Electromagnetic Radiation of Ball Lightning

The intensity of electromagnetic radiation of ball lightning is estimated to the order-of-magnitude precision from general physical considerations and observational data for ball lightning in natural settings: I ~ 10−4 W. Accelerated motion of charged particles constituting the substance of ball lightning (i.e., the motion along nonlinear trajectories) may be the causative factor of its electromagnetic radiation.


INTRODUCTION
Ball lightning (BL) has been described in numerous literature sources, e.g., popular scientific books, reviews, and monographs [1][2][3][4][5][6][7][8][9][10][11][12][13], which cover many of its properties. In this work, we focus only on one aspect of this phenomenon: the ability of BL to cause powered-off electric light bulbs to glow and induce interferences in radio and television sets. This property of BL was mentioned way back in one of the first books entirely dedicated to BL-the monograph by W. Brand [2]. This property was also mentioned in later literature on the subject. The data at our disposal, which includes 5000 witness reports describing the observation of BL in natural conditions, show that this property of BL was observed in around 1.3% of reported cases [12]. Some of them are provided below [12]: Mid-September, 1980, the city of Uchaly, Autonomous Soviet Socialist Republic of Bashkiria (BSSR). Account of L.K. Kudoyarova.
"It happened during a lasting drizzle. As I can remember, there was neither lightning nor thunder. I was watching TV in the evening, and it seemed that I fell asleep. I woke up from a disagreeable feeling and rattling noise coming from the TV set. I looked at the TV set and wanted to get up to switch it off, but shuddered to a halt: a small glowing ball was slowly moving above the TV set past the indoor antenna. In size, it was like a small chicken egg or a tennis ball; it was dim red in color, like the filament of a light bulb with a power of ~15-20 W that is gradually going out. It seemed that the center of the ball was darker, more saturated in color, and somehow oscillating within the brighter shell. Moving past the antenna, while not touching it, the ball caused strong interferences: moving strips appeared on the TV screen, the image was distorted, and rattling noise was heard from the TV set. The ball then made a full circle around the antenna and started floating, while making barely noticeably jerky movements, toward the window. The small window and the window itself were shut, but, as if following the stream of air through a small slit in the window frame, the ball drew near the slit and stopped. That was when I suddenly coughed. The ball grew brighter, spread around the slip, and drew itself into it. But I could not see the ball on the other side of the window, i.e., it was not outside. I thought that the whole episode lasted a few minutes, but it is more likely that a few tens of seconds passed. It was only after the ball disappeared that I was able to get up and come close to the TV set. I could not find any traces that the ball may have left behind on the TV set and the window frame, and I examined everything through a magnifying glass." Mid-July, Kandrykul' Lake, the Republic of Bashkortostan. Account of I. Illarionov, an engineer.
"In that memorable summer, I and a close friend of mine went on a recreational cycling trip around Bashkiria. One day, after a thunderstorm, we made a night stop near Kandrykul' Lake; we pitched a tent and made a fire. It was wet all around, and the fire did not light well. I was sitting on a small chair by a large moist boulder and listening to a radio set. Suddenly, I saw a whole cloud of radiant balls with diameters ranging from three to four millimeters to 50 centimeters. They also glowed differently: some were glowing like bulbs of a pocket torch, while others were more like spotlight bulbs. It grew as bright as day. My radio set did not function and produced only rattling sounds. My friend was busy with fishing rods, but then he shuddered to a halt and slowly raised his head. The last of these accounts suggests that the BL explosion was accompanied by a burst of powerful electromagnetic radiation. From the perspective of impact wave physics, there is nothing surprising, because powerful atmospheric explosions always produce an intensive electromagnetic wave. It is interesting that this type of radiation was generated specifically in the explosion of the BL.
However exotic the mentioned property of BL is, no one has ever tried to look at its compliance with laws of physics. This problem is addressed in the present paper.

ESTIMATING THE ELECTRIC FIELD STRENGTH OF AN ELECTROMAGNETIC WAVE CAPABLE OF LIGHTING UP THE FILAMENT OF A LIGHT BULB
Suppose that a BL is capable of emitting electromagnetic radiation. From general physical considerations, let us estimate the intensity of this type of radiation, regardless of its nature, that can cause the filament of a light bulb to light.
First, starting from parameters of light bulbs, which can be found, e.g., on the internet, we estimate the voltage required for the filament to begin to glow.
The resistance of light bulb filament is R ~ 10 Ω, but it increases as the filament heats up, reaching hundreds of Ohms under operating conditions. As a result, the current passing through the filament decreases and measures fractions of ampere under operating conditions. For the situation at hand, i.e., when the filament only starts to glow, to have an order-of-magnitude estimate, we can take that as i ~ 1 A.
Then, the voltage required to be applied to the bulb to cause its filament to glow is (1) Electromotive force ε ind induced in the filament by a BL radiating electromagnetic wave in its vicinity must have the same value.
To calculate the induced electromagnetic force, we make use of the Faraday's law of induction: where Φ is the magnetic flux through the area defined by a closed path in which the electromotive force arises and t is the time. If we represent the filament as a straight solenoid with number of turns n per unit length and length l, the electromotive force of induction arising in it, when an electromagnetic field passes through its cross-sectional surface, can be estimated by the following formula: (2) where r is the helix radius, μ is the magnetic permeability of the medium, μ 0 is the magnetic constant, ω is the wave frequency, H 0 is the amplitude of magnetic field strength, c is the speed of propagation of electromagnetic signal, and x is the space coordinate in the direction of wave propagation.
For ordinary incandescent light bulbs, the average length l of a median line passing through the filament is on the order of a few centimeters, and the filament radius measures tenths of a millimeter. Typically, the filament is wound twofold: the wire wound in a helix is wound in another helix. Typically, the wire is made of tungsten, and its diameter is on the order of one tenth of a millimeter. To obtain an order-of-magnitude estimate based on the forgoing, we take that μ = 1, μ 0 = 4π × 10 −7 G/m, l = 0.05 m, n = 25000 m −1 , r = 0.0005 m, c = 3 × 10 8 m/s, and ω = 3 × 10 12 Hz.
Assuming that ε ind ≈ , for an electromagnetic wave emitted by the BL, the amplitude of magnetic field strength H 0 near its surface can be estimated to an order-of-magnitude precision using Eq. (2), while taking that the sinus in Eq. (2) equals unity: Volumetric energy density w of the electromagnetic wave is defined by the following expression [14, p. 299 The energy flux of the electromagnetic wave, or the amount of energy transferred by electromagnetic wave per unit time through a unit area situated perpendicular to the direction of wave propagation, can then be written as [14, p. 300]: here, V is the wave propagation velocity, and is the average value for volumetric energy density.
As a result, assuming that μ = 1 and V ≈ c, the energy flux of the electromagnetic wave emitted by the BL can be estimated with an order-of-magnitude precision as follows: The total energy of electromagnetic radiation emitted by the BL per unit time, or the intensity of radiation, is the product of W and the surface of the sphere corresponding to the BL. With R BL ~ 0.15 m, we have: As a result, the estimative value for the intensity of electromagnetic radiation of the BL does not contradict the observational data [12].
The values taken for physical quantities to carry out the order-of-magnitude estimations are fairly close to actual values, yet they are tentative. Nonetheless, from the energetic perspective, we obtained a reasonable estimate. The energy that BL will expend during a period of ~100 s by emitting electromagnetic radiation will be only a tiny fraction of its total energy, which, based on the damage it brings to various objects, was estimated to be ≥10 5 J, a value that can be found in many books dedicated to BL [6,7,11,12,15].

PUTATIVE PHYSICAL MECHANISM OF THE GENERATION OF AN ELECTROMAGNETIC FIELD INSIDE BALL LIGHTNING
While on the subject of possible physical mechanisms leading to the generation of electromagnetic radiation in the BL, we can propose two different mechanisms: one is based on oscillations of BL's charged surface [16] and another is based on the accelerated motion of charged particles along curved orbits within its bulk [17].
As for the first mechanism, estimations of the intensity of this type of radiation give quite a small value for both quadrupole and dipole types of radiation [18], which is not adequate to cause an electric light bulb to glow. We are left with the second mechanism. Thus, the physical mechanism underlying the generation of electromagnetic radiation in BL draws on the motion of charged particles on curved orbits inside the BL, this motion ensuring the presence of centrifugal acceleration. At this stage, it is relevant to raise the question of the internal structure of BL's substance. What do witnesses see inside a BL when it passes at a close distance to them? Around 13% of witnesses who saw a BL from a close distance reported on the presence of the internal BL structure being in motion [12]. They speak of chaotically moving glowing dots, interweaving glowing lines, and glowing little balls moving inside the BL bulk [12]. Sometimes, the visible structure of BL is compared to "a loosely wound wool ball." Here "A thunderstorm was starting in the evening, at around 8 p.m. It was not raining yet, but dark gray heavy clouds were coming from the northern direction. Residents were hanging around in the yard of their house, and children were running. My neighbor and I were standing by an outdoor cooking stove. The neighbor started to stir something on the stove, and I was watching. Suddenly, the neighbor and the stove were illuminated with a bright, kind of electric, light. I turned around and saw a blindly bright ball with the size of a football of a creamy color. It was like a clew of bright threads or, rather, a tangle of thin wire. The ball started to move toward the children in an unhurried manner at a speed of 3 m/s at most. All of us stood motionless; there was silence. The ball floated past the children and disappeared. I did not see where exactly it went, because I was looking at the brightly illuminated children's faces at that time. The whole event lasted no more than 5 seconds." End of June-beginning of July 1962, Nechaevka village, Moshkanskii district, Penza oblast. Account of P.S. Zhuravlev. "In the afternoon, a strong thunderstorm occurred, and the rain was so strong that potatoes were knocked out from the ground. My wife and I looked through a small window in the seni [a room between the living area of the house and the doorsteps] to see how things are with the potatoes. Because of frequent storm discharges, we soon had to step back from the window, and this was when I saw on my right, at a distance of 1.5 meters from me, a ball with a diameter of 20-25 centimeters hovering at a height of 0.5 meters from the earthen floor. The ball was white in color with a faint glow, like a 15-W light bulb. It seemed to be kind of incompact, nonuniform in density, and consisted of stirring small white-red sparks. The ball then started to make jerky movements up and down and back and forth. In distance, jerky movements constituted around 1-1.5 of its diameter. Making these jumps, the ball flew past me at a distance of only around 30 centimeters away from my face. Hovering for a while, it then started to move back in the same manner, while slightly descending, and vanished at a distance of half meter away from the ground. I did not even notice its contact with the ground. I heard no sound or explosion, perhaps, because of the loud roars of the thunderstorm. And after the ball disappeared, there was a strong smell. It is difficult to describe it, but it reminded me of the smell of smoke of a burning match head or perhaps the ozone scent. I got frightened and started looking here and there to see if anything was burning. We put lights on after the thunderstorm, there was no light: the end of a coil on the watt-hour meter had been burnt. We were watching the ball for 10-15 seconds, and I did not sense any heat. In looks, the ball was peaceful and harmless; I wanted to touch it with my hand. It seemed that the ball had a 5mm thick shell." It would appear reasonable that rapidly moving glowing dots can be visually perceived as interweaving glowing lines because of the persistence of vision. Regardless of the nature of this glow, we can assume that they carry electrical charges and can be a source of electromagnetic radiation.
BL was reported to have an uncompensated electric charge Q BL on the order of a few microcoulombs [19]. This charge may be distributed between the moving light dots described above.
For a charge moving at acceleration a, the expression for total intensity of radiation emitted by it at dis-tances much longer than the wavelength and linear dimensions of the region in which it moves can be written, within the dipole approximation and using SI units, as follows [20, p. 436]: (4) where ε 0 is the dielectric constant (≈8.85 × 10 −12 F/m). As was mentioned, expression (4) determines the intensity of a wave at distances much greater than the size of system in which the motion of charged particles takes place, i.e., R BL .
To compare the intensity of radiation given by Eq. (4) with the intensity of electromagnetic radiation near the BL surface that was calculated using Eq. (3), which is I BL ~ 10 −4 W, the latter must be multiplied by factor X that takes into account attenuation of the strength of a spherical wave with distance. The numerical value for this factor can be found from the expression Taking that the distance to the observation point is L 1 ~ 3 m, and the characteristic length of the region in which the charge moves is L 2 ~ 2R BL ~ 0.3 m, we arrive at X ~ 10 −2 .
For charge Q BL ~ 6 × 10 −6 C [19] moving with acceleration to emit radiation with intensity I ~ 10 −4 W in the immediate vicinity of the BL, it must have the acceleration (5) This is a very high acceleration, especially if we take into account the linear dimension of BL. It is does not agree well with laws of physics.
Taking into account that it is centrifugal acceleration, we write From this relation, we can find the velocity of moving dots, i.e., and the numerical value is V ~ 30000 m/s. As a result, the values for acceleration and velocity of charged particles, as estimated above, are quite high, and it can be said that they are not quite plausible and do not agree very well with physics. But the values obtained above for the acceleration and velocity can be considerably reduced and brought closer to common sense if we consider that, in addition to uncompensated charge, BL can carry a balanced quantity of unlike charges, assuming that the substance of BL represents plasma. Put differently, we assume that the BL substance consists of electrons and positively charged ions. Macroscopic electroneutrality is a special feature of plasma. It is maintained owing to a balance between space charge created by electrons and positively charged ions. But this is only an overall balance, it holds for volumes much larger than the Debye length and over fairly long time periods, i.e., much longer  [21]. By taking into account the involvement of these charges in rotational motion, considerably lower estimates for the acceleration and velocity of charged particles in the BL bulk may result. In actuality, the volume of BL with a radius of 0.15 m is around 0.01 m 3 . At normal conditions, the Loschmidt's number is on the order of 3 × 10 25 m −3 . Considering that the BL plasma is also at normal conditions (according to available data, witnesses experience the sensation of heat from a BL in only one case of 50), we find that a BL with a diameter of 0.15 m may contain on the order of 10 23 charged particles of both charge types. Let us assume that each charged particle carries a single elementary charge, then the absolute value for the total charge moving within the BL will be on the order of 10 4 C, but overall the BL is chargeneutral. Accelerated motion of charged particles in the bulk of BL will generate electromagnetic radiation. At the same time, the number of radiating particles moving with acceleration will increase by ten orders of magnitude, and, accordingly, a substantially lower estimated value will be obtained for acceleration of particles inside the BL. As a result, the estimated values for acceleration and velocity of glowing dots in the BL bulk are a ~ 20 m/s 2 and V ~ 1 m/s. The human eye does not discern changes that occur within less than 0.1 s [22]; therefore, a dot moving at a high speed will be visually perceived by a human being as a glowing track. By dividing the length of half of circumference by the charge velocity, we can find observation time τ for the half of BL facing a witness. If τ < 0.1 s, the human eye will perceive the trajectory of motion as a glowing track; and if τ > 0.1 s, the eye will be able to discern a glowing dot in motion. It is these variants of respondent's answers to the question concerning the visible structure of BL that are actually recorded.
With characteristic BL radius R BL ~ 0.15 m, the value for velocity V ~ 1 m/s just falls at the boundary dividing the scenarios described above, i.e., seeing either glowing continuous tracks or moving dots inside BL.
It is worth recalling that emission of electromagnetic radiation by BL is reported quite rarely, and so is the case with reports on its internal structure. We, therefore, warn against having an expectation that all BLs behave in accordance with the described scenario.
We also remark that BL that carries its own charge (i.e., it is not charge-neutral) does not necessarily emit electromagnetic radiation if it is in the plasma state.
For instance, the presence of a magnetic field in the bulk of BL, when it forms, may be a causative factor for the emergence of charged particles moving along curved trajectories in it. This hypothesis is supported by observations of BLs with ring-like shapes. Aside from us, this was reported by Stakhanov [6] and Rayle [9]. According to the data of all of cited studies [6,9,12], the probability to observe a spherical BL is around 0.9, but different studies [6,9,12] report on different values for the probability to observe a ringshaped BL, with the average value being 0.003.

CONCLUSIONS
Drawing on reports on BL observations in natural settings, we estimated the intensity of electromagnetic radiation of a BL with radius R BL ~ 0.15 m within a narrow solid angle in the vicinity of its surface: P 1 0 −3 W/m 2 , and the total intensity of electromagnetic radiation (i.e., in all directions) emitted by the BL is I~ 10 −4 W. The electromagnetic radiation with this intensity may be due to accelerated motion of charged particles (i.e., motion along nonlinear trajectories) in the BL bulk.