|
|
| Line 1: |
Line 1: |
| {{Other uses2|Pinch}}
| | Hi there, I am Alyson Pomerleau [http://Fashionlinked.com/index.php?do=/profile-13453/info/ best psychic readings] and I believe it seems quite great when you say it. I've usually loved residing in Mississippi. The preferred pastime for him and his kids is to perform lacross and he'll be beginning some thing else along with it. Office supervising is exactly where her primary income arrives from but she's already utilized for an additional 1.<br><br>Also visit my webpage :: online [http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies free psychic readings] chat ([http://cspl.postech.ac.kr/zboard/Membersonly/144571 http://cspl.postech.ac.kr/zboard/Membersonly/144571]) |
| [[Image:Lightning over Oradea Romania 3.jpg|thumb|256px|right|[[Lightning discharge|Lightning bolts]] illustrating electromagnetically pinched plasma filaments ]]
| |
| [[Image:plasma-filaments.jpg|thumb|right|300px|Z-pinches constrain the [[plasma (physics)|plasma]] filaments in an [[electrical discharge]] from a [[Tesla coil]]. (Click to enlarge image for detail) ]]
| |
| | |
| A '''pinch''' is the compression of an electrically conducting [[Electrical filament|filament]] by [[magnetic]] forces. The conductor is usually a [[plasma (physics)|plasma]], but could also be a solid or liquid [[metal]]. In a '''[[z-pinch]]''', the current is axial (in the ''z'' direction in a [[cylindrical coordinate system]]) and the [[magnetic field]] [[azimuth]]al; in a '''theta-pinch''', the current is azimuthal (in the theta direction in cylindrical coordinates) and the magnetic field is axial. The phenomenon may also be referred to as a "Bennett pinch"<ref>See for example, Buneman, O., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1961plph.conf..202B&db_key=PHY&data_type=HTML&format=&high=42ca922c9c08281 The Bennett Pinch]" (1961) ''Plasma Physics'', Edited by James E. Drummond. LOC 60-12766. Publ. McGraw-Hill, Inc., New York, 1961, p.202</ref> (after [[Willard Harrison Bennett]]), "electromagnetic pinch",<ref>Lee, S., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1983PlPh...25..571L&db_key=PHY&data_type=HTML&format=&high=42ca922c9c25795 Energy balance and the radius of electromagnetically pinched plasma columns]" (1983) ''Plasma Physics'', Volume 25, Issue 5, pp. 571–576 (1983).</ref> "magnetic pinch",<ref>Schmidt, Helmut, "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1966PhRv..149..564S&db_key=PHY&data_type=HTML&format=&high=42ca922c9c26126 Formation of a Magnetic Pinch in InSb and the Possibility of Population Inversion in the Pinch]" (1966) ''Physical Review'', vol. 149, Issue 2, pp. 564–573</ref> "pinch effect"<ref>Severnyi, A. B., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1959SvA.....3..887S&db_key=AST&data_type=HTML&format=&high=42ca922c9c25243 On the Appearance of Cosmics Rays in the Pinch Effect in Solar Flares]" (1959) ''Soviet Astronomy'', Vol. 3, p.887</ref> or "plasma pinch".<ref>Zueva, N. M.; Solov'ev, L. S.; Morozov, A. I. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1976JETPL..23..256Z&db_key=PHY&data_type=HTML&format=&high=42ca922c9c04156 Nonlinear instability of plasma pinches]" (1976) ''Journal of Experimental and Theoretical Physics Letters'', Vol. 23, p.256</ref>
| |
| | |
| Pinches occur naturally in electrical discharges such as [[lightning|lightning bolt]]s,<ref>Rai, J.; Singh, A. K.; Saha, S. K, "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1973IJRSP...2..240R&db_key=PHY&data_type=HTML&format=&high=42ca922c9c04988 Magnetic field within the return stroke channel of lightning]" (1973) ''Indian Journal of Radio and Space Physics'', vol. 2, Dec. 1973, p. 240-242.</ref> the [[Aurora (astronomy)|aurora]],<ref>Galperin, Iu. I.; Zelenyi, L. M.; Kuznetsova, M. M. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1986CosRe..24..865G&db_key=AST&data_type=HTML&format=&high=42ca922c9c01478 Pinching of field-aligned currents as a possible mechanism for the formation of raylike auroral forms]" (1986) ''Kosmicheskie Issledovaniia'' (ISSN 0023-4206), vol. 24, Nov.-Dec. 1986, p. 865-874. In Russian.</ref> [[current sheet]]s,<ref>Syrovatskii, S. I. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1981ARA%26A..19..163S&db_key=AST&data_type=HTML&format=&high=42ca922c9c06271 Pinch sheets and reconnection in astrophysics]" (1981) In ''Annual review of astronomy and astrophysics''. Volume 19. (A82-11551 02-90) Palo Alto, CA, Annual Reviews, Inc., 1981, p. 163-229</ref> and [[solar flare]]s.<ref>Airapetyan, V. S.; Vikhrev, V. V.; Ivanov, V. V.; Rozanova, G. A. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1990Ap.....32..230A&db_key=AST&data_type=HTML&format=&high=42ca922c9c20133 Pinch Mechanism of Energy Release of Stellar Flares]" (1990) ''Astrophsyics'' (Tr. ''Astrofizika'') v.32 No.3 Nov. p.230 1990</ref> They are also produced in the laboratory, primarily for research into [[fusion power]].
| |
| | |
| ==Pinch production and types==
| |
| [[Image:Crushed rod pollock barraclough.jpg|thumb|128px|A section of the crushed lightning rod studied by Pollock and Barraclough.<ref name="pollock"/> The rod is in the collection of the School of Physics, [[University of Sydney]], [[Australia]].]]
| |
| Pinches are created in the laboratory in equipment related to [[nuclear fusion]], such as the [[Z-pinch|Z-pinch machine]], and high-energy physics, such as the [[dense plasma focus]]. Pinches may also become [[instability|unstable]],<ref>Hardee, P. E., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1982ApJ...257..509H&db_key=AST&data_type=HTML&format=&high=42ca922c9c18868 Helical and pinching instability of supersonic expanding jets in extragalactic radio sources]" (1982) ''Astrophysical Journal'', Part 1, vol. 257, June 15, 1982, p. 509-526</ref> and generate radiation across the [[electromagnetic spectrum]], including [[radio waves]], [[x-rays]]<ref>Pereira, N. R., ''et al.'', "[X rays from z-pinches on relativistic electron-beam generators]" (1988) ''Journal of Applied Physics'' (ISSN 0021-8979), vol. 64, Aug. 1, 1988, p. R1-R27</ref> and [[gamma rays]],<ref>Wu, Mei; Chen, Li; Li, Ti-Pei, "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2005ChJAA...5...57W&db_key=AST&data_type=HTML&format=&high=42ca922c9c03905 Polarization in Gamma-Ray Bursts Produced by Pinch Discharge]" (2005) ''Chinese Journal of Astronomy & Astrophysics'', Vol. 5, p. 57-64</ref> and also [[neutron]]s<ref>Anderson, Oscar A., ''et al.'', "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1958PhRv..110.1375A&db_key=PHY&data_type=HTML&format=&high=42ca922c9c08574 Neutron Production in Linear Deuterium Pinches]" (1958) ''Physical Review'', vol. 110, Issue 6, pp. 1375–1387</ref> and [[synchrotron radiation]].<ref>Peratt, A.L., "[http://books.google.com/books?vid=ISBN079235527X&id=ZSlJRAeL95sC&pg=PA62&lpg=PA62&dq=synchrotron+pinch&sig=hYbqU8FIS6mzVJ9MlkFhleuFNhQ Synchrotron radiation from pinched particle beams]", (1998) Plasma Physics: VII Lawpp 97: Proceedings of the 1997 Latin American Workshop on Plasma Physics, Edited by Pablo Martin, Julio Puerta, Pablo Martmn, with reference to Meierovich, B. E., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1984PhR...104..259M&db_key=PHY&data_type=HTML&format= Electromagnetic collapse. Problems of stability, emission of radiation and evolution of a dense pinch]" (1984) ''Physics Reports'', Volume 104, Issue 5, p. 259-346.</ref> Types of pinches, that may differ in geometry and operating forces,<ref name="adsabs.harvard.edu">Carlqvist, Per, "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1988Ap&SS.144...73C Cosmic electric currents and the generalized Bennett relation]" (1988) ''Astrophysics and Space Science'' (ISSN 0004-640X), vol. 144, no. 1-2, May 1988, p. 73-84</ref> include the cylindrical pinch, inverse pinch, orthogonal pinch effect, [[reversed field pinch]], sheet pinch, screw pinch<ref>Srivastava, K. M.; Vyas, D. N., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1982Ap%26SS..86...71S&db_key=AST&data_type=HTML&format=&high=42ca922c9c09199 Non-linear analysis of the stability of the screw pinch]", (1982) ''Astrophysics and Space Science'', vol. 86, no. 1, Aug. 1982, p. 71-89</ref> (also called ''stabilized z-pinch'', or ''θ-z pinch''),<ref>See "[http://silas.psfc.mit.edu/introplasma/chap4.html#tth_sEc4.7 MHD Equilibria" in Introduction to Plasma Physics by I.H.Hutchinson (2001)]</ref> theta pinch (or ''thetatron''<ref>See ''Dictionary of Material Science and High Energy Physics'' [http://books.google.com/books?vid=ISBN0849328896&id=vvJazZZpB1QC&pg=PA315&lpg=PA315&dq=thetatron+pinch&sig=TCOnE7mqDqYv75nmAgd_knHE9_U p.315] ISBN 0-8493-2889-6</ref>), [[Madison Symmetric Torus#How to make a toroidal pinch|toroidal pinch]], ware pinch<ref>Helander, P. ''et al.'' "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2005PPCF...47B.151H&db_key=PHY&data_type=HTML&format=&high=42ca922c9c04106 The effect of non-inductive current drive on tokamak transport]" (2005) ''Plasma Physics and Controlled Fusion'', Volume 47, Issue 12B, pp. B151-B163</ref> and Z-pinch.
| |
| | |
| Pinches are used to generate [[X-rays]], and the intense magnetic fields generated are used in [[electromagnetic forming]] of metals (they have been demonstrated in crushing aluminium soft drinks cans). They have applications to [[particle beam]]s<ref>Ryutov, D. D.; Derzon, M. S.; Matzen, M. K, "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2000RvMP...72..167R&db_key=PHY&data_type=HTML&format=&high=42ca922c9c24923 The physics of fast Z pinches]" (2000) ''Reviews of Modern Physics'', vol. 72, Issue 1, pp. 167–223</ref> including [[particle beam weapon]]s,<ref>Andre Gsponer, "[http://arxiv.org/abs/physics/0409157 Physics of high-intensity high-energy particle beam propagation in open air and outer-space plasmas]" (2004) http://arxiv.org/abs/physics/0409157</ref> and astrophysics.<ref>Peratt, Anthony L., "[http://adsabs.harvard.edu/abs/1988LaPaB...6..471P The role of particle beams and electrical currents in the plasma universe]" (1988) ''Laser and Particle Beams'' (ISSN 0263-0346), vol. 6, Aug. 1988, p. 471-491.</ref>
| |
| | |
| ==History==
| |
| [[Image:ieee-emblem.jpg|thumb|left|128px|The [[Institute of Electrical and Electronics Engineers]] emblem shows the basic features of an azimuthal magnetic pinch.<ref>See also the IEEE History Center, "[http://www.ieee.org/organizations/history_center/ieee_emblem.html Evolution of the IEEE Logo]" March 1963; see also the comments in "[http://public.lanl.gov/alp/plasma/lab_astro.html Laboratory Astrophysics]"</ref>]]
| |
| The first creation of a z-pinch in the laboratory may have occurred in 1790 in Holland when [[Martin van Marum|Martinus van Marum]] created an explosion by discharging 100 [[Leyden jar]]s into a wire.<ref>van Marum M 1790 ''Proc. 4th Int. Conf. on Dense Z-Pinches'' (Vancouver 1997) (Am. Inst. Phys. Woodbury, New York, 1997) Frontispiece and p ii</ref> The phenomenon was not understood until 1905, when Pollock and Barraclough<ref name="pollock">Pollock J A and Barraclough S, 1905 ''Proc. R. Soc. New South Wales'' 39 131</ref> investigated a compressed and distorted length of copper tube from a [[lightning rod]] after it had been struck by lightning. Their analysis showed that the forces due to the interaction of the large current flow with its own magnetic field could have caused the compression and distortion.<ref>[[Bas Pease|R. S. Pease]], "The Electromagnetic Pinch: From [[James Arthur Pollock|Pollock]] to the [[Joint European Torus]]", "[http://nsw.royalsoc.org.au/journal/118_12.html#pease Pollock Memorial Lecture for 1984 delivered at the University of Sydney, 28 November, 1984"]: ''This review of the electromagnetic pinch starts with an exhibit taken from Pollock's work, carefully preserved and drawn to attention of modern research by [[Charles Norman Watson-Munro|Professor C. Watson-Munro]]. It is a compressed and distorted length of copper tube originally part of the lightning conductor on the Hartley Vale kerosene refinery in New South Wales. It was known to have been struck by lightning. Pollock and Barraclough (1905) from the Department of Mechanical Engineering at Sydney University carried out an analysis to see whether or not the compression could have arisen from the flow of electric current. They concluded that the compressive forces, due to the interaction of the large current flow with its own magnetic field could have been responsible for the compression and distortion. As far as I know, this is the first identified piece of observational data on the electromagnetic pinch; and the first theoretical discussion of the effect.''</ref> A similar, and apparently independent, theoretical analysis of the pinch effect in liquid metals was published by Northrupp in 1907.<ref>Northrupp E F 1907 "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1907PhRvI..24..474N&db_key=PHY&data_type=HTML&format= Some Newly Observed Manifestations of Forces in the Interior of an Electric Conductor]" (1907) ''Phys. Rev''. 24 474. He wrote: "Some months ago, my friend, Carl Hering, described to me a surprising and apparently new phenomenon which he had observed. He found, in passing a relatively large alternating current through a non-electrolytic, liquid conductor contained in a trough, that the liquid contracted in cross-section and flowed up hill lengthwise of the trough... Mr. Hering suggested the idea that this contraction was probably due to the elastic action of the lines of magnetic force which encircle the conductor... As the action of the forces on the conductor is to squeeze or pinch it, he jocosely called it the 'pinch phenomenon'.</ref> The next major development was the publication in 1934 of an analysis of the radial pressure balance in a static z-pinch by [[Willard Harrison Bennett|Bennett]]<ref>W.H.Bennett, "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1934PhRv...45..890B&db_key=PHY&data_type=HTML&format= Magnetically Self-Focussing Streams]", ''Phys. Rev''. '''45''' 890 (1934)</ref> (See the following section for details.)
| |
| | |
| Thereafter, the experimental and theoretical progress on pinches was driven by [[fusion power]] research. In their article on the "Wire-array z-pinch: a powerful x-ray source for [[inertial confinement fusion|ICF]]", M G Haines ''et al.'', wrote on the "Early history of z-pinches":<ref>M G Haines, T W L Sanford and V P Smirnov, "[http://www.iop.org/EJ/abstract/0741-3335/47/12B/S01 Wire-array z-pinch: a powerful x-ray source for ICF]" (2005) ''Plasma Phys. Control. Fusion'' 47 B1-B11 (online in full, click PDF).</ref>
| |
| | |
| :In 1946 Thompson and Blackman submitted a patent for a [[fusion reactor]] based on a toroidal z-pinch<ref>Thompson G P and Blackman M 1946 British Patent 817681. Haines M G 1996 "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1996PPCF...38..643H&db_key=PHY&data_type=HTML&format= Historical Perspective: Fifty years of controlled fusion research]" ''Plasma Phys. Control. Fusion'' 38 643</ref> with an additional vertical magnetic field. But in 1954 Kruskal and Schwarzschild <ref>Kruskal M D and Schwarzschild "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1954RSPSA.223..348K&db_key=GEN&data_type=HTML&format= Some Instabilities of a Completely Ionized Plasma]" 1954 ''Proc. R. Soc. Lond''. A 223 348</ref> published their theory of MHD instabilities in a z-pinch. In 1956 Kurchatov gave his famous Harwell lecture showing nonthermal neutrons and the presence of ''m'' = 0 and ''m'' = 1 instabilities in a deuterium pinch.<ref>Kurchatov I V 1957 ''J. Nucl. Energy'' 4 193</ref> In 1957 Pease<ref>Pease R S "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1957PPSB...70...11P&db_key=PHY&data_type=HTML&format= Equilibrium Characteristics of a Pinched Gas Discharge Cooled by Bremsstrahlung Radiation]" 1957 Proc. ''Phys. Soc. Lond''. 70 11</ref> and Braginskii<ref>Braginskii S I 1957 ''Zh. Eksp. Teor. Fiz'' 33 645; Braginskii S I 1958 ''Sov. Phys.—JETP'' 6 494</ref> independently predicted radiative collapse in a z-pinch under pressure balance when in hydrogen the current exceeds 1.4 MA. (The viscous rather than resistive dissipation of magnetic energy discussed above and in<ref>Haines M G ''et al.'' 2005 ''Phys. Rev. Lett''. submitted; see also EPS Conf. on Plasma Physics 2004 (London, UK) paper 73</ref> would however prevent radiative collapse). Lastly, at Imperial College in 1960, led by R Latham, the [[Plateau-Rayleigh instability]] was shown, and its growth rate measured in a dynamic z-pinch.<ref>Curzon F L ''et al.'' "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1960RSPSA.257..386C&db_key=GEN&data_type=HTML&format= Experiments on the Growth Rate of Surface Instabilities in a Linear Pinched Discharge]" 1960 ''Proc. R. Soc. Lond''. A 257 386</ref>"
| |
| | |
| ==Configurations==
| |
| | |
| ===One-dimensional configurations===
| |
| There are three analytic one dimensional configurations generally studied in plasma physics. These are the θ-pinch, the [[Z-pinch]], and the Screw Pinch. All of the classic one dimensional pinches are cylindrically shaped. Symmetry is assumed in the axial (''z'') direction and in the azimuthal (θ) direction. It is traditional to name a one-dimensional pinch after the direction in which the current travels.
| |
| | |
| '''The θ-pinch'''
| |
| | |
| [[Image:thet pinch.png|right|thumb|200px|A sketch of the θ-Pinch Equilibrium. The z directed magnetic field (shown in purple) corresponds to a θ directed plasma current (shown in yellow).]]
| |
| | |
| The θ-pinch has a magnetic field traveling in the z direction. Using [[Ampère's law]] (discarding the displacement term)
| |
| | |
| : <math>\nabla \times \vec{B} = \mu_0 \vec{J}</math>
| |
| | |
| : <math>\vec{B} = B_{z}(r)\hat{z}</math>
| |
| | |
| : <math>\mu_{0} \vec{J} =\frac{1}{r} \frac{d}{d \theta}B_z \hat{r} - \frac{d}{dr}B_z \hat{\theta} </math>
| |
| | |
| Since ''B'' is only a function of ''r'' we can simplify this to
| |
| | |
| : <math> \mu_0 \vec{J} = -\frac{d}{dr}B_z \hat{\theta}</math>
| |
| | |
| So ''J'' points in the θ direction.
| |
| | |
| Thus, the equilibrium condition ( ∇''p'' = '''j × Β''') for the θ-pinch reads:
| |
| | |
| : <math> \frac{d}{d r} \left( p +\frac{B_z^2}{2 \mu_0 } \right) =0 </math>
| |
| | |
| θ-pinches tend to be resistant to plasma instabilities; This is due in part to the frozen in flux theorem, which is beyond the scope of this article.
| |
| | |
| '''The Z-Pinch'''
| |
| | |
| [[Image:z pinch.png|right|thumb|200px|A sketch of the z-Pinch Equilibrium. A -θ directed magnetic field (shown in purple) corresponds to a z directed plasma current (shown in yellow).]]
| |
| | |
| The Z-Pinch has a magnetic field in the θ direction. Again, by electrostatic Ampere's Law
| |
| | |
| : <math>\nabla \times \vec{B} = \mu_0 \vec{J}</math>
| |
| | |
| : <math>\vec{B} = B_{\theta}(r)\hat{\theta}</math>
| |
| | |
| : <math>\mu_{0} \vec{J} = \frac{1}{r}\frac{d}{dr}(r B_{\theta}) \hat{z} - \frac{d}{dz}B_{\theta} \hat{r}</math>
| |
| | |
| : <math>\mu_{0} \vec{J} = \frac{1}{r}\frac{d}{dr}(r B_{\theta}) \hat{z}</math>
| |
| | |
| So ''J'' points in the ''z'' direction.
| |
| | |
| Thus, the equilibrium condition ( ∇''p'' = '''j × Β''') for the z-pinch reads:
| |
| | |
| : <math> \frac{d}{d r} \left( p +\frac{B_\theta^2}{2 \mu_0 } \right) +\frac{B_\theta^2}{\mu_0 r}=0 </math>
| |
| | |
| Since particles in a plasma basically follow magnetic field lines, Z-pinches lead them around in circles. Therefore, they tend to have excellent confinement properties.
| |
| | |
| '''The screw pinch'''
| |
| The screw pinch is an effort to combine the stability aspects of the θ-pinch and the confinement aspects of the Z-pinch. Referring once again to Ampere's Law
| |
| | |
| : <math>\nabla \times \vec{B} = \mu_0 \vec{J}</math>
| |
| | |
| But this time, the ''B'' field has a θ component ''and'' a ''z'' component
| |
| | |
| : <math>\vec{B} = B_{\theta}(r)\hat{\theta} + B_z (r) \hat{z}</math>
| |
| | |
| : <math>\mu_0 \vec{J} = \frac{1}{r}\frac{d}{dr}(r B_{\theta}) \hat{z} - \frac{d}{dr}B_{z} \hat{\theta}</math>
| |
| | |
| So this time ''J'' has a component in the ''z'' direction and a component in the θ direction.
| |
| | |
| Finally, the equilibrium condition ( ∇''p'' = '''j × Β''') for the screw pinch reads:
| |
| | |
| : <math> \frac{d}{d r} \left( p +\frac{B_z^2+B_\theta^2}{2 \mu_0 } \right) +\frac{B_\theta^2}{\mu_0 r} =0 </math>
| |
| | |
| ===Two-dimensional equilibria===
| |
| [[Image:toroidal coord.png|right|thumb|250px|A ''toroidal coordinate system'' in common use in plasma physics. The red arrow indicates the '''poloidal''' direction (θ) and the blue arrow indicates the '''toroidal''' direction (φ)]]
| |
| | |
| A common problem with one-dimensional equilibria based machines is end losses. As mentioned above, most of the motion of particles in a plasma is directed along the magnetic field. With the θ-pinch and the screw-pinch, this leads particles to the end of the machine very quickly (as the particles are typically moving quite fast). Additionally, the Z-pinch has major stability problems. Though particles can be reflected to some extent with [[magnetic mirror]]s, even these allow many particles to pass. The most common method of mitigating this effect is to bend the cylinder around into a torus. Unfortunately this breaks θ symmetry, as paths on the inner portion (inboard side) of the torus are shorter than similar paths on the outer portion (outboard side). Thus, a new theory is needed. This gives rise to the famous [[Grad–Shafranov equation]].
| |
| | |
| The one dimensional equilibria provide the inspiration for some of the toroidal configurations. An example of this is the ZETA device at Culham England (which also operated as a [[Reversed field pinch|Reversed Field Pinch]]). The most well recognized of these devices is the toroidal version of the screw pinch, the [[Tokamak]].
| |
| | |
| Numerical solutions to the Grad–Shafranov equation have also yielded some equilibria, most notably that of the [[reversed field pinch]].
| |
| | |
| ===Three-dimensional equilibria===
| |
| There does not exist a coherent analytical theory for three-dimensional equilibria. The general approach to finding three dimensional equilibria is to solve the vacuum ideal MHD equations. Numerical solutions have yielded designs for [[stellarator]]s. Some machines take advantage of simplification techniques such as helical symmetry (for example University of Wisconsin's Helically Symmetric eXperiment). However, for an arbitrary three-dimensional configuration an equilibrium relation, similar to that of the 1-D configurations exists:<ref>Ideal Magnetohydrodynamics: Modern perspectives in energy. Jeffrey P. Freidberg. Massachusetts Institute of Technology. Cambridge, Massachusetts. Plenum Press - New York and London - 1987. (Pg.86 & 95)</ref>
| |
| | |
| : <math> \nabla_\perp \left( p +\frac{B^2}{2 \mu_0 } \right) - \frac{B^2}{\mu_0 }\vec{\kappa}=0 </math>
| |
| | |
| Where κ is the curvature vector defined as:
| |
| | |
| : <math> \vec{\kappa} = \vec{b}\cdot \nabla\vec{b}</math> | |
| | |
| with ''b'' the unit vector tangent to ''B''.
| |
| | |
| ==Formal treatment==
| |
| [[Image:water-pinching.jpg|thumb|256px|'''A stream of water pinching''' into droplets has been suggested as an analogy to the electromagnetic pinch.<ref>Trubnikov, Boris A., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1992ITPS...20..898T&db_key=AST&data_type=HTML&format=&high=42ca922c9c06116 A new hypothesis of cosmic ray generation in plasma pinches]" (1992) ''IEEE Transactions on Plasma Science'' (ISSN 0093-3813), vol. 20, no. 6, p. 898-904.</ref> The gravity accelerates free-falling water which causes the water column to constrict. Then [[surface tension]] breaks the narrowing water column into droplets (not shown here) (see [[Plateau-Rayleigh instability]]), which is analogous to the [[magnetic field]] which has been suggested as the cause of pinching in bead lightning.<ref>"The PLASMAK Configuration and Ball Lightning" ([http://www.prometheus2.net/bl-tokyo.pdf PDF]) presented at the International Symposium on Ball Lightning; July 1988</ref> The morphology (shape) is similar to the so-called sausage [[instability]] in plasma.]]
| |
| | |
| ===The Bennett relation===
| |
| Consider a cylindrical column of fully ionized quasineutral <!-- This derivation assumes that the plasma is quasineutral, otherwise N<sub>i</sub> and N<sub>e</sub> could not be expressed as a single N. Of course, one could easily assume that this was implied. Clarification may not be necessary. -->plasma, with an axial electric field, producing an axial current density, '''j''', and associated azimuthal magnetic field, '''B'''. As the current flows through its own magnetic field, a pinch is generated with an inward radial force density of '''j x B'''. In a steady state with forces balancing:
| |
| | |
| :∇''p'' = ∇(''p<sub>e</sub> + p<sub>i</sub>'') = '''j × Β'''
| |
| where ∇''p'' is the magnetic pressure gradient, ''p''<sub>e</sub> and p<sub>i</sub> is the electron and ion pressures. Then using [[Maxwell's equations|Maxwell's equation]] ∇ × '''B''' = μ<sub>0</sub> '''j''' and the [[ideal gas law]] ''p = N k T'', we derive:
| |
| :<math>2 N k(T_e + T_i) = \frac{{\mu_0}} {4 \pi} I^2</math> (the Bennett relation)
| |
| where ''N'' is the number of electrons per unit length along the axis, ''T<sub>e</sub>'' and ''T<sub>i</sub>'' are the electron and ion temperatures, ''I'' is the total beam current, and ''k'' is the [[Boltzmann constant]].
| |
| | |
| ===The generalized Bennett relation===
| |
| [[Image:Generalized Bennett Relation diagram.png|thumb|200px|The generalized Bennett relation considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω]]The ''Generalized Bennett Relation'' considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω. Along the axis of the plasma cylinder flows a current density j<sub>z</sub>, resulting in an azimuthal magnetίc field Β<sub>φ</sub>. Originally derived by Witalis,<ref>Witalis, E. A. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1981PhRvA..24.2758W&db_key=PHY&data_type=HTML&format=&high=42ca922c9c29389 Plasma-physical aspects of charged-particle beams]" (1981) ''Physical Review A - General Physics'', 3rd Series, vol. 24, Nov. 1981, p. 2758–2764</ref> the Generalized Bennett Relation results in:<ref>Anthony L . Peratt, "Physics of the Plasma Universe", 1992 Springer-Verlag, ISBN 0-387-97575-6</ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \frac{1}{4} \frac{\partial^2 J_0}{\partial t^2} & = W_{\perp \text{kin}} + \Delta W_{E_z} + \Delta W_{B_z} + \Delta W_k - \frac{{\mu_0}} {8 \pi} I^2 (a) \\[8pt]
| |
| & {} - \frac{1}{2}G\overline{m}^2 N^2 (a) + \frac{1}{2}\pi a^2 \epsilon_0 \left(E_r^2 (a) - E_\phi^2 (a) \right)
| |
| \end{align}
| |
| </math>
| |
| | |
| *where a current-carrying, magnetic-field-aligned cylindrical plasma has a radius ''a'',
| |
| *''J''<sub>0</sub> is the total moment of inertia with respect to the z axis,
| |
| *''W''<sub>⊥kin</sub> is the [[kinetic energy]] per unit length due to beam motion transverse to the beam axis
| |
| *''W''<sub>B<sub>z</sub></sub> is the self-consistent B<sub>z</sub> energy per unit length
| |
| *''W''<sub>E<sub>z</sub></sub> is the self-consistent E<sub>z</sub> energy per unit length
| |
| *''W''<sub>k</sub> is thermokinetic energy per unit length
| |
| *''I''(''a'') is the axial current inside the radius ''a'' (''r'' in diagram)
| |
| *''N''(''a'') is the total number of particles per unit length
| |
| *''E''<sub>r</sub> is the radial electric field
| |
| *''E''<sub>φ</sub> is the rotational electric field
| |
| The positive terms in the equation are expansional forces while the negative terms represent beam compressional forces.
| |
| | |
| ===The Carlqvist relation===
| |
| The Carlqvist Relation, published by [[Per Carlqvist]] in 1988,<ref name="adsabs.harvard.edu"/> is a specialization of the Generalized Bennett Relation (above), for the case that the kinetic pressure is much smaller at the border of the pinch than in the inner parts. It takes the form
| |
| | |
| :<math>\frac{{\mu_0}} {8 \pi} I^2 (a) +\frac{1}{2}G\overline{m}^2 N^2 (a) = \Delta W_{B_z} + \Delta W_k</math>
| |
| | |
| and is applicable to many space plasmas.
| |
| | |
| [[Image:Bennett Pinch graph.png|thumb|400px|The Bennett pinch showing the total current (I) versus the number of particles per unit length (N). The chart illustrates four physically distinct regions. The plasma temperature is 20 K, the mean particle mass 3×10<sup>−27</sup> kg, and ΔW<sub>Bz</sub> is the excess magnetic energy per unit length due to the axial magnetic field B<sub>z</sub>. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.]]
| |
| | |
| The Carlqvist Relation can be illustrated (see right), showing the total current (''I'') versus the number of particles per unit length (''N'') in a Bennett pinch. The chart illustrates four physically distinct regions. The plasma temperature is quite cold (''T''<sub>i</sub> = ''T''<sub>e</sub> = ''T''<sub>n</sub> = 20 K), containing mainly hydrogen with a mean particle mass 3×10<sup>−27</sup> kg. The thermokinetic energy ''W''<sub>''k''</sub> >> ''πa''<sup>2</sup> ''p''<sub>''k''</sub>(a). The curves, ΔW<sub>Bz</sub> show different amounts of excess magnetic energy per unit length due to the axial magnetic field B<sub>z</sub>. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.
| |
| | |
| '''Chart regions:''' (a) In the top-left region, the pinching force dominates. (b) Towards the bottom, outward kinetic pressures balance inwards magnetic pressure, and the total pressure is constant. (c) To the right of the vertical line Δ''W''<sub>''B''z</sub> = 0, the magnetic pressures balances the gravitational pressure, and the pinching force is negligible. (d) To the left of the sloping curve Δ''W''<sub>''B''z</sub> = 0, the gravitational force is negligible. Note that the chart shows a special case of the Carlqvist relation, and if it is replaced by the more general Bennett relation, then the designated regions of the chart are not valid.
| |
| | |
| Carlqvist further notes that by using the relations above, and a derivative, it is possible to describe the Bennett pinch, the [[Jeans instability|Jeans criterion]] (for gravitational instability,<ref>J. H. Jeans, "[http://adsabs.harvard.edu/abs/1902RSPTA.199....1J The stability of a spherical nebula]" ''Phil. Trans. R. Soc. Lond. A ''199 (1902)</ref> in one and two dimensions), [[Birkeland current|force-free magnetic fields]], gravitationally balanced magnetic pressures, and continuous transitions between these states.
| |
| | |
| ==Crushing cans with the pinch effect==
| |
| [[Image:Aluminium-can-white.jpg|thumb|150px|right|'''Pinched aluminium can''', produced from a [[pulsed power|pulsed]] magnetic field created by rapidly discharging 2 kilojoules from a high voltage [[capacitor]] bank into a 3-turn coil of heavy gauge wire.]]
| |
| {{Main|Electromagnetic forming}}Many high-voltage electronics enthusiasts make their own crude electromagnetic forming devices.<ref name="LaPointe">{{cite web |url=http://members.tm.net/lapointe/Main.html | title=High Voltage Devices and Experiments |accessdate=February 21, 2013 | author=LaPointe, Robert}}</ref><ref name="Tristan">{{cite web | url=http://members.tripod.com/extreme_skier/cancrusher/ | title=Electromagnetic Can Crusher | accessdate=February 21, 2013 | author=Tristan}}</ref><ref name="Borros">{{cite web | url=http://www.powerlabs.org/pssecc.htm | title=Solid State Can Crusher | accessdate=February 21, 2013 | author=Borros, Sam}}</ref> They use [[pulsed power]] techniques to produce a theta pinch capable of crushing an aluminium soft drink can using the [[Lorentz force]]s created when high currents are induced in the can by the strong magnetic field of the primary coil.<ref>{{cite web | url=http://magnet-physik.de/st_magnetopuls.html#method | title=MagnetoPulS | publisher=MAGNET-PHYSIK Dr. Steingroever GmbH | work=web site | year=2002 | archivedate=2003-05-22 | accessdate=February 21, 2013 | archiveurl=http://web.archive.org/web/20030522114102/http://magnet-physik.de/st_magnetopuls.html#method}}
| |
| </ref><ref>{{cite web |url=http://www.english.pstproducts.com/index_htm_files/English%20White%20Paper%20by%20PSTproducts.pdf| title=Industrial Application of the Electromagnetic Pulse Technology | publisher=PSTproducts GmbH |work=white paper | date=June 2009 | accessdate=February 21, 2013}}</ref>
| |
| | |
| An electromagnetic aluminium can crusher consists of four main components (1) A [[high voltage]] [[direct current|DC]] [[power supply]] which provides a source of [[electrical energy]] (2) A large ''energy discharge'' [[capacitor]] to accumulate the electrical energy (3) A high voltage switch or [[spark gap]] and (4) A robust coil (capable of surviving high magnetic pressure) through which the stored electrical energy can be quickly discharged in order to generate a correspondingly strong pinching magnetic field (see diagram below).
| |
| | |
| [[Image:can-pincher.png|thumb|center|300px|Electromagnetic pinch "can crusher": schematic diagram]]
| |
| | |
| In practice, such a device is somewhat more sophisticated than the schematic diagram suggests, including electrical components that control the current in order to maximize the resulting pinch, and to ensure that the device works safely. For more details, see the notes.<ref>Examples of electromagnetic pinch can crushers can be found at (a) Bob LaPointe's site on [http://members.tm.net/lapointe/Main.html High Voltage Devices and Experiments] (b) Tristran's [http://members.tripod.com/extreme_skier/cancrusher/ Electromagnetic Can Crusher] (including schematic) (c) Sam Borros's [http://www.powerlabs.org/pssecc.htm Solid State Can Crusher]</ref>
| |
| | |
| ==Depictions==
| |
| A fictionalized [[Explosively pumped flux compression generator|pinch-generating device]] was used in ''[[Ocean's Eleven (2001 film)|Ocean's Eleven]]'', where it was used to disrupt Las Vegas's power grid just long enough for the characters to begin their heist.<ref>{{cite news | publisher=American Physical Society| title= The Con-Artist Physics of 'Ocean's Eleven'.| date=March 2002 | url=http://www.aps.org/publications/apsnews/200203/oceans-eleven.cfm}}</ref>
| |
| | |
| ==See also==
| |
| *[[Madison Symmetric Torus]] (Reversed field pinch, How to make a toroidal pinch)
| |
| *[[Explosively pumped flux compression generator]]
| |
| *[[Magneforming]]
| |
| *[[List of plasma (physics) articles]]
| |
| | |
| ==References==
| |
| {{Reflist|2}}
| |
| | |
| ==External links==
| |
| *[http://205.243.100.155/frames/shrinkergallery.html Examples of electromagnetically shrunken coins and crushed cans.]
| |
| *[http://205.243.100.155/frames/shrinker.html Theory of electromagnetic coin shrinking]
| |
| *[http://205.243.100.155/frames/Shrinking_History.htm The Known History of "Quarter Shrinking"]
| |
| *[http://tesladownunder.iinet.net.au/CanCrushing.htm Can crushing info using electromagnetism among other things.]
| |
| *[http://dorland.pp.ph.ic.ac.uk/magpie/ The MAGPIE project at Imperial College London] is used to study wire array Z-pinch implosions.
| |
| | |
| [[Category:Electromagnetism]]
| |
| [[Category:Fusion power]]
| |
| [[Category:Plasma physics]]
| |