C3D. Conductivity In Less Than Three Dimensions

Transcription

1 ADANCED UNDERGRADUATE LABORATORY C3D Conductivity In Less Thn Three Dimensions Revisions: July 2016: Dvid Biley Jnury 2008: Dvid Biley : Json Hrlow 2003: Tetyn Antimirov Originl: Dvid Biley 2002 Copyright 2016 University of Toronto This work is licensed under the Cretive Commons Attribution-NonCommercil-ShreAlike 4.0 Unported License. (

2 Introduction Any field is constrined by the geometry of the spce in which it exists. Mesuring such fields llows one to determine wide rnge of physicl properties: everything from the depth of the locl wter tble to setting limits on the size nd existence of extr spce-time dimensions. If current I is introduced from n externl source into the interior of medium of constnt resistivity, the current will flow rdilly wy from the source point. The potentil due to this source (reltive to the potentil t infinite distnce) will vry s = 1 (1) 4π R where R is the distnce from the source to the point where the potentil is mesured. This is the fmilir 1/R potentil of ny non-self-intercting 1 field (e.g. electric, grvittionl, coustic, ) tht cn fill ll spce nd is generted from point source. In mny systems, such s wires, thin films, or quntum chromodynmic gluon strings, the field is effectively constrined to less thn 3 dimensions nd the potentil no longer follows 1/R potentil. In modern superstring (nd relted) theories, there re fields tht extend in more thn 3 spce dimensions nd they similrly hve different rdil potentil dependences. Resistivity Surveying is populr nd successful geophysicl prospecting method. Electric current is pssed into the ground, nd the resulting electric potentil ptterns on the ground re mesured, reveling subsurfce resistivity vritions cused by geologicl, hydrogeologicl nd mnmde vritions. Resistivity surveys re used tody for groundwter investigtions, depth to bedrock mesurements, contmintion investigtions, minerl explortion nd rcheology. In this experiment students investigte electric potentil ptterns cused by current flow in simple objects. Two suggested objects re sheet of luminum foil nd slb of grphite of unknown thickness. By using power supply cpble of delivering 1 Ampere of current, nd DC voltmeter which cn smple vrious points on surfce, vrious properties of these objects cn be mesured. Theory When direct current flows through rel conducting object, it is generlly introduced by positive nd negtive electrode, which re connected to bttery or power supply. While current is stedily flowing through the object, there is non-zero electric field within the conductor. Note tht this is not possible in electrosttics. The electric field t ny point within the conductor is relted to the current density by:! E =! J σ (2) 1 We ignore higher-order or strong-field self-interction effects tht re usully unimportnt in everydy mesurements.. 2

3 where E is the electric field in m -1, J is the current density in A m 2 nd σ is the conductivity of the object in Ω -1 m -1. Or, in terms of the resistivity,! E = ρ! J (3) where ρ is the resistivity, defined s ρ=1/σ, with units of Ω m. The electric field is relted to the grdient of the electric potentil by:! E =! (4) Figure 1 shows the typicl dipole potentil pttern. Equipotentils re lines or surfces upon which the electric potentil is constnt. E! -fields follow current flow lines, which re everywhere perpendiculr to the equipotentils. 3 Figure 1: A typicl equipotentil pttern set up by DC current flowing from C 1 (positive) to C 2 (negtive). Current flow lines (dshed) follow the Electric Field vectors, which re perpendiculr to the equipotentils (solid lines). Edge Effects Current flow lines t the boundry between conductor nd n insultor must run prllel to the boundry, since current cnnot flow into the insultor. The equipotentils just inside the edges of the conductor must therefore be perpendiculr to the edge. This will hve effects on the overll ptterns of equipotentils within the conductor. Four-Wire Method Typiclly resistivity mesurements re done using some configurtion of four wires. Two deliver current: C 1 nd C 2. Two re used to mesure potentil difference between vrious points: P 1 nd

4 P 2. There re mny wys to rrnge these four wires, ech with vrious dvntges nd disdvntges in resistivity surveys. Two of the most common liner rrngements re the Wenner nd Schlumberger rrys. Ech sets P 1 nd P 2 between C 1 nd C 2 long line connecting C 1 nd C 2, s shown in Figure 2. P1 P2 Figure 2: Electrode configurtions used in resistivity surveys: A Wenner rry hs =b; Schlumberger rry hs b</4. Let the totl current flowing from C 1 to C 2 be I, the voltge mesured between P 1 nd P 2 be. How cn we use these mesurements in the bove configurtions to mke resistivity mesurements of rel or imgined objects? b 1-D: Infinite Long, Nrrow Strip (wire) P1 P2 I b Figure 3. Schlumberger rry on long, nrrow rectngulr cylinder of cross-sectionl re A, e.g. nrrow strip of Al foil. Consider the theoreticl sitution depicted in Figure 3. Since the cylinder is nrrow, like long thin wire, we cn ssume the current flows in homogeneous mnner through it, with current flow lines ll prllel to the length of the cylinder. The constnt current density between C 1 nd C 2 is simply J=I/A. The Electric field cn be estimted between P 1 nd P 2. From Eqution 3 we know tht the mgnitude of the electric field is equl to the grdient in electric potentil. Along the short line of length b between P 1 nd P 2 the grdient in electric potentil is pproximtely E=/b, where is the mesured potentil difference. We now my insert these expressions for E nd J into Eqution 3 to yield: 4 A

5 b = ρi A So our estimte for resistivity of the mteril in the cylinder is: ρ = A bi (5) (6) 2-D: Infinite Thin Conducting Sheet (foil) P1 P2 d 5 Figure 4. Schlumberger rry set up on n infinite sheet of thickness d. b Consider the theoreticl sitution depicted in Figure 4 for n infinite sheet. The current, I, spreds out wy from the positive electrode t C 1 nd flows out to infinity. Assuming the sheet hs uniform resistivity, the current should spred out evenly, creting concentric circulr equipotentils round C 1. The re tht this current psses through t ech circle is the circumference of the circle times the thickness of the sheet, d, so the current density is J=I/2πrd. Ner the points P 1 nd P 2, where the voltge is being mesured, the current density due to C 1 will be pproximtely J 1 =I 1 /2πd (ssuming tht d <<b<<). Similrly, negtive electrode t C 2 will contribute n pproximte current density J 2 =I 2 /2πd, so the totl current density will be J=J 1 +J 2 =(I 1 +I 2 )/2πd. If the sme current I flows in t C 1 nd out t C 2, then I 1 =I 2 =I nd J= I/πd. Ner the points P 1 nd P 2, where the voltge is being mesured, the current density would be pproximtely J=I/πd (ssuming tht b<<). The mgnitude of the electric field in this region is found from the grdient in potentil, E=/b. We now my insert these expressions for E nd J into Eqution 2 to yield: b = ρi (7) πd So our estimte for resistivity of the mteril in the infinite sheet is: ρ = πd (8) bi QUESTION: Does this result hold for very lrge but finite sheet? One might think it should if the sheet dimensions re much lrger thn, b, & d nd the probes re fr from the edges. But think bout this: For n infinite sheet the current flowing in t C 1 cn flow out t C 2 or out t infinity, but for finite sheet the current cn only flow out t C 2 unless the edges of the sheet re grounded. The symmetry rguments re unffected by whether the edges re grounded, but if the

6 current cn t leve t the edges, then t the edges the current flowing out from C 1 must be the sme s the current flowing into C 2, so only hlf s much current will flow for given voltge. So for lrge but finite sheet with ungrounded edges we expect ρ = 2πd (9) bi (Note: This is good point to remind you tht you should never ssume tht the person writing these mnuls re lwys correct, so you will wnt to convince yourself by your mesurements whether eqution 8 or 9, or something in between if the finite sheet is not lrge enough to ignore edge effects.) 3-D: Infinite Hlf-Universe Filled with Conductor P1 b P2 I Figure 5. Schlumberger rry set up on the surfce of n infinite hlf-universe of homogeneous conductor. Consider the theoreticl sitution depicted in Figure 5. The current, I, spreds downwrd wy from the positive electrode t C 1. Assuming the current spreds out evenly, there will be concentric hemisphericl equipotentils centered on C 1, descending into the conductor. The re tht this current psses through t distnce r from C 1 is the re of the hemisphere, so the current density is J=I/2πr 2. Following the sme rguments s for the infinite sheet, our estimte for resistivity of the mteril in very lrge but finite hlf-universe conductor will be ρ = 2π2 (10) bi Once gin, you my wnt to think bout how this result my differ between infinite nd very lrge conductors, nd the possible boundry conditions for the ltter. 6

7 Two-Lyer Models P1 P2 I! 1 d! 2 Figure 6. Wenner rry set up on two-lyer conductor. The top lyer hs resistivity ρ 1 nd depth, d. The bottom lyer hs resistivity ρ 2 nd infinite depth. Where rocks nd soils occur in more or less horizontl lyers nd the lyers hve differing electricl resistivities, resistivity surveying my be used to find the depths nd resistivities of the vrious lyers. One simple model is the two-lyer Erth, in which the top lyer hs resistivity ρ 1 nd depth d, nd the lyer just below it hs resistivity ρ 2 nd infinite depth. Numericl methods cn be used to compre observtions with such model. Theoreticl mster curves hve been computed for the sitution shown in Figure 6. Current nd potentil electrodes re rrnged in Wenner Arry. Eqution 10, derived for n infinite hlfuniverse, cn be used with b= to yield n eqution for pprent resistivity, ρ : ρ = 2π (11) I If the lower lyer is more resistive thn the upper lyer, more current flows in the upper lyer thn for uniform erth, thereby incresing the mesured voltge. The pprent resistivity will be greter thn ρ 1. If the lower lyer is less resistive thn the upper lyer, more current flows in the lower lyer, reducing the mesured voltge t the surfce. The pprent resistivity will be less thn ρ 1. For best results, the rry spcing must be vried over wide rnge. If is much smller thn the depth d, then the bottom lyer will hve very little effect on the mesured resistivity. Thus ρ ρ 1 when <<d. As is incresed, one cn plot ρ versus, nd more is lerned bout the depth, d, nd the resistivity of the lower lyer. See Appendix 1 for description of the curve-mtching method often used in exploring the two-lyer model; Orelln 1966 or Mooney 1956 my lso be useful references. 7

8 Sfety Reminders Tret the 1 A constnt current supply with respect: 100 ma current cn kill you nd 10 ma cn cuse pinful shock. The reson this supply it is not lethl is becuse you re protected by your skin, but you cn still be shocked if you re creless or silly, so keep reding. The internl resistnce of the humn body is typiclly few hundred ohms depending on the pth so pplying 20 internlly cn generte lethl severl hundred ma currents. The resistnce of humn skin is typiclly 10 5 Ω, so for the mximum voltge of the power supply of 20, the mximum current through your body should typiclly be less thn ~0.1 ma, unless the skin resistnce is reduced or bypssed. Do not touch the experiment if your hnds re wet! The resistnce of wet skin cn be s low s ~1000 Ω, which could llow the constnt current supply to generte pinful shocks. Never touch more thn one probe t time. Electricins sometimes work on electricl equipment with only one hnd nd put the other hnd in their pocket to prevent ny unexpected currents flowing from one hnd to the other cross their chests nd stopping their hert. Touching the current probes to your tongue will produce very pinful shock tht could cuse permnent dmge, grde of zero for the experiment, nd possible expulsion from the course. Do not stb the current probes through your skin. This could kill you nd the resulting posthumous Drwin Awrd ( would embrrss your fmily, friends, nd techers. Don t do ny other silly things with the pprtus. NOTE: This is not complete list of ll hzrds; we cnnot wrn ginst every possible dngerous stupidity, e.g. opening plugged-in electricl equipment, juggling cryostts,. Experimenters must constntly use common sense to ssess nd void risks, e.g. if you spill liquid on the floor it will become slippery, shrp edges my cut you,. If you re unsure whether something is sfe, sk the supervising professor, the lb technologist, or the lb coordintor. If n ccident or incident hppens, you must let us know. More sfety informtion is vilble t 8

9 Experiment Two current nd two voltge electrodes re provided. The power supply for the current electrodes provides constnt current with mximum voltge of 20. First mke thin (i.e. one-dimensionl) strip of luminum foil nd mesure its resistivity to confirm tht everything is working well. The two suggested objects you my then mesure re sheet of luminum foil nd lrge grphite tble. The grphite tble consists of slb of unknown thickness with ir underneth. Here re some questions you my ddress. Clculte the resistivity of the luminum foil from your sheet mesurements. Does this vlue gree with the vlue from the thin strip, nd does it gree with the literture vlue, e.g. see NDT Wht re the ptterns of equipotentils? Are they wht you would expect for dipole current source? How re the ptterns ffected ner the edges of the conductor? Are equipotentils indeed perpendiculr to the edges s expected from the theoreticl boundry conditions? Mesure the resistivity nd depth of the grphite slb, using the curve-mtching method. Compre with literture vlues for grphite conductivity, e.g. Deprez How does the electric potentil vry long rdil line ner one of the current-crrying electrodes? Is the functionl dependence of versus r wht you would expect? When the current electrodes re set up ner the edge of the conductor, does this chnge the functionl dependence of versus r ner the electrodes? Would you expect it to? How do the edges ffect the curve-mtching method, which ssumes you hve two lyers of infinite size? Cn you figure wy to mke mesurements with the edges of the sheet or slb grounded. Consult with the supervising professor before doing nything with the grphite slb, to be sure you won t dmge it. Cn you understnd the properties of conducting objects with other shpes or composition? Bibliogrphy N. Deprez nd D.S. McLchln, The nlysis of the electricl conductivity of grphite powders during compction, Journl of Physics D: Applied Physics 21 (1988) ; H.M. Mooney nd W.W. Wetzel, The Potentils About Point Electrode nd Apprent Resistivity Curves for 2, 3 nd 4-lyered Erth, University of Minnesot Press, 1956 NDT Eduction Collbortion, Conductivity nd Resistivity lues for Aluminum & Alloys, Mrch 2002; E. Orelln nd H.M. Mooney, Mster Tbles nd Curves for erticl Electricl Sounding (ES) Over Lyered Structures, Mdrid Intercieci 1966, QC809.E15 O7 (Physics Librry Reference). 9

10 Appendix 1: The curve mtching method The interprettion of Wenner sounding using curve mtching method requires logrithmic plotting for both pprent resistivity ρ nd spcing. When both the observed nd theoreticl curves re plotted on the sme type of logrithmic pper, the effect of scle is eliminted while curve shpe is mintined. Once stisfctory mtch between the observed nd theoreticl curves is obtined, prmeters like the resistivities ρ 1 nd ρ 2 nd the thickness d, of the best fitting model cn be determined. We plot the pprent resistivity (verticl xis) clculted from the observtions s 2π/I ginst the spcing (horizontl xis). The set of theoreticl mster curves for the Wenner rry is shown t Figure 7. The individul curves re described in terms of reflection coefficient κ given by ρ2 ρ1 κ = ρ2 + ρ1 (12) which hs mximum nd minimum vlues of +1 nd Figure 7. Set of theoreticl mster curves for the Wenner rry.