The Discovery of the Pinch Effect

Brian James
Honorary Associate Professor, School of Physics, University of Sydney.

(This article is reproduced with permission from Australian Physics, vol.44 no.1, p9, 2007).

In 2005, when the Australian Physics community was celebrating the International Year of Physics in recognition of the centenary of Einstein's golden year, another centenary of local interest passed virtually unnoticed. In 1905 the Royal Society of NSW published in its journal a paper entitled Note on a Hollow Lightning Conductor Crushed by the Discharge by J.A. Pollock, Professor of Physics at the University of Sydney (see figure 1) and S.H. Barraclough, Lecturer in Mechanical Engineering at the same institution (see figure 2) [1]. This paper is now acknowledged as being the first explanation of the electromagnetic phenomenon now known as the pinch effect.

Figure 1:
J.A Pollock, FRS 1865-1922; Professor of Physics, University of Sydney 1899-1922. Date of photograph unknown. (University of Sydney Archives)








.

Figure 2:
S.H.E. Barraclough 1871-1958; Lecturer in Mechanical Engineering at the time of the work described here; subsequently Professor of Mechanical engineering, Dean of the Faculty of Engineering and Fellow of the Senate of the University. Date of photograph ~ 1940s. (University of Sydney Archives)








.

The lightning conductor referred to was installed on the chimney stack of the Hartley Vale Kerosene Refinery (near Lithgow, NSW). Following a lightning strike to the conductor, a crushed tubular section was sent to Professor Pollock by Mr G.H. Clark, under whose supervision the lightning conductor has been erected. The tubular section, near the top of the lightning conductor, joined a short solid rod attached to a ball with one large spike at the top and four smaller radiating ones to the main solid conductor to ground. The crushed conductor (see figure 3) remains in the School of Physics at the University of Sydney where it is used regularly to exemplify the pinch effect in undergraduate lectures.

Figure 3:
The crushed tubular section of the Hartley Vale lightning rod. (School of Physics, University of Sydney)

The authors show that for a current Io in a thin tube of radius a, the inward acting force per unit area, due to the magnetic force on the current-carrying conductor, is given by (in SI notation)

The derivation is straightforward. Assuming an axially symmetric conductor, the average magnetic field within the thin wall of the tube is readily calculated

and from this the force per unit area on the tube. The discussion is however unsettling as it appears to ignore the vector nature of force. A solid conductor is considered first. Although it is noted that the magnetic "forces at all points [are] directed radially inwards in planes at right angles to the axis of the cylinder", the first quantity derived is "the sum of the mechanical forces acting throughout the matter contained in unit length of the cylinder". To a modern student of Physics this apparent sum of the magnitudes of forces acting in different directions is non-sensical. A little calculus indeed confirms that the quoted result for a solid (in SI notation)

is, as suggested, the integral over the cross-section of the rod of the magnitude of the force on unit length of a filament of infinitesimal cross-section.

The derivation is repeated for a tube of non-negligible thickness, from which the limiting case of a thin wall is obtained. The final result (equation 1) used for further analysis is easily confirmed to be the radial inwards force per unit area on the wall of a thin tube. This is not surprising since, for the purposes of calculation, the problem may be approached by considering the tube to be split along its length and flattened out, in which case the vector nature of the problem becomes irrelevant.

The authors express this in terms of a relationship between the current Io and the equivalent external pressure increase to produce the same radial stress. Substituting the radius of the tube (a = 9.0 mm), the result is given (in SI units) as

where n is the equivalent excess external pressure (in atmospheres) which would produce the same radial stress in the copper tube.

The authors consider the possibility that for a brief lightning current pulse, skin effects may be present but are able to show than this would be negligible in the circumstances under consideration.

The main problem in determining the effect of this compressive stress on the copper tube is a lack of knowledge of any elevation of the temperature of the copper due to the current pulse, and if so how this would affect the compressive strength of the tube. The authors point out that while the tensile and elastic limit for copper are known, copper has no definite compressive strength. There is however a well-established elastic limit in compression. The latter quantity is know to decrease with temperature, with measurements available, at the time, up to approximately 600 degrees C, well short of the melting temperature of 1054 degrees C. The authors are able to draw on extensive investigations of the external pressures necessary to produce collapse of tubes (needed for the design of boilers).

As an example, they authors calculate that if the temperature of the tube were "as low as 500 degrees C", the required current to collapse the tube would be about 100,000 A. From a detailed examination of the collapsed section of the tube, however, the authors believe that the tube was probably plastic at the time of collapse, due to heating by the discharge. In this state they suggest that the tube "...gave way under forces equivalent to an excess pressure outside of the order of not more than an atmosphere; this would indicate a current of about 20,000 amperes, ..."

The possible range of current deduced, 20-100 kA, with a most probable value at the lower end of this range is consistent with the knowledge of lightning discharges obtained from extensive studies later in the 20th century. In fact statistics based on observations of many lightning discharges yield typical peak currents of ~30 kA, with only a few percent above 100 kA [2].

Following a talk I gave on this subject to the Royal Society of NSW in May 2006, Geoffrey Walsh [3] pointed out that a simple calculation requiring only values for the density, resistivity and specific heat of copper shows that even for a current as high as 100 kA the predicted temperature rise would be less than 1 K for a typical return stroke duration of 50 µs. Even if there were multiple return stokes, the effect on the properties of copper would still be negligible. If we assume therefore that the copper remains at ambient temperature the compressive elastic limit of copper would be around 9 times larger than the value they used for 500 degrees C, leading to a current of around 300 kA. Lightning currents as large as 500 kA are rare, but not unknown. In the light of the discussion above, perhaps a more reasonable conclusion would be that the tube collapsed due to the pinch effect resulting from an unusually large lightning strike, most likely involving a current of several hundred kiloamps.

It is possible that the skin effect during current rise could produce surface heating, resulting in surface modification. A simple calculation, however, suggests that this would not be significant.

It is puzzling that the authors did not make a similar calculation, although it should be mentioned that their paper, at the beginning of the 20th century, pre-dated detailed experimental studies of the nature of lightning. They were thus not in a position to conclude that the current values they proposed were reasonable, and presumably had little information on the duration of the high current component of a lightning discharge.

Pollock and Barraclough did not coin the term pinch effect. This occurred 2 years later in a paper by Northrup [4], which described the contraction of the cross-section when a large alternating current flowed in a liquid conductor. Northrup did not reference Pollock and Barraclough's paper. (In fact, there is only one reference in Northrups's paper - to J.C. Maxwell!)

The most common textbook references to the pinch effect occur in plasma physics texts were the earliest reference given is usually a paper by W.H. Bennett [5] in 1934, which describes the magnetic self-focussing of a fast stream of electrons in a neutralising ion background. The magnetically-pinched linear discharge (also called a z-pinch or Bennett pinch) was one of the first devices used in fusion research. The description of the MHD equilibrium of the linear pinch, and associated instabilities came later. Bennett's paper did not reference Pollock and Barraclough.

Thus for a period Pollock and Barraclough did not receive appropriate recognition for their pioneering work. This has been corrected now. In a recent article reviewing the use of wire-array z-pinches as x-ray sources, Haines et al [6], acknowledge that "...Pollack and Barraclough in Australia proposed the pinch effect to explain the implosion of a copper rod used as a lightning conductor"; and at the time of writing the Wikipedia entry for Pinch (plasma physics) [7], notes that "The phenomenon was not understood until 1905, when Pollock and Barraclough investigated a compressed and distorted length of copper tube from a lightning rod after it had been struck by lightning."

I would like to acknowledge that my interest in looking more closely at Pollack and Barraclough's paper was the result of an email from Dr Michael Coppins of Imperial College in early 2006. Michael was seeking biographical information on Pollock and Barraclough for a talk he was giving on the history of the pinch effect. In passing he asked if we had recognised the centenary of the publication in 2005. Regrettably, I had to respond in the negative, and to admit that, to my knowledge, it had passed without notice.

References

1. J.A. Pollock and S.H. Barraclough, Proc R Soc NSW, 39 131 (1905)

2. V.A. Rakov and M.A. Uman, Lightning: Physics and Effects, Cambridge University Press (Cambridge 2003)

3. Geoffrey Walsh, private communication

4. E.F. Northrup, Phys. Rev. 24 474 (1907)

5. W.H. Bennett, Phys. Rev. 45 90 (1934)

6. M.G. Haines, T.W.I. Sanford and V.P. Smirnov, Plasma Phys Control Fusion, 47 B1-B11 (2005)

7. http://en.wikipedia.org/wiki/Plasma_pinch