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IOP PUBLISHING                                                                  
                                             PHYSICA SCRIPTA

Phys. Scr. 77 (2008) 035504 (5pp)                                               
                         doi:10.1088/0031-8949/77/03/035504

Electrostatic charge bounds for ball
lightning models

Karl D Stephan

Department of Technology, Texas State University, San Marcos, TX 78666, USA

E-mail: kdstephan@txstate.edu

Received 18 October 2007
Accepted for publication 22 January 2008
Published 12 February 2008
Online at stacks.iop.org/PhysScr/77/035504

Abstract.

Several current theories concerning the nature of ball lightning predict a
substantial electrostatic charge in order to account for its observed
motion and shape (Turner 1998 Phys. Rep. 293 1; Abrahamson and Dinniss
2000 Nature 403 519). Using charged soap bubbles as a physical model for
ball lightning, we show that the magnitude of charge predicted by some of
these theories is too high to allow for the types of motion commonly
observed in natural ball lightning, which includes horizontal motion above
the ground and movement near grounded conductors. Experiments show that at
charge levels of only 10­15 nC, 3-cm-diameter soap bubbles tend to be
attracted by induced charges to the nearest grounded conductor and
rupture. We conclude with a scaling rule that can be used to extrapolate
these results to larger objects and surroundings.

PACS numbers: 52.80.mg, 41.20.cv

1. Introduction

Ball lightning is a rare phenomenon of atmospheric physics for which no
satisfactory theory has yet been found. One of its most problematic
observed characteristics is the way it moves. In a 1966 survey by Rayle
[1] covering more than 100 accounts of ball lightning, a majority of the
interviewees reported mainly horizontal motion, while only 19% reported
vertical motion. Numerous accounts [2, 3] describe ball lightning as often
moving more or less parallel to the earth's surface a meter or two above
the ground. Any theory claiming to give a satisfactory account of ball
lightning must also account for the unusual nature of its observed motion.
Theories of ball lightning fall into two broad categories- external-energy
theories and internal-energy theories--with many subdivisions within each
[4]. External-energy theories assume that the visible light emission and
other energetic manifestations of ball lightning are powered by an
external energy source such as radio-frequency or microwave radiation.
This idea was first proposed by Kapitsa in 1955 [5]. The object's location
and movement in these theories is typically determined by the nature of
the assumed electromagnetic field, not solely by the physics or dynamics
of the ball itself. On the other hand, internal-energy theories assume
that ball lightning is a more or less self-contained structure containing
an electrical, chemical, or other type of energy source within it. In
these theories, the object's motion must result from interactions between
the object and its surrounding environment, where only `normal' physics is
presumed to occur. In this paper, we analyze certain assumptions about
electrostatic charge made in two internal-energy theories, and find
discrepancies between the consequences of these assumptions and ball
lightning's observed behavior. Several internal-energy theorists have
hypothesized a substantial net electrostatic charge for ball lightning in
order to explain its motion and shape. Turner [6, 7] has proposed a model
of ball lightning which involves an electrochemical reaction among
nitrogen compounds in humid air. The model predicts a net positive charge
for the overall object of as much as 3 µC. Abrahamson and Dinniss [8]
assumed a net electrostatic charge for ball lightning when they
hypothesized an interconnected network of slowly oxidizing silicon
nanoparticles. According to their theory, repulsion of like charges may be
responsible for maintaining the approximately spherical shape of the
network. Since these leading theories of ball lightning include the
assumption of a significant electrostatic charge, an experimental
investigation of how ball-lightning-like objects move when charged can
test the plausibility of this assumption. As we will show below,
surprisingly small amounts of net charge are likely to lead to the
object's

0031-8949/08/035504+05$30.00 1 © 2008 The Royal Swedish Academy of
Sciences Printed in the UK

Phys. Scr. 77 (2008) 035504
                                                             Karl D
Stephan

destruction in a short time. If ball lightning does carry a net
electrostatic charge, its level must be well below those proposed by
Turner, and may be so small as to exert an insignificant influence on its
motion compared to other forces.

2. Experimental approach

In this study, we used soap bubbles as a physical model for ball lightning
in order to investigate how ball lightning's electrostatic charge can
affect its interaction with its environment. Both individual witnesses of
the natural phenomenon1 and numerous studies examining multiple reports
[10, 11]2 have noted that ball lightning changes shape, moves, and
disappears in ways that sometimes resemble a soap bubble. The structure of
ball lightning is in all likelihood much more complex than that of a
bubble. However, soap bubbles share with ball lightning the following
characteristics: (i) approximately spherical shape, (ii) diameter ranging
from 1 cm or less than 50 cm or more, (iii) an apparent density
approximating that of ambient air, (iv) a duration in air from a few
seconds up to a minute or more (consistent with the observed lifetime of
ball lightning [4]), and (v) a surface tension which tends to restore the
object's spherical shape when perturbed away from it. In soap bubbles, the
surface tension is provided by a solution of soap in water. Such films are
stable against perturbations that increase the film area, since an
increase in area decreases the soap concentration, leading to higher
surface tension which tends to restore the surface to its original area
[12]. This is known as the Marangoni effect. In ball lightning, the origin
of apparent surface tension is uncertain, although plasmas can show
surface-tension-like effects [13]. These characteristics shared by bubbles
and ball lightning make bubbles useful as physical models for natural ball
lightning, provided any limitations of the model are taken into account.
One such limitation is the fact that an electrostatic charge on a bubble
is predominantly surface charge on the conductive liquid film, while
charge in ball lightning may be distributed throughout the entire volume
of the ball in an unknown way. We will consider two simplified models of
the electrostatic conductivity of ball lightning to show how this
limitation is probably not a serious one. In the first model, suppose that
ball lightning is an insulator which somehow holds excess charge in a
fixed volume distribution. (A physical model for this situation would be a
solid sphere of a good insulator such as glass, with excess charge
embedded inside it.) Assume that this distribution is given by the
function V (r, , ) in C m-3 inside a sphere of radius ro, and V = 0
outside the sphere, where (r, , ) are the radial distance, polar angle,
and azimuthal angle, respectively, of a spherical coordinate system whose
origin coincides with the center of the sphere. The electrostatic
potential (r, , ) everywhere outside the sphere can be expressed as the
sum of spherical surface harmonic functions Yn (, ), each weighted by a
moment constant pn,

(r, , ) = 1

n=0
pn Yn(, )
r n+1      ,         (1)

1 Report by Nezamaikin V N in [9].
2 Nauer H, cited in [2], pp 164­5.

where is the dielectric constant of the medium [14]. The first term in
this sum decreases with radius as r -1. It is identical to the field
produced by a point charge at the origin whose magnitude equals the net
charge of the sphere. The second term decreases as r -2 and corresponds to
the field of a dipole, the third term to a quadrupole, and so on. Since
the higher order terms decrease more rapidly with increasing r due to the
larger exponent of the higher-order terms, at a distance of a few times
the sphere radius ro most of the significant force effects will be a
function only of the total charge inside the sphere (the leading term),
and not to its particular distribution inside (the higher-order terms).
Therefore, if the electrical behavior of a spherical shell of net charge
(i.e. a soap bubble) is equated to that of an insulating ball lightning
object of equal radius and net charge, the equivalence will be good except
for distances less than a few sphere radii, and approximately correct for
distances less than this. Consider a second (and more likely) model for
ball lightning's conductivity: that it is a conductor to the extent that
an excess electrostatic charge placed on the sphere will be free to move
to a steady-state distribution with a time constant on the order of
seconds or less. If the charge within a solid conducting sphere obeys
Coulomb's inverse-square force law, a well-known result from
electrostatics [15] shows that all excess charge will move to the surface
of the sphere, and none will remain inside. If this conductivity model is
correct, the analogy between charge on ball lightning and the same
quantity of charge on a soap bubble will be essentially exact. Even if the
Coulomb potential is replaced by the Yukawa potential (also known as the
screened Coulomb potential or the Debye­Hückel potential), we reach a
similar conclusion. This potential is sometimes used in describing the
forces between charged particles in dusty plasmas. The Yukawa potential
between two charges of magnitude q is given by

q2
4    r exp 
r
D ,                  (2)

where D is the Debye length in the plasma, a measure of how well the
plasma screens isolated charges [16]. Spencer [17] has shown that assuming
a Yukawa potential for the charges in a conducting sphere leads to a
situation where some of the charge is on the surface and the remainder is
distributed uniformly inside. While this distribution differs from the
soap bubble's surface-only distribution, the argument above given for the
insulating sphere applies equally well to a sphere with uniform charge
inside, in that significant differences between the bubble case and the
ball-lightning case will appear only for distances within a few sphere
radii. Therefore, whatever the details of conductivity for electrostatic
charges within ball lightning, we believe our experimental results are
likely to be a good model of the forces on an actual ball-lightning object
for which electrostatic forces are an important, if not wholly
determinative, factor in its motion. Our experimental setup is shown in
figure 1. A Van de Graaff generator 48 cm high with a spherical
high-voltage electrode 22.5 cm in diameter was connected to a small copper
loop through a ball chain. Corona discharge from these connections limited
the attainable voltage to 15 kV. Compressed air from a 2.5-cm-diameter
tube behind the loop formed bubbles when the loop was periodically dipped
in

Phys. Scr. 77 (2008) 035504                                                     
                                                                    Karl D 
Stephan

Figure 1.  Layout of the charged-bubble experiment (not to scale) showing
Van de Graaff generator and compressed-air bubble-producing mechanism on
left, and trough with grounded bottom and side partitions on right. The
trough was 40 cm wide by 60 cm high by 120 cm in length.

a soap solution. About 10% of the bubbles generated by the setup fell
between the walls of a trough placed about 1 m away. The trough was formed
of plastic-foam insulating material coated with aluminum foil. A video
camera at one end of the trough photographed the bubbles in flight against
a background of 2.5-cm spaced white gridlines on black cardboard at the
other end of the trough. A second video camera (not shown) was placed at
right angles to the long axis of the trough to locate the bubble's
position along that axis. The inside surface of the trough wall nearest
the Van de Graaff generator was electrically isolated and formed the input
electrode of an electrometer. The electrometer input circuit was a 103 M
resistor in parallel with a 3.6-nF capacitor to ground. A high-impedance
voltage follower (TL071CP) transmitted the electrometer voltage to a
LabViewTM 12-bit A /D converter sampling at 1 kHz. This voltage was
corrected for current offset errors and integrated in software to find the
charge received at the electrometer input as a function of time.
Preliminary trials were performed using a captive bubble in a can-shaped
electrode to verify that electrically isolated bubbles could retain their
charge over the 30 s time span of the experiment, and no observable
leakage was detected. The raw electrometer output voltage during a trial
in which five separate bubbles (or bubble clusters) collided with the
electrometer wall is shown in figure 2. As a bubble approaches the wall,
it induces charge of a polarity opposite to that of its own charge in the
electrode by virtue of the increasing mutual capacitance between the
bubble and the electrode. When the bubble contacts the wall, the
capacitance goes to infinity and a peak occurs in the voltage, identifying
the exact time of collision with a time resolution of 10 ms or better. The
R­C parallel circuit's time constant = 0.33 s shapes the discharge
waveform as an exponential tail as shown. As long as bubble events are
separated by 1 s or more, the voltage waveform v(t) can be integrated
during the time interval from t1 to t2 when the voltage v exceeds 10% of
its peak value to obtain an estimate of the total bubble charge Q in
accordance with the equation

Q  C[v(t2) - v(t1)] + 1
R
t2
=t1
v() d.    (3)
­1

0         5        10       15        20       25        30
Voltage (V)

Time (s)

Figure 2.                   Raw voltage obtained from electrometer circuit 
during
charged-bubble test. Bubble impacts are clearly visible at t = 11.06,
18.44, 24.93, 25.70 and 28.96 s, all correlating with videotaped
impacts of single or multiple bubbles.

Although this procedure does not capture all the bubble's charge, it
produces an estimate for Q that is accurate to within ±10% to ±20%,
depending on the waveform shape. During the experiment, a significant
difference was noted between the motion of charged and uncharged bubbles
(made with the Van de Graaff generator off). Uncharged bubbles drifted
aimlessly around the laboratory for as long as 30 s before either
rupturing because of film evaporation or because of a collision with an
object (typically the floor). All bubbles had a negative buoyancy,
drifting straight down in the absence of other forces. However, charged
bubbles moved more rapidly than uncharged ones. The charged bubbles were
vigorously repelled from the region of the Van de Graaff generator and
attracted to any grounded object such as a metal cabinet or stanchion.
Although bubble lifetimes were not measured precisely, the average
lifetime of the charged bubbles was probably less than half that of the
uncharged bubbles. To show that we can accurately model all forces
involved in bubble motion in this experiment, we obtained three
dimensional position data on a bubble with a charge of Q = -4.1 nC as
measured by its electrometer signal and a radius r0 = 1.6 cm. Position
data was obtained for the last 0.5 s or so of its lifetime up to the point
when it collided with the electrode and ruptured. Independent measurements
of the accuracy of our video system and data analysis procedure indicate
that the probable rms measurement error is about 6 mm in the x-direction
and 1.3 cm in the z-direction. To see why bubbles are attracted to
grounded surfaces, consider the idealized situation shown in figure 3,
which shows a trough that is infinitely long in the y-direction (into the
page) and infinitely tall in the z-direction (h goes to infinity). If a
bubble of radius r0 has a charge Q, that charge will induce a surface
charge of the opposite polarity on the grounded surfaces surrounding the
bubble. These charges in turn attract the bubble with a force that is
balanced horizontally when x = 0 (in the center of the trough), but is
always directed downward vertically, since there is no upper boundary to
the trough. As the bubble approaches a side wall, the induced surface
charge becomes unbalanced, becoming

Phys. Scr. 77 (2008) 035504                                                     
                                                            Karl D Stephan

Figure 3.    Charged bubble of radius r0 in grounded conductive
trough (trough is infinitely long in y-direction).

greater at the nearest wall and less at the opposite wall. This results in
a net horizontal force which pushes the bubble toward the nearest wall
until it collides with the wall and ruptures. A concise summary of the
forces on the bubble supports these ideas. The gravitational force on the
bubble (in newtons) is FG = -^z(4r20Ftg), (4)

where g = 9.8 m s-2 (the acceleration of gravity) and ^z is the unit
vector directed vertically upward. The net gravitational force is due only
to the mass of the bubble film of thickness t (m) and density F (kg m-3),
since the mass of the entrapped air is canceled out by the buoyancy due to
displacement of an approximately equal mass of air.

The electrostatic force on the bubble can be derived from a function C (r)
which is the capacitance between the bubble and the trough as a function
of the bubble center's position vector r . Since the bubble's
electrostatic potential energy is

UE    (r) = C(r)V 2(r)
2               ,    (5)

where V      (r) is the potential of the sphere (in volts), and for a
constant charge Q = C(r)V (r),

UE    (r) = Q2
2 ·
C    (r),           (6)

we use the fact that the negative gradient of the electrostatic
potential energy is the electrostatic force

FE    (r) = - Q2
C     (r) .     (7)

In this way, the problem of finding the electrostatic force reduces to the
problem of finding the bubble's capacitance to ground as a function of
position. This analysis was performed with COMSOL finite-element software
for the specific case of a trough with width w = 40 cm. The open trough
was modeled by a closed prism 2 m high (z) by 1.9 m deep (y) by 40 cm wide
(x). As long as the bubble was centered in the y-direction in the COMSOL
model, we found that further increases of dimensions of the trough beyond
those stated made less than 1% difference in the capacitance indicated by
the model. This indicated that the model capacitance in the closed prism
is a close approximation to the capacitance in the open laboratory trough.
We explain below how we used data from this closed prism model in the
open-trough experimental situation. We then fit empirical analytical
expressions to the COMSOL data for use in a differential-equation solution
described below. A bubble moving in air experiences a drag force

FD    (v) = -v r20AIR|v|
2       CD    (Rey(|v|)),             (8)

where the velocity vector v (m s-1) is the time derivative of the position
vector r , the air density is AIR, and the dimensionless drag coefficient
CD is a tabulated function of the Reynolds number

Rey    (|v|) = 2r0AIR|v|
µ        ,                    (9)

in which the dynamic viscosity of air is µ (N-s m-2) [18]. Analytic
expressions approximating these functions in the region of interest were
derived and used together with the electrostatic-force expressions to
solve the overall second order differential equation of motion for the
bubble:

FG + FE    (r) + FD(v) = mB d2r
dt 2 .              (10)

In equation (10), the mass mB includes both the film mass and
the mass of the gas inside the bubble:

mB = 4r20Ft + 43 r30AIR.                       (11)

Equation (10) was integrated using a Runge­Kutta algorithm
and a time step of 11.1 ms for comparison with experimental
data, which we will now describe.

3. Results

The theoretical and experimental results of the analysis of a charged
bubble moving in the grounded trough are shown in figure 4. As the figure
shows, we obtain good agreement between the actual measured path of the
bubble as it is attracted to the electrode wall, and the theoretical
prediction of the path based on the three forces in equation (10).
Definite integration of equation (10) requires values for the bubble mass
mB and initial conditions for v and r at an initial time t0. We obtained a
value for the bubble mass by measuring its average velocity outside the
trough from video data, since a bubble's terminal velocity (free-fall in a
gravitational field only) depends sensitively on its mass. By the time
bubbles reach the trough, they have lost any horizontal motion, so the
initial velocity vector was known. The initial positions x0 and y0 were
known from the video recordings. We chose the initial position for the
theoretical curve to be z0 = 56.7 cm, which yielded the best fit to the
experimental trajectory. Since our COMSOL electrostatic model assumed an
infinitely deep trough, it does not accurately model the situation above
the edges of the laboratory trough, where the electrostatic force
eventually goes to zero above the edges located at z = 60 cm. The choice
of z0 = 56.7 cm `turns on' the electrostatic force at that height, and
models the actual situation well.

Phys. Scr. 77 (2008) 035504                                                     
                                                                            
Karl D Stephan

4. Conclusions

Assuming a ball-lightning object is self-contained, and is not simply the
visible manifestation of a larger independent field or current structure,
its trajectory will be determined by the sum of the forces acting on it.
This assumption is made in the two internal-energy theories summarized
earlier in this paper. We have shown that soap bubbles charged to
relatively small levels tend to be rapidly attracted to grounded surfaces
and destroy themselves upon impact. We have also shown that the trajectory
of a bubble can be accurately modeled by considering only drag, gravity,
and the electrostatic force due to the bubble's capacitance to ground.
While care should be taken in extrapolating these results to larger
objects farther away from grounded surfaces, the invariance of
electrostatic solutions to Maxwell's equation under dilation
(multiplication of all space dimensions by a constant) means that the same
voltages and forces will apply to a scaled-up version of our experiment,
as long as the charge on the object is scaled by the same factor. Scaling
a 1.6-cm-radius bubble to 16-cm-radius (a typical size for ball lightning)
means that a charge of the order of 40­120 nC will result in behavior
similar to what we observed, although drag does not scale linearly with
dilation as the electrostatic solution does. Therefore, a net
electrostatic charge on a ball lightning object of less than 0 .2 µC or so
is consistent with many observations, although charges near the upper end
of this range would tend to shorten the phenomenon's lifetime through
attraction to grounded objects. But a charge of 3 µC, as proposed by
Turner, may approach the Rayleigh limit [19] at which an object held
together by surface tension flies apart because of electrostatic
repulsion. Calculating the Rayleigh limit requires a knowledge of the
object's surface tension, which is not known for ball lightning. However,
the charge limits we found were far below the Rayleigh limit for soap
bubbles. The essential conclusion of this paper is that for ball lightning
in the neighborhood of the earth's surface or other grounded objects, the
upper limit for net electrostatic charge is established not by the
Rayleigh limit at which the object self-destructs due to mutual repulsion,
but by attraction to grounded objects through the mechanism of induced
charge. Even if a spherical object has no net charge, it can develop a
dipole moment in a uniform electric field and experience a net attractive
force in a non-uniform field. Since ball lightning is generally associated
with thunderstorms and the electric field on the ground beneath such
storms can greatly exceed the fair-weather field of about 100 V m-1, it is
possible that electrostatic forces are involved in ball-lightning motion
near objects that would tend to concentrate electric field lines. This
concentration leads to non-uniformities in the field and would produce a
net force on a neutral conducting object such as ball lightning, assuming
ball lightning has a non-zero conductivity. While we did not investigate
this possibility, such problems may serve as a focus for further
investigations of the problem of ball lightning motion in electrostatic
fields.

Acknowledgments

I thank Professor John A Pearce of the Department of Electrical and
Computer Engineering at the University of Texas at Austin for his
hospitality in allowing me to use the facilities in Professor Pearce's
Process Energetics Laboratory. This work was partially supported by a
Texas State University Research Enhancement Grant. Gratitude is also
expressed to the reviewers, who pointed out several matters requiring
elucidation.

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