ and complex wave
vector ␬ in a nonconducting medium, we have7
␬2
ϭ␮⑀␻2
, ͑1͒
where ␮ is the magnetic permeability and ⑀ is the electric
permittivity of the medium. If we assume that ␻ is real and
␬ϭkϩi
␣
2
, ͑2͒
where k is the real propagating wave vector and ␣ is the
absorption coefficient, we have
nϭ
ck
␻
ϭReͱ ␮⑀
␮0⑀0
, ͑3͒
␣ϭ2␻ Im ͱ␮⑀, ͑4͒
with cϭ1/ͱ␮0⑀0 being the speed of light in vacuum and ␮0
and ⑀0 the vacuum permeability and permittivity, respec-
tively. In most cases, we have ␮Ӎ␮0 , a condition that is
assumed here. From the above relation, we find the phase
velocity of the wave,
vpϭ
␻
k
ϭ
c
n
, ͑5͒
which characterizes how fast the wave changes its phase. So
the field propagating along the z direction is proportional to
eikzϪ␣z/2Ϫi␻t
.
Now let us consider that the electric field is a nondecaying
(␣ϭ0) wave packet with a range of frequencies ␻ϭ␻(k):
E͑z,t͒ϭ ͵dk A͑k͒eikzϪi␻t
, ͑6͒
where A(k) is a narrow function peaked at kϭk0 . We can
then expand ␻(k) as
␻͑k͒ϭ␻͑k0͒ϩ͑kϪk0͒
d␻
dk
ͯkϭk0
ϩO͓͑kϪk0͒2
͔, ͑7͒
if it is well behaved, that is, a smooth function around the
given wave vector k0 . If only the zeroth- and first-order
terms are kept in the expansion, we have
E͑z,t͒Ӎei͓k0vgϪ␻͑k0͔͒t
͵dk A͑k͒eikzϪikvgt
ϭei͓k0vgϪ␻͑k0͔͒t
E͑zϪvgt,0͒, ͑8͒
where
vgϭ
d␻
dk
ͯkϭk0
͑9͒
is termed the group velocity of the wave at kϭk0 because
the packet acts if it is traveling in space with such a velocity
without changing its shape, with an overall phase change.
Using Eq. ͑3͒, we obtain
vgϭ
vp
1ϩ͑␻/n͒͑dn/d␻͒
. ͑10͒
Note that we have assumed that n is a function of ␻ through
k. We can consider the group velocity to be the velocity of
the wave packet, that is, the velocity of the energy and in-
formation contained in the packet, if the linear term in the
above Taylor expansion is the dominant term. However, the
meaning of the group velocity can change if the wave packet
becomes incoherent. Care must be taken when the angular
frequency of the wave is near a resonance or dn/d␻Ͻ0.3
Quantum mechanically, the resonance occurs when the
frequency of light matches the energy difference between
two allowed quantum levels in the system and is typically
accompanied by strong absorption under normal circum-
stances. This is why normal matter that can be well approxi-
mated by a two-level model can never slow the light very
much. For laser-dressed atom clouds, the coupling and inter-
action between a three-level atom and two lasers can drasti-
cally alter the behavior of the system, including effectively
604 604Am. J. Phys. 69 ͑5͒, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers

5. eliminating the absorption at the resonance frequency and
therefore creating electromagnetically induced transparency,
as shown in the problems given here.
II. PROBLEMS
A. Coherent population trapping
The key to keeping the group velocity at the vicinity of a
resonance frequency meaningful lies in the properties of the
laser-dressed atomic cloud. Without such an effect, absorp-
tion would be too strong to have any transmitted light.
Consider that each atom in the medium has three levels.
The presence of a coupling ͑dressing͒ laser (␻cӍ␻2Ϫ␻1)
and a probe laser (␻Ӎ␻2Ϫ␻0) causes a mixing of the three
levels, ͉0͘, ͉1͘, and ͉2͘.
The Hamiltonian of such a system is
HϭH0ϩH1 . ͑11͒
Here the unperturbed Hamiltonian H0 is given by
͗l͉H0͉lЈ͘ϭប␻l␦llЈ , ͑12͒
with l, lЈϭ0,1, 2. The perturbation H1 is restricted to be
͗l͉H1͉lЈ͘ϭ͗lЈ͉H1͉l͘*ϭប⍀llЈeϪi␻llЈt
, ͑13͒
with ␻llЈϭ␻lϪ␻lЈ and ⍀llϭ⍀01ϭ⍀10ϭ0. Note that ␻2
Ͼ␻1Ͼ␻0ϭ0 and ␻l0ϭ␻l . This is a so-called ‘‘⌳’’ system
with the highest level coupled to two lower levels.
For the Hamiltonian given, find the time-dependent wave
function
͉␺͑t͒͘ϭ͚lϭ0
2
cl͑t͉͒l͘, ͑14͒
if ͉␺(0)͘ϭc0(0)͉0͘ϩc1(0)͉1͘ with ͉c0(0)͉2
ϩ͉c1(0)͉2
ϭ1.
Discuss the condition for c2(t)ϵ0 and its implication.
B. Electromagnetically induced transparency
If we define a density matrix
␳͑t͒ϭ͉␺͑t͒͗͘␺͑t͉͒, ͑15͒
whose diagonal elements are the probabilities of occupying
specific states and off-diagonal elements represent the tran-
sition rates between two given states, we have
iប
‫␳ץ‬
‫ץ‬t
ϭ͓H,␳͔, ͑16͒
from the Schro¨dinger equation. The interactions between at-
oms in the cloud can cause a finite linewidth and decay of
each level, which can be accounted for by a relaxation ma-
trix:
͗l͉⌫͉lЈ͘ϭ2␥l␦llЈ , ͑17͒
and change Eq. ͑16͒ into
iប
‫␳ץ‬
‫ץ‬t
ϭ͓H,␳͔Ϫ
iប
2
͑⌫␳ϩ␳⌫͒. ͑18͒
Assuming that only the dominant decaying factor is nonzero,
that is, ␥2ϭ␥ and ␥0,1ϭ0, and that the atom is in the ground
state at tϭ0, show that
⑀͑␻͒ϭ⑀0ͫ1ϩ
␻d͑␻Ϫ␻2͒
␻R
2
/4Ϫ͑␻Ϫ␻2͒2
Ϫi␥͑␻Ϫ␻2͒ͬ, ͑19͒
where ␻dϭna͉p20͉2
/ប⑀0 with ͉p20͉ being the coupling dipole
strength between ͉2͘ and ͉0͘ and ␻Rϭ2͉⍀21͉ the Rabi angu-
lar frequency between ͉2͘ and ͉1͘.
C. The slowest light
In the recent experiment, Hau and co-workers have suc-
cessfully reduced the group velocity of light in a cold, laser-
dressed sodium atom cloud to 1 mile per hour ͑0.45 m/s͒.1
Each sodium atom can be approximated well by a three-level
system. Assume that the permittivity of such a laser-dressed
atom cloud is given by Eq. ͑19͒ and the frequency of the
probe laser (␻/2␲) is near the resonance frequency ͑␻2/2␲
Ӎ5.1ϫ1014
Hz for sodium atom͒. Estimate the number den-
sity of the atom cloud needed in order to have vg
Ӎ0.45 m/s. Assume that the Rabi angular frequency is about
␻Rϭ3.5ϫ107
rad/s and the coupling dipole strength is about
͉p20͉Ӎ2.5ϫ10Ϫ29
C m.
III. SOLUTIONS
A. Coherent population trapping
From the time-dependent Schro¨dinger equation
iប
‫ץ‬͉␺͑t͒͘
‫ץ‬t
ϭH͉␺͑t͒͘, ͑20͒
we have
ic˙0͑t͒ϭ␻0c0͑t͒ϩ⍀20ei␻2t
c2͑t͒, ͑21͒
ic˙1͑t͒ϭ␻1c1͑t͒ϩ⍀21ei␻21t
c2͑t͒, ͑22͒
ic˙2͑t͒ϭ␻2c2͑t͒ϩ⍀02eϪi␻2t
c0͑t͒ϩ⍀12eϪi␻21t
c1͑t͒.
͑23͒
If we redefine the coefficients by
cl͑t͒ϭeϪi␻lt
bl͑t͒, ͑24͒
the equation set is simplified to
ib˙ 0͑t͒ϭ⍀20b2͑t͒, ͑25͒
ib˙ 1͑t͒ϭ⍀21b2͑t͒, ͑26͒
ib˙ 2͑t͒ϭ⍀02b0͑t͒ϩ⍀12b1͑t͒. ͑27͒
Multiplying Eq. ͑25͒ with ⍀02 and Eq. ͑26͒ with ⍀12 and
adding them together, and substituting the resulting equation
into Eq. ͑27͒ after taking one more time derivative, we ob-
tain
b¨ 2͑t͒ϭϪ͉͑⍀20͉2
ϩ͉⍀21͉2
͒b2͑t͒. ͑28͒
We have used ⍀02ϭ⍀20* and ⍀12ϭ⍀21* . So we have
b2͑t͒ϭAei⍀t
ϩBeϪi⍀t
, ͑29͒
with ⍀ϭͱ͉⍀20͉2
ϩ͉⍀21͉2
. Taking the initial condition
b2(0)ϭc2(0)ϭ0, we arrive at
b2͑t͒ϭC sin ⍀t, ͑30͒
with C being a constant. Substituting this result back into
Eqs. ͑25͒ and ͑26͒, we have
b0͑t͒ϭ͓c0͑0͒Ϫ␣͔cos ⍀tϩ␣, ͑31͒
b1͑t͒ϭ͓c1͑0͒Ϫ␤͔cos ⍀tϩ␤, ͑32͒
where ␣ and ␤ are constants constrained by
605 605Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

6. ⍀02␣ϩ⍀12␤ϭ0. ͑33͒
We have used the initial conditions b0(0)ϭc0(0) and
b1(0)ϭc1(0). The coefficient C is given by
Cϭ
i
⍀
͓⍀02c0͑0͒ϩ⍀12c1͑0͔͒. ͑34͒
If c0(0) and c1(0) are such that Cϵ0, we have c2(t)ϵ0 all
the time. A typical case is ͉⍀02͉ϭ͉⍀12͉ and ͉c0(0)͉
ϭ͉c1(0)͉ϭ1/&, with the total phase difference between the
two terms being ␲. So the state ͉2͘ will stay empty and the
atoms are trapped in the lower states. The effect of such a
coherent population trapping is that the absorption or emis-
sion of light is completely eliminated.
B. Electromagnetically induced transparency
Consider that the traveling ͑probing͒ laser is described by
a time-dependent electric field E(t)ϭE0eϪi␻t
with ␻ very
close to ␻2 . The perturbation from such a field is
͗2͉H1͉0͘ϭϪ͗2͉p͉0͘E0eϪi␻t
ϭប⍀20eϪi␻t
, ͑35͒
where p is the dipole moment induced by the field. Now if
we examine the density matrix elements between two states,
␳llЈϭ͗l͉␳͉lЈ͘, we have
i
‫␳ץ‬20
‫ץ‬t
ϭ͑␻2Ϫi␥͒␳20ϩ⍀21eϪi␻21t
␳10
ϩ⍀20eϪi␻t
͑␳00Ϫ␳22͒, ͑36͒
i
‫␳ץ‬10
‫ץ‬t
ϭ␻1␳10ϩ⍀12eϪi␻12t
␳20Ϫ⍀20eϪi␻2t
␳12 . ͑37͒
We have used
͚lϭ0
2
͉l͗͘l͉ϭ1 ͑38͒
in deriving the above equations. We can then replace ␳00 ,
␳22 , and ␳12 by their values at tϭ0, that is, ␳00ϭ1, ␳22ϭ0,
and ␳12ϭ0, and change a variable with ␨10ϭ␳10eϪi␻21t
, be-
cause we are only looking for the linear solution. Then we
have
i
‫␳ץ‬20
‫ץ‬t
ϭ͑␻2Ϫi␥͒␳20ϩ⍀21␨10ϩ⍀20eϪi␻t
, ͑39͒
i
‫␨ץ‬10
‫ץ‬t
ϭ␻2␨10ϩ⍀12␳20 . ͑40͒
This equation set resembles a harmonic oscillator under
damping and driving forces. The steady solutions are there-
fore given by
␳20͑t͒ϭAeϪi␻t
, ͑41͒
␨10͑t͒ϭBeϪi␻t
. ͑42͒
Substituting the above solutions into the equations, we obtain
Aϭ
⍀20͑␻Ϫ␻2͒
͑␻Ϫ␻2ϩi␥͒͑␻Ϫ␻2͒Ϫ͉⍀21͉2 . ͑43͒
Because ␳20 represents the dipole transition rate between ͉2͘
and 0͘, the polarization of the system is given by P
ϭna␳20p02ϭ(⑀Ϫ⑀0)E(t) with p02ϭ͗0͉p͉2͘ϭp20* . Then we
reach Eq. ͑19͒.
C. The slowest light
We know that the group velocity is given by
vgϭ
d␻
dk
ϭ
c
nϩ␻͑dn/d␻͒
. ͑44͒
For all known materials, nϳO(1). So if vgӶc, we must
have
vgӍ
c
␻͑dn/d␻͒
. ͑45͒
For vgϭ0.45 m/s, as observed in the experiment by Hau’s
group,1
one must have
␻͑dn/d␻͒Ӎ6.7ϫ108
. ͑46͒
From the given permittivity, we have
nϩi
c␣
2␻
Ӎ1ϩ
1
2
␻d͑␻Ϫ␻2͒
␻R
2
/4Ϫ͑␻Ϫ␻2͒2
Ϫi␥͑␻Ϫ␻2͒
. ͑47͒
Considering that ␻ is very close to ␻2 , we have
nϩi
c␣
2␻
Ӎ1ϩ
2␻d͑␻Ϫ␻2͒
␻R
2
ϫͫ1ϩ
4͑␻Ϫ␻2͒2
␻R
2 ϩ
i4␥͑␻Ϫ␻2͒
␻R
2 ϩ¯ͬ,
͑48͒
which gives
␻͑dn/d␻͒Ӎ
2␻
ប⑀0
na͉p20͉2
␻R
2 . ͑49͒
We have used ␻dϭna͉p20͉2
/ប⑀0 . With the numerical values
of the quantities given, we then obtain naӍ2ϫ1020
mϪ3
, a
density quite difficult to achieve experimentally.
Note that the absorption coefficient ␣ is zero at the reso-
nance frequency. This is the essence of the electromagneti-
cally induced transparency, a condition that must be met in
order to have a significant light transmission at the resonance
frequency. Otherwise, the drastically slowed group velocity
of light observed by Hau’s group would not have been pos-
sible.
a͒
Electronic mail: pang@nevada.edu
1
L. V. Hau, presentation at the American Association for the Advancement
of Science, February 2000, Washington, DC.
2
L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, ‘‘Light speed
reduction to 17 meters per second in an ultracold atomic gas,’’ Nature
͑London͒ 397, 594–598 ͑1999͒.
3
For a recent review, see R. Y. Chiao and A. M. Steinberg, Tunneling
Times and Superluminality, Progress in Optics Vol. 37, edited by E. Wolf
͑Elsevier, Amsterdam, 1997͒, pp. 347–405.
4
A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, ‘‘Electromagnetically
introduced transparency: Propagation dynamics,’’ Phys. Rev. Lett. 74,
2447–2450 ͑1995͒.
5
S. E. Harris, ‘‘Electromagnetically induced transparency,’’ Phys. Today
50 ͑7͒, 36–42 ͑1997͒.
6
M. O. Scully and M. S. Zubairy, Quantum Optics ͑Cambridge U.P., Cam-
bridge, 1997͒, Secs. 7.2 and 7.3.
7
D. J. Jackson, Classical Electrodynamics ͑Wiley, New York, 1999͒, 3rd
ed., Secs. 7.5 and 7.8.
606 606Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

7. Negative group velocity
Kirk T. McDonalda)
Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544
͑Received 21 August 2000; accepted 21 September 2000͒
͓DOI: 10.1119/1.1331304͔
I. PROBLEM
Consider a variant on the physical situation of ‘‘slow
light’’ 1,2
in which two closely spaced spectral lines are now
both optically pumped to show that the group velocity can be
negative at the central frequency, which leads to apparent
superluminal behavior.
A. Negative group velocity
In more detail, consider a classical model of matter in
which spectral lines are associated with oscillators. In par-
ticular, consider a gas with two closely spaced spectral lines
of angular frequencies ␻1,2ϭ␻0Ϯ⌬/2, where ⌬Ӷ␻0 . Each
line has the same damping constant ͑and spectral width͒ ␥.
Ordinarily, the gas would exhibit strong absorption of
light in the vicinity of the spectral lines. But suppose that
lasers of frequencies ␻1 and ␻2 pump both oscillators into
inverted populations. This can be described classically by
assigning negative oscillator strengths to these oscillators.3
Deduce an expression for the group velocity vg(␻0) of a
pulse of light centered on frequency ␻0 in terms of the ͑uni-
valent͒ plasma frequency ␻p of the medium, given by
␻p
2
ϭ
4␲Ne2
m
, ͑1͒
where N is the number density of atoms, and e and m are the
charge and mass of an electron. Give a condition on the line
separation ⌬ compared to the linewidth ␥ such that the group
velocity vg(␻0) is negative.
In a recent experiment by Wang et al.,4
a group velocity of
vgϭϪc/310, where c is the speed of light in vacuum, was
demonstrated in cesium vapor using a pair of spectral lines
with separation ⌬/2␲Ϸ2 MHz and linewidth ␥/2␲
Ϸ0.8 MHz.
B. Propagation of a monochromatic plane wave
Consider a wave with electric field E0ei␻(z/cϪt)
that is
incident from zϽ0 on a medium that extends from zϭ0 to a.
Ignore reflection at the boundaries, as is reasonable if the
index of refraction n(␻) is near unity. Particularly simple
results can be obtained when you make the ͑unphysical͒ as-
sumption that the ␻n(␻) varies linearly with frequency
about a central frequency ␻0 . Deduce a transformation that
has a frequency-dependent part and a frequency-independent
part between the phase of the wave for zϽ0 to that of the
wave inside the medium, and to that of the wave in the
region aϽz.
C. Fourier analysis
Apply the transformations between an incident monochro-
matic wave and the wave in and beyond the medium to the
Fourier analysis of an incident pulse of form f(z/cϪt).
D. Propagation of a sharp wave front
In the approximation that ␻n varies linearly with ␻, de-
duce the waveforms in the regions 0ϽzϽa and aϽz for an
incident pulse ␦(z/cϪt), where ␦ is the Dirac delta function.
Show that the pulse emerges out of the gain region at zϭa at
time tϭa/vg , which appears to be earlier than when it enters
this region if the group velocity is negative. Show also that
inside the negative group velocity medium a pulse propa-
gates backwards from zϭa at time tϭa/vgϽ0 to zϭ0 at t
ϭ0, at which time it appears to annihilate the incident pulse.
E. Propagation of a Gaussian pulse
As a more physical example, deduce the waveforms in the
regions 0ϽzϽa and aϽz for a Gaussian incident pulse
E0eϪ(z/cϪt)2/2␶2
ei␻0(z/cϪt)
. Carry the frequency expansion of
␻n(␻) to second order to obtain conditions of validity of the
analysis such as maximum pulse width ␶, maximum length a
of the gain region, and maximum time of advance of the
emerging pulse. Consider the time required to generate a
pulse of rise time ␶ when assessing whether the time advance
in a negative group velocity medium can lead to superlumi-
nal signal propagation.
II. SOLUTION
The concept of group velocity appears to have been first
enunciated by Hamilton in 1839 in lectures of which only
abstracts were published.5
The first recorded observation of
the group velocity of a ͑water͒ wave is due to Russell in
1844.6
However, widespread awareness of the group velocity
dates from 1876 when Stokes used it as the topic of a
Smith’s Prize examination paper.7
The early history of group
velocity has been reviewed by Havelock.8
H. Lamb9
credits A. Schuster with noting in 1904 that a
negative group velocity, i.e., a group velocity of opposite
sign to that of the phase velocity, is possible due to anoma-
lous dispersion. Von Laue10
made a similar comment in
1905. Lamb gave two examples of strings subject to external
potentials that exhibit negative group velocities. These early
considerations assumed that in case of a wave with positive
group and phase velocities incident on the anomalous me-
dium, energy would be transported into the medium with a
positive group velocity, and so there would be waves with
negative phase velocity inside the medium. Such negative
phase velocity waves are formally consistent with Snell’s
607 607Am. J. Phys. 69 ͑5͒, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers

8. law11
͑since ␪tϭsinϪ1
͓(ni /nt)sin ␪i͔ can be in either the first
or second quadrant͒, but they seemed physically implausible
and the topic was largely dropped.
Present interest in negative group velocity is based on
anomalous dispersion in a gain medium, where the sign of
the phase velocity is the same for incident and transmitted
waves, and energy flows inside the gain medium in the op-
posite direction to the incident energy flow in vacuum.
The propagation of electromagnetic waves at frequencies
near those of spectral lines of a medium was first extensively
discussed by Sommerfeld and Brillouin,12
with emphasis on
the distinction between signal velocity and group velocity
when the latter exceeds c. The solution presented here is
based on the work of Garrett and McCumber,13
as extended
by Chiao et al.14,15
A discussion of negative group velocity
in electronic circuits has been given by Mitchell and Chiao.16
A. Negative group velocity
In a medium of index of refraction n(␻), the dispersion
relation can be written
kϭ
␻n
c
, ͑2͒
where k is the wave number. The group velocity is then
given by
vgϭReͫd␻
dk ͬϭ
1
Re͓dk/d␻͔
ϭ
c
Re͓d͑␻n͒/d␻͔
ϭ
c
nϩ␻ Re͓dn/d␻͔
.
͑3͒
We see from Eq. ͑3͒ that if the index of refraction de-
creases rapidly enough with frequency, the group velocity
can be negative. It is well known that the index of refraction
decreases rapidly with frequency near an absorption line,
where ‘‘anomalous’’ wave propagation effects can occur.12
However, the absorption makes it difficult to study these
effects. The insight of Garrett and McCumber13
and of Chiao
et al.14,15,17–19
is that demonstrations of negative group ve-
locity are possible in media with inverted populations, so
that gain rather than absorption occurs at the frequencies of
interest. This was dramatically realized in the experiment of
Wang et al.4
by use of a closely spaced pair of gain lines, as
perhaps first suggested by Steinberg and Chiao.17
We use a classical oscillator model for the index of refrac-
tion. The index n is the square root of the dielectric constant
⑀, which is in turn related to the atomic polarizability ␣ ac-
cording to
Dϭ⑀EϭEϩ4␲PϭE͑1ϩ4␲N␣͒ ͑4͒
͑in Gaussian units͒, where D is the electric displacement, E is
the electric field, and P is the polarization density. Then, the
index of refraction of a dilute gas is
nϭͱ⑀Ϸ1ϩ2␲N␣. ͑5͒
The polarizability ␣ is obtained from the electric dipole
moment pϭexϭ␣E induced by electric field E. In the case
of a single spectral line of frequency ␻j , we say that an
electron is bound to the ͑fixed͒ nucleus by a spring of con-
stant Kϭm␻j
2
, and that the motion is subject to the damping
force Ϫm␥jx˙, where the dot indicates differentiation with
respect to time. The equation of motion in the presence of an
electromagnetic wave of frequency ␻ is
x¨ϩ␥jx˙ϩ␻j
2
xϭ
eE
m
ϭ
eE0
m
ei␻t
. ͑6͒
Hence,
xϭ
eE
m
1
␻j
2
Ϫ␻2
Ϫi␥j␻
ϭ
eE
m
␻j
2
Ϫ␻2
ϩi␥j␻
͑␻j
2
Ϫ␻2
͒2
ϩ␥j
2
␻2 , ͑7͒
and the polarizability is
␣ϭ
e2
m
␻j
2
Ϫ␻2
ϩi␥j␻
͑␻j
2
Ϫ␻2
͒2
ϩ␥j
2
␻2 . ͑8͒
In the present problem we have two spectral lines, ␻1,2
ϭ␻0Ϯ⌬/2, both of oscillator strength Ϫ1 to indicate that the
populations of both lines are inverted, with damping con-
stants ␥1ϭ␥2ϭ␥. In this case, the polarizability is given by
␣ϭϪ
e2
m
͑␻0Ϫ⌬/2͒2
Ϫ␻2
ϩi␥␻
͑͑␻0Ϫ⌬/2͒2
Ϫ␻2
͒2
ϩ␥2
␻2
Ϫ
e2
m
͑␻0ϩ⌬/2͒2
Ϫ␻2
ϩi␥␻
͑͑␻0ϩ⌬/2͒2
Ϫ␻2
͒2
ϩ␥2
␻2
ϷϪ
e2
m
␻0
2
Ϫ⌬␻0Ϫ␻2
ϩi␥␻
͑␻0
2
Ϫ⌬␻0Ϫ␻2
͒2
ϩ␥2
␻2
Ϫ
e2
m
␻0
2
ϩ2⌬␻0Ϫ␻2
ϩi␥␻
͑␻0
2
ϩ⌬␻0Ϫ␻2
͒2
ϩ␥2
␻2 , ͑9͒
where the approximation is obtained by the neglect of terms
in ⌬2
compared to those in ⌬␻0 .
For a probe beam at frequency ␻, the index of refraction
͑5͒ has the form
n͑␻͒Ϸ1Ϫ
␻p
2
2 ͫ ␻0
2
Ϫ⌬␻0Ϫ␻2
ϩi␥␻
͑␻0
2
Ϫ⌬␻0Ϫ␻2
͒2
ϩ␥2
␻2
ϩ
␻0
2
ϩ⌬␻0Ϫ␻2
ϩi␥␻
͑␻0
2
ϩ⌬␻0Ϫ␻2
͒2
ϩ␥2
␻2ͬ, ͑10͒
where ␻p is the plasma frequency given by Eq. ͑1͒. This is
illustrated in Fig. 1.
The index at the central frequency ␻0 is
Fig. 1. The real and imaginary parts of the index of refraction in a medium
with two spectral lines that have been pumped to inverted populations. The
lines are separated by angular frequency ⌬ and have widths ␥ϭ0.4⌬.
608 608Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

9. n͑␻0͒Ϸ1Ϫi
␻p
2
␥
͑⌬2
ϩ␥2
͒␻0
Ϸ1Ϫi
␻p
2
⌬2
␥
␻0
, ͑11͒
where the second approximation holds when ␥Ӷ⌬. The
electric field of a continuous probe wave then propagates
according to
E͑z,t͒ϭei͑kzϪ␻0t͒
ϭei␻͑n͑␻0͒z/cϪt͒
Ϸez/͓⌬2c/␥␻͑2/p͔͒
ei␻0͑z/cϪt͒
. ͑12͒
From this we see that at frequency ␻0 the phase velocity is c,
and the medium has an amplitude gain length ⌬2
c/␥␻p
2
.
To obtain the group velocity ͑3͒ at frequency ␻0 , we need
the derivative
d͑␻n͒
d␻
ͯ␻0
Ϸ1Ϫ
2␻p
2
͑⌬2
Ϫ␥2
͒
͑⌬2
ϩ␥2
͒2 , ͑13͒
where we have neglected terms in ⌬ and ␥ compared to ␻0 .
From Eq. ͑3͒, we see that the group velocity can be negative
if
⌬2
␻p
2Ϫ
␥2
␻p
2 у
1
2 ͩ⌬2
␻p
2 ϩ
␥2
␻p
2 ͪ2
. ͑14͒
The boundary of the allowed region ͑14͒ in (⌬2
,␥2
) space is
a parabola whose axis is along the line ␥2
ϭϪ⌬2
, as shown
in Fig. 2. For the physical region ␥2
у0, the boundary is
given by
␥2
␻p
2 ϭͱ1ϩ4
⌬2
␻p
2Ϫ1Ϫ
⌬2
␻p
2 . ͑15͒
Thus, to have a negative group velocity, we must have
⌬р&␻p , ͑16͒
which limit is achieved when ␥ϭ0; the maximum value of ␥
is 0.5␻p when ⌬ϭ0.866␻p .
Near the boundary of the negative group velocity region,
͉vg͉ exceeds c, which alerts us to concerns of superluminal
behavior. However, as will be seen in the following sections,
the effect of a negative group velocity is more dramatic when
͉vg͉ is small rather than large.
The region of recent experimental interest is ␥Ӷ⌬Ӷ␻p ,
for which Eqs. ͑3͒ and ͑13͒ predict that
vgϷϪ
c
2
⌬2
␻p
2 . ͑17͒
A value of vgϷϪc/310 as in the experiment of Wang cor-
responds to ⌬/␻pϷ1/12. In this case, the gain length
⌬2
c/␥␻p
2
was approximately 40 cm.
For later use we record the second derivative,
d2
͑␻n͒
d␻2 ͯ␻0
Ϸ8i
␻p
2
␥͑3⌬2
Ϫ␥2
͒
͑⌬2
ϩ␥2
͒3 Ϸ24i
␻p
2
⌬2
␥
⌬2 , ͑18͒
where the second approximation holds if ␥Ӷ⌬.
B. Propagation of a monochromatic plane wave
To illustrate the optical properties of a medium with nega-
tive group velocity, we consider the propagation of an elec-
tromagnetic wave through it. The medium extends from z
ϭ0 to a, and is surrounded by vacuum. Because the index of
refraction ͑10͒ is near unity in the frequency range of inter-
est, we ignore reflections at the boundaries of the medium.
A monochromatic plane wave of frequency ␻ and incident
from zϽ0 propagates with phase velocity c in vacuum. Its
electric field can be written
E␻͑z,t͒ϭE0ei␻z/c
eϪi␻t
͑zϽ0͒. ͑19͒
Inside the medium this wave propagates with phase velocity
c/n(␻) according to
E␻͑z,t͒ϭE0ei␻nz/c
eϪi␻t
͑0ϽzϽa͒, ͑20͒
where the amplitude is unchanged since we neglect the small
reflection at the boundary zϭ0. When the wave emerges into
vacuum at zϭa, the phase velocity is again c, but it has
accumulated a phase lag of (␻/c)(nϪ1)a, and so appears as
E␻͑z,t͒ϭE0ei␻a͑nϪ1͒/c
ei␻z/c
eϪi␻t
ϭE0ei␻an/c
eϪi␻͑tϪ͑zϪa͒/c͒
͑aϽz͒. ͑21͒
It is noteworthy that a monochromatic wave for zϾa has the
same form as that inside the medium if we make the
frequency-independent substitutions
z→a, t→tϪ
zϪa
c
. ͑22͒
Since an arbitrary waveform can be expressed in terms of
monochromatic plane waves via Fourier analysis, we can use
these substitutions to convert any wave in the region 0Ͻz
Ͻa to its continuation in the region aϽz.
A general relation can be deduced in the case where the
second and higher derivatives of ␻n(␻) are very small. We
can then write
␻n͑␻͒Ϸ␻0n͑␻0͒ϩ
c
vg
͑␻Ϫ␻0͒, ͑23͒
where vg is the group velocity for a pulse with central fre-
quency ␻0 . Using this in Eq. ͑20͒, we have
E␻͑z,t͒ϷE0ei␻0z͑n͑␻0͒/cϪ1/vg͒
ei␻z/vgeϪi␻t
͑0ϽzϽa͒.
͑24͒
In this approximation, the Fourier component E␻(z) at fre-
quency ␻ of a wave inside the gain medium is related to that
of the incident wave by replacing the frequency dependence
Fig. 2. The allowed region ͑14͒ in (⌬2
,␥2
) space such that the group ve-
locity is negative.
609 609Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

10. ei␻z/c
by ei␻z/vg, i.e., by replacing z/c by z/vg , and multi-
plying by the frequency-independent phase factor
ei␻0z(n(␻0)/cϪ1/vg)
. Then, using transformation ͑22͒, the wave
that emerges into vacuum beyond the medium is
E␻͑z,t͒ϷE0ei␻0a͑n͑␻0͒/cϪ1/vg͒
ϫei␻͑z/cϪa͑1/cϪ1/vg͒͒
eϪi␻t
͑aϽz͒. ͑25͒
The wave beyond the medium is related to the incident wave
by multiplying by a frequency-independent phase, and by
replacing z/c by z/cϪa(1/cϪ1/vg) in the frequency-
dependent part of the phase.
The effect of the medium on the wave as described by
Eqs. ͑24͒ and ͑25͒ has been called ‘‘rephasing.’’ 4
C. Fourier analysis and ‘‘rephasing’’
The transformations between the monochromatic incident
wave ͑19͒ and its continuation in and beyond the medium,
͑24͒ and ͑25͒, imply that an incident wave
E͑z,t͒ϭf͑z/cϪt͒ϭ ͵Ϫϱ
ϱ
E␻͑z͒eϪi␻t
d␻ ͑zϽ0͒, ͑26͒
whose Fourier components are given by
E␻͑z͒ϭ
1
2␲
͵Ϫϱ
ϱ
E͑z,t͒ei␻t
dt, ͑27͒
propagates as
E͑z,t͒Ϸ
Ά
f͑z/cϪt͒ ͑zϽ0͒
ei␻0z͑n͑␻0͒/cϪ1/vg͒
f͑z/vgϪt͒ ͑0ϽzϽa͒
ei␻0a͑n͑␻0͒/cϪ1/vg͒
f͑z/cϪtϪa͑1/cϪ1/vg͒͒
͑aϽz͒.
͑28͒
An interpretation of Eq. ͑28͒ in terms of ‘‘rephasing’’ is as
follows. Fourier analysis tells us that the maximum ampli-
tude of a pulse made of waves of many frequencies, each of
the form E␻(z,t)ϭE0(␻)ei␾(␻)
ϭE0(␻)ei(k(␻)zϪ␻tϩ␾0(␻))
with E0у0, is determined by adding the amplitudes E0(␻).
This maximum is achieved only if there exist points ͑z,t͒
such that all phases ␾͑␻͒ have the same value.
For example, we consider a pulse in the region zϽ0
whose maximum occurs when the phases of all component
frequencies vanish, as shown at the left of Fig. 3. Referring
to Eq. ͑19͒, we see that the peak occurs when zϭct. As
usual, we say that the group velocity of this wave is c in
vacuum.
Inside the medium, Eq. ͑24͒ describes the phases of the
components, which all have a common frequency-
independent phase ␻0z(n(␻0)/cϪ1/vg) at a given z, as well
as a frequency-dependent part ␻(z/vgϪt). The peak of the
pulse occurs when all the frequency-dependent phases van-
ish; the overall frequency-independent phase does not affect
the pulse size. Thus, the peak of the pulse propagates within
the medium according to zϭvgt. The velocity of the peak is
vg , the group velocity of the medium, which can be nega-
tive.
The ‘‘rephasing’’ ͑24͒ within the medium changes the
wavelengths of the component waves. Typically the wave-
length increases, and by greater amounts at longer wave-
lengths. A longer time is required before the phases of the
waves all become the same at some point z inside the me-
dium, so in a normal medium the velocity of the peak ap-
pears to be slowed down. But in a negative group velocity
medium, wavelengths short compared to ␭0 are lengthened,
long waves are shortened, and the velocity of the peak ap-
pears to be reversed.
By a similar argument, Eq. ͑25͒ tells us that in the vacuum
region beyond the medium the peak of the pulse propagates
according to zϭctϩa(1/cϪ1/vg). The group velocity is
again c, but the ‘‘rephasing’’ within the medium results in a
shift of the position of the peak by the amount a(1/c
Ϫ1/vg). In a normal medium where 0Ͻvgрc the shift is
negative; the pulse appears to have been delayed during its
passage through the medium. But after a negative group ve-
locity medium, the pulse appears to have advanced!
This advance is possible because, in the Fourier view,
each component wave extends over all space, even if the
pulse appears to be restricted. The unusual ‘‘rephasing’’ in a
negative group velocity medium shifts the phases of the fre-
quency components of the wave train in the region ahead of
the nominal peak such that the phases all coincide, and a
peak is observed, at times earlier than expected at points
beyond the medium.
As shown in Fig. 3 and further illustrated in the examples
in the following, the ‘‘rephasing’’ can result in the simulta-
neous appearance of peaks in all three regions.
Fig. 3. A snapshot of three Fourier components of a pulse in the vicinity of a negative group velocity medium. The component at the central wavelength ␭0
is unaltered by the medium, but the wavelength of a longer wavelength component is shortened, and that of a shorter wavelength component is lengthened.
Then, even when the incident pulse has not yet reached the medium, there can be a point inside the medium at which all components have the same phase,
and a peak appears. Simultaneously, there can be a point in the vacuum region beyond the medium at which the Fourier components are again all in phase,
and a third peak appears. The peaks in the vacuum regions move with group velocity vgϭc, but the peak inside the medium moves with a negative group
velocity, shown as vgϭϪc/2. The phase velocity vp is c in vacuum, and close to c in the medium.
610 610Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

11. D. Propagation of a sharp wave front
To assess the effect of a medium with negative group ve-
locity on the propagation of a signal, we first consider a
waveform with a sharp front, as recommended by Sommer-
feld and Brillouin.12
As an extreme but convenient example, we take the inci-
dent pulse to be a Dirac delta function, E(z,t)ϭE0␦(z/c
Ϫt). Inserting this in Eq. ͑28͒, which is based on the linear
approximation ͑23͒, we find
E͑z,t͒Ϸ
Ά
E0␦͑z/cϪt͒ ͑zϽ0͒
E0ei␻0z͑n͑␻0͒/cϪ1/vg͒
␦͑z/vgϪt͒ ͑0ϽzϽa͒
E0ei␻0a͑n͑␻0͒/cϪ1/vg͒
␦͑z/cϪtϪa͑1/cϪ1/vg͒͒
͑aϽz͒.
͑29͒
According to Eq. ͑29͒, the delta-function pulse emerges
from the medium at zϭa at time tϭa/vg . If the group ve-
locity is negative, the pulse emerges from the medium before
it enters at tϭ0!
A sample history of ͑Gaussian͒ pulse propagation is illus-
trated in Fig. 4. Inside the negative group velocity medium,
an ͑anti͒pulse propagates backwards in space from zϭa at
time tϭa/vgϽ0 to zϭ0 at time tϭ0, at which point it ap-
pears to annihilate the incident pulse.
This behavior is analogous to barrier penetration by a rela-
tivistic electron20
in which an electron can emerge from the
far side of the barrier earlier than it hits the near side, if the
electron emission at the far side is accompanied by positron
emission, and the positron propagates within the barrier so as
to annihilate the incident electron at the near side. In the
Wheeler–Feynman view, this process involves only a single
electron which propagates backwards in time when inside
the barrier. In this spirit, we might say that pulses propagate
backwards in time ͑but forward in space͒ inside a negative
group velocity medium.
The Fourier components of the delta function are indepen-
dent of frequency, so the advanced appearance of the sharp
wave front as described by Eq. ͑29͒ can occur only for a gain
medium such that the index of refraction varies linearly at all
frequencies. If such a medium existed with negative slope
dn/d␻, then Eq. ͑29͒ would constitute superluminal signal
propagation.
However, from Fig. 1 we see that a linear approximation
to the index of refraction is reasonable in the negative group
velocity medium only for ͉␻Ϫ␻0͉Շ⌬/2. The sharpest wave
front that can be supported within this bandwidth has char-
acteristic rise time ␶Ϸ1/⌬.
For the experiment of Wang et al. where ⌬/2␲Ϸ106
Hz,
an analysis based on Eq. ͑23͒ would be valid only for pulses
with ␶տ0.1 ␮s. Wang et al. used a pulse with ␶Ϸ1 ␮s,
close to the minimum value for which Eq. ͑23͒ is a reason-
able approximation.
Since a negative group velocity can only be experienced
over a limited bandwidth, very sharp wave fronts must be
excluded from the discussion of signal propagation. How-
ever, it is well known12
that great care must be taken when
discussing the signal velocity if the waveform is not sharp.
E. Propagation of a Gaussian pulse
We now consider a Gaussian pulse of temporal length ␶
centered on frequency ␻0 ͑the carrier frequency͒, for which
the incident waveform is
Fig. 4. Ten ‘‘snapshots’’ of a Gaussian pulse as it traverses a negative group
velocity region (0ϽzϽ50), according to Eq. ͑31͒. The group velocity in the
gain medium is vgϭϪc/2, and c has been set to 1.
611 611Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

12. E͑z,t͒ϭE0eϪ͑z/cϪt͒2/2␶2
ei␻0z/c
eϪi␻0t
͑zϽ0͒. ͑30͒
Inserting this in Eq. ͑28͒ we find
E͑z,t͒ϭ
Ά
E0eϪ͑z/cϪt͒2/2␶2
ei␻0͑z/cϪt͒
͑zϽ0͒
E0eϪ͑z/vgϪt͒2/2␶2
ei␻0͑n͑␻0͒z/cϪt͒
͑0ϽzϽa͒
E0ei␻0a͑n͑␻0͒Ϫ1͒/c
eϪ͑z/cϪa͑1/cϪ1/vg͒Ϫt͒2/2␶2
ϫei␻0͑z/cϪt͒
͑aϽz͒.
͑31͒
The factor ei␻0a(n(␻0)Ϫ1)/c
in Eq. ͑31͒ for aϽz becomes
e␻p
2
␥a/⌬2c
using Eq. ͑11͒, and represents a small gain due to
traversing the negative group velocity medium. In the experi-
ment of Wang et al., this factor was only 1.16.
We have already noted in the previous section that the
linear approximation to ␻n(␻) is only good over a fre-
quency interval about ␻0 of order ⌬, and so Eq. ͑31͒ for the
pulse after the gain medium applies only for pulse widths
␶տ
1
⌬
. ͑32͒
Further constraints on the validity of Eq. ͑31͒ can be ob-
tained using the expansion of ␻n(␻) to second order. For
this, we repeat the derivation of Eq. ͑31͒ in slightly more
detail. The incident Gaussian pulse ͑30͒ has the Fourier de-
composition ͑27͒,
E␻͑z͒ϭ
␶
ͱ2␲
E0eϪ␶2
͑␻Ϫ␻0͒2/2
ei␻z/c
͑zϽ0͒. ͑33͒
We again extrapolate the Fourier component at frequency ␻
into the region zϾ0 using Eq. ͑20͒, which yields
E␻͑z͒ϭ
␶
ͱ2␲
E0eϪ␶2
͑␻Ϫ␻0͒2/2
ei␻nz/c
͑0ϽzϽa͒. ͑34͒
We now approximate the factor ␻n(␻) by its Taylor ex-
pansion through second order:
␻n͑␻͒Ϸ␻0n͑␻0͒ϩ
c
vg
͑␻Ϫ␻0͒
ϩ
1
2
d2
͑␻n͒
d␻2 ͯ␻0
͑␻Ϫ␻0͒2
. ͑35͒
With this, we find from Eqs. ͑26͒ and ͑34͒ that
E͑z,t͒ϭ
E0
A
eϪ͑z/vgϪt͒2/2A2␶2
ei␻0n͑␻0͒z/c
eϪi␻0t
͑0ϽzϽa͒. ͑36͒
where
A2
͑z͒ϭ1Ϫi
z
c␶2
d2
͑␻n͒
d␻2 ͯ␻0
. ͑37͒
The waveform for zϾa is obtained from that for 0ϽzϽa by
the substitutions ͑22͒ with the result
E͑z,t͒ϭ
E0
A
ei␻0a͑n͑␻0͒Ϫ1͒/c
eϪ͑z/cϪa͑1/cϪ1/vg͒Ϫt͒2/2A2␶2
ϫei␻0z/c
eϪi␻0t
͑aϽz͒, ͑38͒
where A is evaluated at zϭa here. As expected, the forms
͑36͒ and ͑38͒ revert to those of Eq. ͑31͒ when
d2
(␻n(␻0))/d␻2
ϭ0.
So long as the factor A(a) is not greatly different from
unity, the pulse emerges from the medium essentially undis-
torted, which requires
a
c␶
Ӷ
1
24
⌬2
␻p
2
⌬
␥
⌬␶, ͑39͒
using Eqs. ͑18͒ and ͑37͒. In the experiment of Wang et al.,
this condition is that a/c␶Ӷ1/120, which was well satisfied
with aϭ6 cm and c␶ϭ300 m.
As in the case of the delta function, the centroid of a
Gaussian pulse emerges from a negative group velocity me-
dium at time
tϭ
a
vg
Ͻ0, ͑40͒
which is earlier than the time tϭ0 when the centroid enters
the medium. In the experiment of Wang et al., the time ad-
vance of the pulse was a/͉vg͉Ϸ300a/cϷ6ϫ10Ϫ8
s
Ϸ0.06␶.
If one attempts to observe the negative group velocity
pulse inside the medium, the incident wave would be per-
turbed and the backwards-moving pulse would not be de-
tected. Rather, one must deduce the effect of the negative
group velocity medium by observation of the pulse that
emerges into the region zϾa beyond that medium, where the
significance of the time advance ͑40͒ is the main issue.
The time advance caused by a negative group velocity
medium is larger when ͉vg͉ is smaller. It is possible that
͉vg͉Ͼc, but this gives a smaller time advance than when the
negative group velocity is such that ͉vg͉Ͻc. Hence, there is
no special concern as to the meaning of negative group ve-
locity when ͉vg͉Ͼc.
The maximum possible time advance tmax by this tech-
nique can be estimated from Eqs. ͑17͒, ͑39͒, and ͑40͒ as
tmax
␶
Ϸ
1
12
⌬
␥
⌬␶Ϸ1. ͑41͒
The pulse can advance by at most a few rise times due to
passage through the negative group velocity medium.
While this aspect of the pulse propagation appears to be
superluminal, it does not imply superluminal signal propaga-
tion.
In accounting for signal propagation time, the time needed
to generate the signal must be included as well. A pulse with
a finite frequency bandwidth ⌬ takes at least time ␶Ϸ1/⌬ to
be generated, and so is delayed by a time of order of its rise
time ␶ compared to the case of an idealized sharp wave front.
Thus, the advance of a pulse front in a negative group veloc-
ity medium by Շ␶ can at most compensate for the original
delay in generating that pulse. The signal velocity, as defined
by the path length between the source and detector divided
by the overall time from onset of signal generation to signal
detection, remains bounded by c.
As has been emphasized by Garrett and McCumber13
and
by Chiao,18,19
the time advance of a pulse emerging from a
gain medium is possible because the forward tail of a smooth
pulse gives advance warning of the later arrival of the peak.
The leading edge of the pulse can be amplified by the gain
medium, which gives the appearance of superluminal pulse
612 612Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

13. velocities. However, the medium is merely using information
stored in the early part of the pulse during its ͑lengthy͒ time
of generation to bring the apparent velocity of the pulse
closer to c.
The effect of the negative group velocity medium can be
dramatized in a calculation based on Eq. ͑31͒ in which the
pulse width is narrower than the gain region ͓in violation of
condition ͑39͔͒, as shown in Fig. 4. Here, the gain region is
0ϽzϽ50, the group velocity is taken to be Ϫc/2, and c is
defined to be unity. The behavior illustrated in Fig. 4 is per-
haps less surprising when the pulse amplitude is plotted on a
logarithmic scale, as in Fig. 5. Although the overall gain of
the system is near unity, the leading edge of the pulse is
amplified by about 70 orders of magnitude in this example
͓the implausibility of which underscores that condition ͑39͒
cannot be evaded͔, while the trailing edge of the pulse is
attenuated by the same amount. The gain medium has tem-
porarily loaned some of its energy to the pulse permitting the
leading edge of the pulse to appear to advance faster than the
speed of light.
Our discussion of the pulse has been based on a classical
analysis of interference, but, as remarked by Dirac,21
classi-
cal optical interference describes the behavior of the wave
functions of individual photons, not of interference between
photons. Therefore, we expect that the behavior discussed
above will soon be demonstrated for a ‘‘pulse’’ consisting of
a single photon with a Gaussian wave packet.
ACKNOWLEDGMENTS
The author thanks Lijun Wang for discussions of his ex-
periment, and Alex Granik for references to the early history
of negative group velocity and for the analysis contained in
Eqs. ͑14͒–͑16͒.
a͒
Electronic mail: mcdonald@puphed.princeton.edu
1
L. V. Hau et al., ‘‘Light speed reduction to 17 metres per second in an
ultracold atomic gas,’’ Nature ͑London͒ 397, 594–598 ͑1999͒.
2
K. T. McDonald, ‘‘Slow light,’’ Am. J. Phys. 68, 293–294 ͑2000͒. A
figure to be compared with Fig. 1 of the present paper has been added in
the version at http://arxiv.org/abs/physics/0007097
3
This is in contrast to the ‘‘⌳’’ configuration of the three-level atomic
system required for slow light ͑Ref. 2͒ where the pump laser does not
produce an inverted population, in which case an adequate classical de-
scription is simply to reverse the sign of the damping constant for the
pumped oscillator.
4
L. J. Wang, A. Kuzmich, and A. Dogariu, ‘‘Gain-assisted superluminal
light propagation,’’ Nature ͑London͒ 406, 277–279 ͑2000͒. Their website,
http://www.neci.nj.nec.com/homepages/lwan/gas.htm, contains additional
material, including an animation much like Fig. 4 of the present paper.
5
W. R. Hamilton, ‘‘Researches respecting vibration, connected with the
theory of light,’’ Proc. R. Ir. Acad. 1, 267,341 ͑1839͒.
6
J. S. Russell, ‘‘Report on waves,’’ Br. Assoc. Reports ͑1844͒, pp. 311–
390. This report features the first recorded observations of solitary waves
͑p. 321͒ and of group velocity ͑p. 369͒.
7
G. G. Stokes, Problem 11 of the Smith’s Prize examination papers ͑2
February 1876͒, in Mathematical and Physical Papers ͑Johnson Reprint
Co., New York, 1966͒, Vol. 5, p. 362.
8
T. H. Havelock, The Propagation of Disturbances in Dispersive Media
͑Cambridge U.P., Cambridge, 1914͒.
9
H. Lamb, ‘‘On Group-Velocity,’’ Proc. London Math. Soc. 1, 473–479
͑1904͒.
10
See p. 551 of M. Laue, ‘‘Die Fortpflanzung der Strahlung in Dispergier-
enden und Absorbierenden Medien,’’ Ann. Phys. ͑Leipzig͒ 18, 523–566
͑1905͒.
11
L. Mandelstam, Lectures on Optics, Relativity and Quantum Mechanics
͑Nauka, Moscow, 1972͒; in Russian.
12
L. Brillouin, Wave Propagation and Group Velocity ͑Academic, New
York, 1960͒. That the group velocity can be negative is mentioned on p.
122.
13
C. G. B. Garrett and D. E. McCumber, ‘‘Propagation of a Gaussian Light
Pulse through an Anomalous Dispersion Medium,’’ Phys. Rev. A 1, 305–
313 ͑1970͒.
14
R. Y. Chiao, ‘‘Superluminal ͑but causal͒ propagation of wave packets in
transparent media with inverted atomic populations,’’ Phys. Rev. A 48,
R34–R37 ͑1993͒.
Fig. 5. The same as Fig. 4, but with the electric field plotted on a logarith-
mic scale from 1 to 10Ϫ65
.
613 613Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

14. 15
E. L. Bolda, J. C. Garrison, and R. Y. Chiao, ‘‘Optical pulse propagation
at negative group velocities due to a nearby gain line,’’ Phys. Rev. A 49,
2938–2947 ͑1994͒.
16
M. W. Mitchell and R. Y. Chiao, ‘‘Causality and negative group delays in
a simple bandpass amplifier,’’ Am. J. Phys. 68, 14–19 ͑1998͒.
17
A. M. Steinberg and R. Y. Chiao, ‘‘Dispersionless, highly superluminal
propagation in a medium with a gain doublet,’’ Phys. Rev. A 49, 2071–
2075 ͑1994͒.
18
R. Y. Chiao, ‘‘Population Inversion and Superluminality,’’ in Amazing
Light, edited by R. Y. Chiao ͑Springer-Verlag, New York, 1996͒, pp.
91–108.
19
R. Y. Chiao and A. M. Steinberg, ‘‘Tunneling Times and Superluminal-
ity,’’ in Progress in Optics, edited by E. Wolf ͑Elsevier, Amsterdam,
1997͒, Vol. 37, pp. 347–405.
20
See p. 943 of R. P. Feynman, ‘‘A Relativistic Cut-Off for Classical Elec-
trodynamics,’’ Phys. Rev. 74, 939–946 ͑1948͒.
21
P. A. M. Dirac, The Principles of Quantum Mechanics ͑Clarendon, Ox-
ford, 1958͒, 4th ed., Sec. 4.
Forces in complex fluids
Bruce J. Ackersona)
and Anitra N. Novyb)
Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078-3072
͑Received 25 August 2000; accepted 18 December 2000͒
͓DOI: 10.1119/1.1351151͔
I. PROBLEM
Ferrofluids1
are stable suspensions of magnetic particles
having linear dimension on the order of 10 nm. Due to vig-
orous Brownian motion the magnetic particles assume ran-
dom orientations rendering the suspension as a whole para-
magnetic. These complex fluids show a variety of phe-
nomena and instabilities that amuse and delight students and
teachers alike.2
Because these fluids are used in a variety of
applications including rotary seals, sensors, and actuators,3
they are commercially available.4
Figure 1 shows the experimental apparatus for viewing
and recording the response of a ferrofluid film trapped at an
air–water interface. Figure 2 shows recorded images for a
drop ͑ϳ40 ␮l͒ of mineral-oil-based ferrofluid5
introduced to
the surface of clean, filtered, de-ionized ͑18 M⍀͒ water. The
hydrophobic ferrofluid spreads uniformly over the surface of
water contained in a Petri dish. We gently stir the surface to
emulsify the film, creating a collection of dark flat circular
drops of ferrofluid as recorded in Fig. 2͑a͒. Figure 2͑b͒
shows the film 1 min after a cylindrical magnet having a
radius of 1 cm is introduced with the axis of symmetry ver-
tical and the lower end 3.3 cm above the ferrofluid film. The
ferrofluid film clears from directly beneath the magnet but
moves radially inward at large distances, forming tear-
shaped drops with the clearer regions streaming outward.
The ferrofluid collects in a ring structure at a finite radius
͑which is most dense at radius ϳ1.0 cm͒ from the center of
the magnetic field symmetry axis. As the ferrofluid builds up,
clumps or cone-shaped structures develop. As the cones
grow, they become unstable and migrate one at a time into
the central region. Figure 2͑c͒ taken at 31
4 min shows the
clumping in the ring-shaped structure with one cone at two
o’clock escaping to the central region. Finally in Fig. 2͑d͒,
taken 21 min after introducing the magnet, a regular ‘‘crys-
talline’’ array of well-separated ferrofluid cones has formed.
Yet there remains a ferrofluid film ring surrounding this crys-
talline structure.
How is it possible that the ferrofluid is both attracted to
͑cones͒ and repelled from ͑film͒ the region directly below the
cylindrical magnet?
II. SOLUTION
Magnetic body forces, surface tension, viscous drag, and
gravity combine to produce the peculiar behavior observed
here. We focus on the magnetic body forces as the primary
explanation for the question posed above. Figure 3 shows a
side view of one of the ‘‘cones’’ of ferrofluid which form the
two-dimensional crystalline array beneath the magnet. These
clumps have a nearly ellipsoidal shape above the water sur-
face with the symmetry axis parallel to the applied field.
Other studies show that ferrofluid droplets submerged in an
immiscible fluid deform into ellipsoids and align parallel to
the direction of a uniform external magnetic field.6
Solutions
have long existed for the magnetic ͑electric͒ field of an el-
lipsoid of permeability ␮ ͑dielectric constant ⑀͒ subjected to
a uniform external field in a surrounding medium of perme-
ability ␮0 ͑dielectric constant ⑀0͒.7
When the symmetry axis
of the ellipse is aligned parallel to the external field direction,
the field inside the ellipse is uniform and parallel to the ap-
plied field ͑at large distances͒. Thus, to first order the ‘‘ellip-
soidal cones’’ behave like little magnets oriented parallel to
the external field and so move along the water surface to the
strongest field regions located directly beneath the cylindri-
cal magnet. However, the magnetic field induced in each of
these cones is aligned parallel with the neighboring cone
fields; therefore, the cones repel one another in the plane of
the interface just like parallel oriented permanent magnets. A
crystal lattice results.8
These results are qualitative, but in-
tuitive, given our experience playing with permanent mag-
nets.
How do we understand the quite different behavior of the
film? Rosensweig1
gives a general derivation of the body
force f or force per unit volume, which reduces for ferrofluid
suspensions to
fϭ␮0͑M"ٌ͒Hϭ␮0M“H, ͑1͒
where ␮0 is the vacuum permeability, H is the magnetic field
strength, and M is the magnetization in the film volume el-
ement. This functional form suggests the Kelvin force den-
sity on an isolated body, except that the local field H re-
places the applied field H0 . Intuitively, we understand this
body force to be like the force acting on a magnetic dipole.
614 614Am. J. Phys. 69 ͑5͒, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers

15. The force depends on the orientation of the dipole ͑repre-
sented by M͒ in the local field ͑represented by H͒ and on the
gradient of the local field in the vicinity of the dipole. There
is no force, only a torque, acting on a dipole in a uniform
field. The derivation of the last equality in Eq. ͑1͒ reasonably
assumes that “ÃHϭ0 and that the magnetic flux density B
is linearly related to the magnetic field strength and to the
magnetization as follows:
Bϭ␮Hϭ␮0͑HϩM͒ ͑2͒
with ␮ the permeability of the film. Using Eq. ͑2͒ to repre-
sent the body force in terms of the magnetic flux density in
the second equality of Eq. ͑1͒ gives
fϭ
1
␮ ͩ1Ϫ
␮0
␮ ͪB“Bϭ
1
2␮ ͩ1Ϫ
␮0
␮ ͪ“B2
. ͑3͒
Consequently, the negative square of the magnetic flux den-
sity within the film corresponds to a ‘‘potential field’’ to
which the film responds.
Consider the film to be of constant thickness and spread
uniformly over the surface of the water when in the presence
of the external magnetic field. In the absence of magnetic
charge and surface current density, consideration of the Max-
well equations leads to the following boundary conditions
for the magnetic field strength and magnetic flux density:
͑BϪB0͒"nϭ0, ͑HϪH0͒Ãnϭ0, ͑4͒
where the terms with a subscript refer to the air ͑or vacuum͒
side of the film boundary and the terms without a subscript to
the ferrofluid. The vector n is a unit normal to the film sur-
face. Note that these are the same boundary conditions and
field relationships used to find the field inside an ellipsoid
subjected to a uniform external field. For the film problem,
we take the normal to the film, the zˆ direction, to be parallel
to the magnet axis and assume azimuthal symmetry. The unit
vector ␳ˆ designates the radial direction with respect to the
Fig. 3. Ellipsoidal ‘‘clump’’ or cone of ferrofluid.
Fig. 1. Experimental setup for viewing the dynamics of a ferrofluid film.
Fig. 2. Ferrofluid film before the magnet was introduced ͑a͒, after 1 min ͑b͒, after 3
1
4 min ͑c͒, and after 21 min ͑d͒. Note that any dark spots that appear the
same in all frames are due to imperfections in the optical train.
615 615Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

16. magnet axis. Using the relationships given in Eqs. ͑2͒ and
͑4͒, the magnetic flux density and magnetic field strength just
inside the film are represented in terms of the magnetic flux
density just outside the film as
BϭB0zzˆϩ
␮
␮0
B0␳␳ˆ,
͑5͒
Hϭ
1
␮
B0zzˆϩ
1
␮0
B0␳␳ˆ.
Since the film is thin, we assume these boundary results to
represent the field within the film.
Next we approximate the external field at the surface of
the film by a dipole field with moment m directed parallel
to zˆ,
B0ϭ
␮0
4␲r3 ͑3͑m"rˆ͒rˆϪm͒
ϭ
␮0m
4␲ ͫ 3␳z
͑␳2
ϩz2
͒5/2 ␳ˆ ϩ
2z2
Ϫ␳2
͑␳2
ϩz2
͒5/2 zˆ ͬ, ͑6͒
where z measures the distance from the film to the dipole
source ͑cylindrical magnet͒, ␳ is the radial position in the
film, rϭͱ␳2
ϩz2
is the position vector magnitude in spheri-
cal coordinates, and the vectors with a caret are unit vectors.
This magnetic flux density, when plugged into the right-hand
side of Eq. ͑5͒, gives an estimate of the magnetic flux density
inside the film. Directly beneath the dipole the magnetic flux
density is normal to the film and has the same magnitude on
both sides of the film surface. As ␳ increases from zero,
however, there is a component of the magnetic flux density
which is parallel to the surface and larger inside the film than
outside for ␮/␮0Ͼ1. While the magnitude of the external
magnetic flux density decreases monotonically with increas-
ing ␳ for this dipole field, the magnetic flux density inside
the film will increase for sufficiently large permeability ratio,
␮/␮0 , before decreasing to zero. The increasing magnetic
flux density results in a force ͓see Eq. ͑3͔͒ that pushes the
film outward radially to collect in a ring where the flux den-
sity reaches a maximum. Note that we neglected contribu-
tions from the film to the external magnetic field on the
assumption that this field is small for a very thin film. While
an exact solution to this problem is beyond the scope of a
typical undergraduate course in electromagnetism, we
checked our results using the method of images to find the
magnetic flux density produced by a dipole near a thin plane
of material with permeability ␮. In the limit that the thick-
ness goes to zero, the fields inside and outside the film be-
come identical to those approximated here. There is one ca-
veat. The body force must be calculated using the second
equality in Eq. ͑1͒. The first equality involves a derivative
with respect to z, which must be performed before the film
thickness is taken to zero.
Figure 4 shows the reduced potential, estimated for our
ferrofluid film using the dipole field given in Eq. ͑6͒, as a
function of ␳ for zϭ1, and different values of the permeabil-
ity ratio ␮/␮0 . The functional form of this reduced potential
is
⌽ϭϪ
B2
͑␮0m/4␲͒2
ϭϪͩ␳4
ϩ͑Ϫ4ϩ9͑␮/␮0͒2
͒␳2
z2
ϩ4z4
͑␳2
ϩz2
͒5 ͪ. ͑7͒
For permeability ratios greater than ͱ8/3Ϸ1.63 a potential
minimum obtains at finite radius. The minimum position ␳min
obeys the following linear relationship with respect to z:
␳min /zϭͱϪ6͑␮/␮0͒2
ϩ3ϩͱ36͑␮/␮0͒4
Ϫ33͑␮/␮0͒2
ϩ1.
͑8͒
This ratio ranges between zero and one half as the perme-
ability ratio increases from 1.63 to infinity. Since the initial
magnetic susceptibility of our ferrofluid is ␹ϭ1.9, the per-
meability ratio is 2.9, and Eq. ͑8͒ predicts ␳min /zϭ0.43. Ap-
proximating the magnet by a dipole placed at the lower end
of the magnet, the end closest to the ferrofluid, gives ␳min /z
ϳ1.0 cm/3.3 cmϳ0.30 or placing the dipole at the physical
center of the magnet gives ␳min /zϳ1.0 cm/5.8 cmϳ0.17.
Considering that the magnet is an extended source rather
than a point dipole, the agreement is quite good.
The response of a ferrofluid film to a nonuniform external
field is complex, but a fairly straightforward undergraduate
boundary value calculation explains puzzling observations.
The large susceptibility of the magnetic fluid and fluid sur-
face orientation beneath a magnet determines the net force
on the ferrofluid, giving attraction for the cones and repul-
sion for the film.
a͒
Author to whom correspondence should be addressed; electronic mail:
bjack@okstate.edu
b͒
This communication originated as an honors thesis project of A.N., who is
presently a graduate student in physics at Colorado State University.
1
R. E. Rosensweig, Ferrohydrodynamics ͑Cambridge U.P., Cambridge,
1985͒, p. 110 ff.
2
R. E. Rosensweig, ‘‘Magnetic Fluids,’’ Sci. Am. 247 ͑4͒, 136–145 ͑1982͒.
3
B. M. Berkovsky, V. F. Medvedev, and M. S. Krakov, Magnetic Fluids
Engineering Applications ͑Oxford U.P., Oxford, 1993͒, Chap. 6.
4
Ferrofluidics Corporation, 40 Simon Street, Nashua, NH 03060-3075.
5
Ferrofluidics: Catalog No. EMG 905, Lot No. F8193A.
6
V. I. Arkhipenko, Yu. D. Barkov, and V. G. Bashtovoi, ‘‘Study of a
magnetized fluid drop shape in a homogeneous magnetic field,’’ Magn.
Gidrodin. ͑3͒, 131–134 ͑1978͒.
7
G. Arfken, Mathematical Methods for Physicists ͑Academic, New York,
1970͒, p. 603; S. D. Poisson, ‘‘Seconde me´moire sur la the´orie du mag-
ne´tisme,’’ Mem. Acad. R. Sci. Inst. France 5, 488–533 ͑1821–1822͒.
8
A. T. Skjeltorp, ‘‘One- and two-dimensional crystallization of magnetic
holes,’’ Phys. Rev. Lett. 51 ͑25͒, 2306–2309 ͑1983͒.
Fig. 4. Reduced potential for an element of ferrofluid film for zϭ1 as a
function of the distance ␳ from the magnet axis and shown for different
values of the permeability ratio ␮/␮0ϭ1, 2, 2.9, and 5 from top to bottom.
616 616Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

17. A mechanical model that exhibits a gravitational critical radius
Kirk T. McDonalda)
Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544
͑Received 7 July 1999; accepted 21 December 2000͒
͓DOI: 10.1119/1.1351152͔
I. PROBLEM
A popular model at science museums ͑and also a science
toy1
͒ that illustrates how curvature can be associated with
gravity consists of a surface of revolution rϭϪk/z with z
Ͻ0 about a vertical axis z. The curvature of the surface,
combined with the vertical force of Earth’s gravity, leads to
an inward horizontal acceleration of kg/r2
for a particle that
slides freely on the surface in a circular, horizontal orbit.
Consider the motion of a particle that slides freely on an
arbitrary surface of revolution, rϭr(z)у0, defined by a con-
tinuous and differentiable function on some interval of z. The
surface may have a nonzero minimum radius R at which the
slope dr/dz is infinite. Discuss the character of oscillations
of the particle about circular orbits to deduce a condition that
there be a critical radius rcritϾR, below which the orbits are
unstable. That is, the motion of a particle with rϽrcrit rapidly
leads to excursions to the minimum radius R, after which the
particle falls off the surface.
Give one or more examples of analytic functions r(z) that
exhibit a critical radius as defined above. These examples
provide a mechanical analogy as to how departures of gravi-
tational curvature from that associated with a 1/r2
force can
lead to a characteristic radius inside which all motion tends
toward a singularity.
II. SOLUTION
We work in a cylindrical coordinate system (r,␪,z) with
the z axis vertical. It suffices to consider a particle of unit
mass.
In the absence of friction, there is no torque on a particle
about the z axis, so the angular momentum component J
ϭr2
␪˙ about that axis is a constant of the motion, where the
dot ͑•͒ indicates differentiation with respect to time.
For motion on a surface of revolution rϭr(z), we have
r˙ϭrЈz˙, where the prime ͑
Ј͒ indicates differentiation with
respect to z. Hence, the kinetic energy can be written
Tϭ 1
2͑r˙2
ϩr2
␪˙ 2
ϩz˙2
͒ϭ 1
2͓z˙2
͑1ϩrЈ2
͒ϩr2
␪˙ 2
͔. ͑1͒
The potential energy is Vϭgz. Using Lagrange’s method,
the equation of motion associated with the z coordinate is
z¨͑1ϩrЈ2
͒ϩz˙2
rrЉϭϪgϩ
JrЈ
r3 . ͑2͒
For a circular orbit at radius r0 , we have
r0
3
ϭ
J2
r0Ј
g
. ͑3͒
We write ␪˙ 0ϭ⍀, so that Jϭr0
2
⍀.
For a perturbation about this orbit of the form
zϭz0ϩ⑀ sin ␻t, ͑4͒
we have, to order ⑀,
r͑z͒Ϸr͑z0͒ϩrЈ͑z0͒͑zϪz0͒ϭr0ϩ⑀r0Ј sin ␻t, ͑5͒
rЈϷr0Јϩ⑀r0Љ sin ␻t, ͑6͒
1
r3 Ϸ
1
r0
3 ͩ1Ϫ3⑀
r0Ј
r0
sin ␻tͪ. ͑7͒
Inserting ͑4͒–͑7͒ into ͑2͒ and keeping terms only to order
⑀, we obtain
Ϫ⑀␻2
͑1ϩr0Ј2
͒sin ␻t
ϷϪgϩ
J2
r0
3 ͩr0ЈϪ3⑀
r0Ј2
r0
sin ␻tϩ⑀ r0Љ sin ␻tͪ. ͑8͒
From the zero’th-order terms we recover ͑3͒, and from the
order-⑀ terms we find that
␻2
ϭ⍀2
3r0Ј2
Ϫr0r0Љ
1ϩr0Ј2 . ͑9͒
The orbit is unstable when ␻2
Ͻ0, i.e., when
r0r0ЉϾ3r0Ј2
. ͑10͒
This condition has the interesting geometrical interpretation
͑noted by a referee͒ that the orbit is unstable wherever
͑1/r2
͒ЉϽ0, ͑11͒
i.e., where the function 1/r2
is concave inwards.
For example, if rϭϪk/z, then 1/r2
ϭz2
/k2
is concave
outwards, ␻2
ϭJ2
/(k2
ϩr0
4
), and there is no regime of insta-
bility.
We give three examples of surfaces of revolution that sat-
isfy condition ͑11͒.
First, the hyperboloid of revolution defined by
r2
Ϫz2
ϭR2
, ͑12͒
where R is a constant. Here, r0Јϭz0 /r0 , r0ЉϭR2
/r0
3
, and
␻2
ϭ⍀2
3z0
2
ϪR2
2z0
2
ϩR2 ϭ⍀2
3r0
2
Ϫ4R2
2r0
2
ϪR2 . ͑13͒
The orbits are unstable for
z0Ͻ)R, ͑14͒
or equivalently, for
r0Ͻ
2)
3
Rϭ1.1547Rϵrcrit. ͑15͒
As r0 approaches R, the instability growth time approaches
an orbital period.
Another example is the Gaussian surface of revolution,
617 617Am. J. Phys. 69 ͑5͒, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers

18. r2
ϭR2
ez2
, ͑16͒
which has a minimum radius R, and a critical radius rcrit
ϭRͱ4
eϭ1.28R.
Our final example is the surface
rϭϪ
k
zͱ1Ϫz2
͑Ϫ1ϽzϽ0͒, ͑17͒
which has a minimum radius of Rϭ2k, approaches the sur-
face rϭϪk/z at large r ͑small z͒, and has a critical radius of
rcritϭ6k/ͱ5ϭ1.34R.
These examples arise in a 2ϩ1 geometry with curved
space but flat time. As such, they are not fully analogous to
black holes in 3ϩ1 geometry with both curved space and
curved time. Still, they provide a glimpse as to how a particle
in curved space–time can undergo considerably more com-
plex motion than in flat space–time.
ACKNOWLEDGMENTS
The author wishes to thank Ori Ganor and Vipul Periwal
for discussions of this problem.
a͒
Electronic mail: mcdonald@puphed.princeton.edu
1
The Vortx͑tm͒ Miniature Wishing Well, Divnick International, Inc., 321 S.
Alexander Road, Miamisburg, OH 45342, http://www.divnick.com/
AWESTRUCK SCIENTISTS
The second feature of science is that it shows that the world is simple. Even many scientists do
not appreciate that they are hewers of simplicity from complexity. They are often more deluded
than those they aim to tell. Scientists are often overawed by the complexity of detecting simplicity.
They look at the latest fundamental particle experiment, see that it involves a thousand kilograms
of apparatus and a discernible percentage of a gross national product, and become thunderstruck.
They see the complexity of the apparatus and the intensity of the effort needed to construct and
operate it, and confuse that with the simplicity that the experiment, if successful, will expose.
Some scientists are so awestruck that they even turn to religion! Others keep a cool head, and
marvel not at an implied design but at the richness of simplicity.
P. W. Atkins, ‘‘The Limitless Power of Science,’’ in Nature’s Imagination—The Frontiers of Scientific Vision, edited by
John Cornwell ͑Oxford University Press, New York, 1995͒.
618 618Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems