One century later: Remarks on the Barnett experiment
Alexander L. Kholmetskiia)
Department of Physics, Belarus State University, 4, F. Skorina Avenue, 220080 Minsk, Belarus
共Received 16 May 2002; accepted 10 January 2003兲
The Barnett experiment 共the rotation of either a solenoid or a short-circuited cylindrical capacitor
about a common axis兲 was performed at the beginning of the 20th century to choose between the
Hertz, Maxwell, or Lorentz electrodynamics theories. The results obtained by Barnett led him to
conclude that both the electrodynamics of Hertz and Maxwell should be rejected, while Lorentz was
solely correct. We analyze this experiment from a modern point of view and show that the results
can be explained within the framework of Maxwell electrodynamics and relativity theory. However,
the explanation still has some physical difficulties. © 2003 American Association of Physics Teachers.
关DOI: 10.1119/1.1557300兴
I. INTRODUCTION
The experiment by Barnett1,2 was carried out at the beginning of the 20th century to resolve the problem of unipolar
induction. At that time it was an exciting problem closely
related to the question: do the lines of magnetic induction
move as if they were rigidly attached to the magnet, or do the
lines remain fixed in space under the rotation of magnet? It
was expected that experimental resolution of this problem
共formulated in the archaic language of the beginning of the
20th century兲 would select either the Hertz, Maxwell, or Lorentz electrodynamic theory. The Hertz theory was associated
with a fully entrained ether, the Lorentz theory was associated with nonentrained ether, while Maxwell’s electrodynamics was based on special relativity, independent of any ether
model. It seemed that Barnett devised the most interesting
way of experimentally resolving the problem, and provided
the most convincing results in comparison with other workers. He concluded that the lines of magnetic induction remain fixed in space under the rotation of a magnet, in agreement only with Lorentz electrodynamics and Lorentz ether
theory.
Later it was recognized that Barnett’s experiment does not
contradict the special theory of relativity: Einstein’s relativity
principle is valid for inertial motion solely, and in general, it
does not imply a relativity of rotational motion. During the
20th century the problems of classical electrodynamics with
rotational motion and related problems of unipolar induction
were
extensively
investigated
theoretically
and
experimentally.3–14 Nevertheless, Barnett’s experiment still
has no consistent modern explanation.
II. DESCRIPTION OF THE EXPERIMENT BY
BARNETT
A simplified schematic of the experiment is shown in Figs.
1 and 2. There is a solenoid with a current I and a cylindrical
ជ
capacitor inside the solenoid, where a magnetic induction B
is parallel to the axis z. For an elongated solenoid the value
of Bជ can be taken to be constant in the inner region far from
the boundaries of the solenoid. The capacitor is shortcircuited by a conducting wire A. Both the solenoid and the
capacitor are mounted coaxially and can rotate about the
common axis z. An electrical connection between facings of
the capacitor breaks during rotation, and after the current
goes to zero and the rotation stops, the charge on the capaci558
Am. J. Phys. 71 共6兲, June 2003
http://ojps.aip.org/ajp/
tor is measured. As the capacitor rotates, it acquires a charge
Q due to the Lorentz force applied to the wire A moving in
ជ . Barnett did not perform the experiment
the magnetic field B
with a rotating capacitor 关Fig. 2共a兲兴, because similar experiments had been already carried out by Faraday and many
others. For the rotating capacitor of Fig. 2共a兲 the charge Q
can be easy calculated. The goal of Barnett’s experiment is to
measure the charge on the capacitor after the rotation of the
solenoid 关Fig. 2共b兲兴 and to compare it with the calculated
value of Q.
The inner armature of the capacitor was a brass tube 14.9
cm long and an external diameter of 3.97 cm. The outer
armature was a brass tube 28.0 cm long and an internal diameter of 6.67 cm. The length of the solenoid exceeded the
length of the outer armature. During the rotation of the solenoid about the z axis at 20 rev/s, a connection between the
armatures is broken by a special key system after which the
field is reduced to zero. Then a sensitive electrometer was
used to measure the charge on the capacitor. The results of
this experiment and other modified Barnett experiments
showed that the capacitor acquired a charge that was not
more than 1.4% of the calculated Q within the precision of
the experiment. Hence, he concluded that the relativity principle is violated, and only Lorentz electrodynamics is able to
explain the experimental results.
III. MODERN EXPLANATION OF BARNETT
EXPERIMENT
In the following calculations we assume for simplicity that
aⰆr, where a is the length of the wire A in the laboratory
frame, and r is the modulus of the radius vector of any point
on the wire A. We will carry out all calculations to order
␻ r/c 共in Barnett’s experiment this ratio was about 10⫺8 ); we
will use units such that c⫽1 共c is the speed of light in
vacuum兲. To this order we first consider the experiment in
Fig. 2共a兲. If the capacitor rotates with frequency ␻ in the
ជ ⫻rជ ,
laboratory frame, the wire A has a linear velocity uជ ⬇ ␻
ជ
which is orthogonal to the magnetic field B at any moment.
Hence, the conduction electrons in A are subject to the Lorentz force
F⫽euB
共1兲
in the radial direction 共e is the charge of electron兲. This force
charges the capacitor to the potential difference
© 2003 American Association of Physics Teachers
558
It is interesting to analyze these experiments from the
viewpoint of an observer in an accelerated frame, attached to
either the capacitor 关Fig. 2共a兲, case 1: Q⫽0] or to the solenoid 关Fig. 2共b兲, case 2: Q⫽0]. 共The various cases discussed
in the following are summarized in Table I.兲 Under rotation
about the axis z at the constant angular frequency ␻, the
relation between the spatial coordinates of the laboratory
共rest兲 frame (x ⬘ ,y ⬘ ,z ⬘ ) and the spatial coordinates of the
rotating frame 共x, y, z兲 has the form
x ⬘ ⫽x cos共 ␻ t 兲 ⫺y sin共 ␻ t 兲 ,
Fig. 1. Simplified schematic of the Barnett experiment: 共a兲 top view and 共b兲
side view.
y ⬘ ⫽x sin共 ␻ t 兲 ⫹y cos共 ␻ t 兲 ,
共3兲
z ⬘ ⫽z.
The nonzero metric coefficients in such a rotating frame are
R0
R0
⌬V⫽auB ln ⫽ ␻ raB ln .
r0
r0
The measured charge is
Q⫽C⌬V⫽C ␻ raB ln
R0
.
r0
共2兲
Now let us calculate the charge on the capacitor for the
experiment in Fig. 2共b兲. Because of the rotation of the soleជ and magnetic Bជ field change inside.
noid, the electric field E
The magnetic field is not relevant because the capacitor is at
rest in the laboratory frame. To determine the electric field Eជ
inside the solenoid in the laboratory frame, we note that a
moving conductor with nonzero current acquires a nonzero
charge density ␳. A physical reason is the different scale
contraction for the systems of negatively and positively
charged particles in a conductor, because when I⫽0, they
have different circular speeds. 共Formally, ␳ can be found by
the transformation of the four-vector of the current density.兲
At the same time, due to the symmetry of Barnett’s experimental apparatus to rotations and translations with respect to
the axis z, the charge should be distributed homogeneously
on the surface of the solenoid. In the region far from the
boundaries of the solenoid, the electric field inside is equal to
zero for any value of the charge density ␳ according to electrostatics. Hence, we conclude that inside the solenoid the
electric field is equal to zero. Therefore, there are no electrical forces acting on the electrons in wire A, and the capacitor
does not acquire any charge. This reasoning explains the result of Barnett’s experiment. The results of the experiment
might imply that the concept of relative motion is not applicable, in general, to accelerated motion.
g 00⫽1⫺ ␻ 2 r 2 ,
g 11⫽g 22⫽g 33⫽⫺1,
g 01⫽g 10⫽ ␻ y,
g 02⫽g 20⫽⫺ ␻ x.
To determine the electric and magnetic fields in the rotating
frame, it is necessary to apply a corresponding transformation to the electromagnetic field tensor
F ik ⫽
⳵ x ⬘l ⳵ x ⬘m
F⬘ ,
⳵ x i ⳵ x k lm
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Am. J. Phys., Vol. 71, No. 6, June 2003
共5兲
taking into account its form in the laboratory frame with a
solenoid at rest:
⬘ ⫽
F lm
冦
0
0
0
0
0
0
⫺B
0
0
B
0
0
0
0
0
0
冧
共6兲
.
By substituting Eq. 共6兲 into Eq. 共5兲 and taking into account
Eq. 共3兲, we obtain the nonzero components of F in the rotating frame to the assumed accuracy of the calculations:
F 01⫽⫺F 10⫽⫺Bx ␻ ,
F 02⫽⫺F 20⫽⫺By ␻ ,
共7兲
F 12⫽⫺F 21⫽⫺B.
The nonzero components of the electric and magnetic fields
are found according to 共see, for example, Ref. 15兲
冑␥
E 1 ⫽⫺F 01g 11
11
g 00
,
E 2⫽
F 02
冑g 00
,
B 3 ⫽ 共 g 11F 12⫹g 01F 02兲 冑␥ 11,
共8兲
where ␥ is the metric tensor of three-dimensional space, related to the metric tensor of four-dimensional space–time g
by the relation
␥ ␣␤ ⫽⫺g ␣␤ ⫹
Fig. 2. 共a兲 Cylindrical capacitor rotates around the axis z at a constant
angular frequency ␻; 共b兲 solenoid rotates around the same axis at the angular frequency ⫺␻.
共4兲
g 0␣g 0␤
g 00
共 ␣ , ␤ ⫽1,...,3 兲 .
共9兲
Now let us consider the experiment in Fig. 2共a兲 共case 1,
Q⫽0 for a laboratory observer兲 from the viewpoint of an
observer rotating with the capacitor. 共We designate this case
CR.兲 For this observer the wire A does not move, and the
magnetic field is not relevant. Hence, the capacitor can only
be charged due to the action of the electric field on the conduction electrons in the wire A. If we use Eq. 共4兲 and Eqs.
共7兲–共9兲, we obtain E x ⫽⫺Bx ␻ and E y ⫽⫺By ␻ to the given
order of approximation. Thus, the electric field vector is in
Alexander L. Kholmetskii
559
Table I. Summary of Barnett’s experiment.
Conditions
Place of observer
Charge
Case 1.
Faraday’s experiment
共rotating capacitor兲
Case 2.
Barnett’s experiment
共rotating solenoid兲
Case CR.
Faraday’s experiment
Observer in laboratory
frame
Q⫽0
ជ ).
Capacitor is charged by the force e(uជ ÃB
Observer in laboratory
frame
Q⫽0
Capacitor is uncharged, because E⫽0 inside the solenoid for a
uniform charge distribution over the surface of the solenoid.
Observer attached to
rotating capacitor
Q⫽0
Case SR.
Barnett’s experiment
Observer attached to
rotating solenoid
Q⫽0
Case CRG. Faraday’s experiment
in a gravitation field
Observer attached to
rotating capacitor; it is
equivalent to case 2
according to local
Lorentz invariance
Observer attached to
rotating solenoid; it is
equivalent to case 1
according to local
Lorentz invariance
Observer in laboratory
frame
Q⫽0
ជ . The electric field E results
Capacitor is charged by the force eE
from the nonuniform charge distribution over the surface
of the solenoid.
Capacitor is uncharged, because the resultant Lorentz force
ជ ⫹e(uជ ÃBជ )⫽0 inside the solenoid. Here Eជ ,Bជ ⫽0. The
eE
ជ is induced by the nonuniform
compensating electric field E
charge distribution over the surface of the solenoid.
Capacitor should be uncharged as in the equivalent case 2.
Q⫽0
Capacitor should be charged as in the equivalent case 1.
Q⫽0
Observer in laboratory
frame
Q⫽0
Capacitor should be uncharged as in case CRG. Hence, the resultant
ជ ⫹e(uជ ÃBជ )⫽0 inside the solenoid.
Lorentz force eE
ជ ,Bជ ⫽0. At the same time, the gravitational field with the
Here E
axial symmetry creates a uniform charge distribution over the
surface of the solenoid and, according to the laws of electrostatics,
ជ should be equal to zero inside the solenoid.
the field E
Capacitor should be charged as in case SRG. A single possible
force for its charge is eEជ . At the same time, the gravitational
field with axial symmetry creates a uniform charge distribution
over the surface of the solenoid and, according to electrostatics,
ជ should be equal to zero inside the solenoid.
the field E
Case SRG. Barnett’s experiment
in a gravitation field
Case CG.
Faraday’s experiment in a
gravitation field
Case SG.
Barnett’s experiment in a
gravitation field
the radial direction, and its magnitude is E⫽uB. Hence, we
obtain the same expression, Eq. 共1兲, for the force charging
the capacitor, F⫽eE⫽euB.
It is interesting to find the origin of this electric field. The
reason for the creation of the electric field inside of the solenoid is the nonzero charge density ␳ on its surface. Its
value can be found using the transformation of the fourvector of current density
j i⫽
冉
␳
,
␳u␣
冑⫺g 冑⫺g
冊
from the laboratory to the rotating frame, applying the transformation law
j i⫽
⳵ x ⬘k
j⬘
⳵xi k
for the transformation 共3兲. 共Here g⫽det g.) We omit the
simple calculations and present the final result to the accuracy of the calculations. The charge density at any fixed point
on the solenoid is equal to ␳ ⫽ j ␻ R cos ␣, where j is the
current density of the solenoid in the laboratory frame, R is
the radius of the solenoid, and ␣ is the angle between the
axis of the wire A and the radius-vector in the rotating frame.
We conclude that ␳ is distributed nonuniformly over the surface of the solenoid.
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Comment
According to electrostatics, such a distribution of charge
creates a constant electric field in the inner volume of the
solenoid, which is collinear to the axis of the wire A. The
calculated electric field is E⫽uB, which coincides with the
value obtained by the direct transformation of the electric
and magnetic fields to the rotating frame. Thus, we conclude
that for the case CR, the electric field, which is responsible
for the charge of the capacitor, results from the nonuniform
distribution of charge over the surface of the solenoid.
Now let us consider the experiment in Fig. 2共b兲 共rotating
solenoid in a laboratory frame, case 2兲, and explain the result
(Q⫽0) for an observer attached to the solenoid 共case SR兲. In
its reference frame the capacitor rotates with the angular velocity ␻ and both the electric and magnetic fields can act on
the wire A. We use Eqs. 共4兲, 共7兲, and 共8兲 and obtain the
nonzero components of these fields, E x ⫽⫺ ␻ xB, E y
⫽⫺ ␻ yB, and B z ⫽B, to the desired accuracy. In such a case
ជ lies in the radial direction, and the forces eEជ and e(uជ
E
ជ ) act in opposite directions. The magnitude of the electric
⫻B
field vector is uB. Hence, we see that the total Lorentz force
per each electron of the wire A is
ជ ⫹e 共 uជ ÃBជ 兲 ⫽0.
qFជ ⫽eE
共10兲
These considerations mean that for case SR the capacitor
remains uncharged. As for the case CR, the electric field
Alexander L. Kholmetskii
560
inside of the solenoid is induced by the nonuniform distribution of charge over its surface.
The result of Barnett’s experiment is also useful for an
analysis of the principle of local Lorentz invariance if we
imagine that in this experiment we are able to create a gravitational field that is axially symmetrical with respect to the
axis z. Under rotation of the capacitor 共case 1兲, we choose
the gravitational potential to be equal to the centripetal acceleration of an observer on the outer surface of the capacitor
共case CRG兲. For rotation of the solenoid 共case 2兲, the gravitational potential is equal to the centripetal acceleration of an
observer on the surface of the solenoid 共case SRG兲. If the
radii of the solenoid and the outer armature of the capacitor
are close to each other, we can assume the same gravitational
field in both cases CRG and SRG. Then, according to local
Lorentz invariance, cases 2 and CRG, as well as cases 1 and
SRG should be equivalent. This equivalence means that for
an observer in the laboratory frame, the gravitational field
reverses the results of Barnett experiment: under rotation of
the capacitor it remains uncharged 共case CG兲, while under
rotation of the solenoid the capacitor is charged 共case SG兲.
Such a reversal of the Barnett experiment can only be
explained by the redistribution of currents and charges in
space in the presence of the gravitation field. In particular,
we require the appearance of the electric field in the solenoid, which compensates the action of the magnetic force on
the wire A under rotation of the capacitor 共case CG兲, and
charges the capacitor under rotation of the solenoid 共case
SG兲. At the same time, the gravitational field with axial symmetry induces a uniform distribution of the charge on the
surface of the solenoid in the Barnett experiment. In such a
case, according to the previously mentioned result of electrostatics, the electric field cannot appear inside the solenoid.
IV. CONCLUSION
We have seen that Barnett’s experiment can be fully explained within the framework of Maxwell’s electrodynamics
theory. A consideration of this experiment for the observers
attached to either the rotating capacitor or the rotating solenoid also explains the experimental observations: the rotating
capacitor acquires a charge, while it remains uncharged under the rotation of the solenoid. According to the principle of
local Lorentz invariance, the application of an external gravitational field with an axial symmetry relative to the rotational
axis should reverse the result of the experiment: the rotating
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Am. J. Phys., Vol. 71, No. 6, June 2003
capacitor remains uncharged, while under rotation of the solenoid the capacitor acquires a charge. However, the electric
field, which is responsible for such an inversion of the results
of Barnett’s experiment, cannot appear inside the solenoid
for any uniform distribution of charge over its surface, induced by the axially symmetrical gravitational field.
It would be impossible in the near future to realize the
Barnett experiment in such a strong gravitation field for an
experimental resolution of the calculated contradiction between the requirements of local Lorentz invariance and electrostatic laws. Nevertheless, it would be interesting to do a
further analysis of the local Lorentz invariance principle.
ACKNOWLEDGMENTS
The author warmly thanks Vladimir Onoochin 共Moscow兲
and Oleg Missevitch 共Institute of Nuclear Problems, Belarus
State University, Minsk兲 for helpful discussions.
a兲
Electronic mail: kholm@bsu.by
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