The electrified interface.
|
|
- Ilene Shelton
- 6 years ago
- Views:
Transcription
1 Physial and Intefaial Eletohemisty 3 Exess negative Eletode Solution + + Potential elmholtz Laye s Letue 5 Eletode/solution Intefae Exess positive x a..6 nm istane L The eletified intefae. The intefae between two dissimila intefaes is eletified. Almost all sufaes ay an exess eleti hage. ene when two dissimila phases ome into ontat, hage sepaation ous in the intefaial egion whih esults in the geneation of an intefaial potential diffeene o eleti field. ow does this happen? E field geneated Bottom line : an eletial ouble laye is set up at M/S intefae. Exess positive Exess negative
2 Eletoneutality is valid in bulk solution. onside a metal eletode in ontat with an aqueous solution ontaining salt (e.g Kl (aq)). The solution ontains solvated haged ions and solvent dipoles. Foes expeiened by ions and solvent moleules in bulk of solution ae isotopi : spheial symmety opeates. Ions and wate moleules (on a time aveage) expeiene foes whih ae position and dietion independent. Thee is no net alignment of solvent dipoles, and positive and negative ions ae equally distibuted thoughout any volume element of the solution. Eletoneutality opeates in bulk solution egion vey fa fom eletode sufae. Eletoneutality beaks down in sufae egion. What about the solution egion next to the eletode sufae? In this egion foes expeiened by ions and solvent dipoles ae no longe isotopi and homogeneous. The foes ae anisotopi beause of the pesene of the eletode phase. New solvent stutue, diffeent fom that of the bulk, develops beause of the phase bounday. Eletoneutality beaks down on the solution side of the intefae. Thee will be a net oientation of solvent dipoles and a net exess hage in any volume element of the solution adaent to the eletode sufae. The solution side of the intefae beomes eletified.
3 Intefaial hage sepaation geneates high intefaial Efield. One the solution side of the intefae beomes eletified (aquies a net o exess hage), an eleti field will opeate aoss the phase bounday. Sine the metalli phase ontains haged patiles, the latte will espond to this E field. The fee eletons will move away fom o move towads the intefae depending on the dietion of the E field. Thus a net hage will be indued on the metal, whih will be equal in magnitude and opposite in sign to that on the solution side of the phase bounday. Thus hage sepaation ous aoss the M/S intefae, and this gives ise to an intefaial potential diffeene. Typially the potential diffeene is a.. V. oweve the spatial dimensions of the intefae egion ae vey small, typially nm thik. Thus the eleti field stength pesent at the M/S intefae will be typially 7 Vm whih is vey lage. Eletified intefae. Fomal definition. Tem eletified intefae used to desibe the aangement of hages and oiented solvent dipoles at the intefae between an eletode and an eletolyte solution. We now pesent some simple appoximate models to desibe the popeties of the eletified intefae. We examine thee simple models : elmholtz ompat laye model Gouyhapman diffuse laye model Sten model. A key idea whih we develop involves the epesentation of intefaial stutue in tems of an eletial equivalent iuit element (speifially a single apaito o seies of apaitos). We initially adopt a simple pitoial desiption. This will be followed by a moe quantitative mathematial analysis. 3
4 Eletode Exess positive L Exess negative Solution Eleti field pesent at intefae x Simple epesentation of eletode/solution intephase egion : elmholtz ompat laye model. Eletode Exess positive Exess negative Solution Stutue of thin double laye egion modelled as a paallel plate apaito with a plate sepaation of moleula dimension. L Potential x a..6 nm elmholtz Laye s istane L 6 F m Numeial alulations Using the elmholtz Model. We need the following Relationships fom basi Physis. Sufae on metal (m ) x istane between plates of apaito elmholtz apaitane (Fm ) Intefaial Potential diffeene (V) = pemittivity of vauum = x Fm = dieleti onstant of solution. ( bulk) 78 ( elmholtz) 5 6 E x Eleti field Stength (Vm ) Exess negative Eletode Exess positive Solution L Potential x a..6 nm elmholtz Laye s istane A fundamental poblem is assigning a value fo the dieleti onstant of the solvent in the thin elmholtz egion. Solvent stutue in this egion diffes onsideably fom that of the bulk solution. ave onsideable dieleti satuation effets and so dieleti onstant will be muh lowe than that assoiated with the bulk solution. The dieleti onstant may also vay apidly with distane in intefae egion. 4
5 Potential istibution in elmholtz ompat Laye. Potential distibution obtained using the PoissonBoltzmann equation whih elates hage density and eletostati potential. d dx Ions teated as point hages. ene an assume that exess between eletode sufae and OP is zeo, hene =. x M S M x x (x) Radius of solvated ion = x M x d dx d dx x Linea potential pofile in ompat laye. S x x x M x M M S x S Simple models ae not always good ones. The simple elmholtz pitue is not omplete sine it pedits: The double laye apaitane is a onstant independent of ion onentation and eletode potential Wheeas: Expeiment indiates that the double laye apaitane vaies with both of these quantities in a definite manne. A moe elaboate model is equied. 5
6 The diffuse double laye. ow does a 3 distibution of hage aise? We have negleted the disodeing effet of the themal motion of the ions in the solution. This opposes the odeing tendeny due to opeation of eletostati foes in the intefae egion. Themal and eletostati foes esults in an equilibium. The exess s S ountebalaning the exess hage density s M on the metal, is at a maximum lose to the eletode sufae. It diminishes in an appoximate exponential manne with ineasing distane fom the eletode sufae, giving ise to a diffuse spae hage laye adaent to the eletode. etailed analysis indiates that the thikness of the diffuse laye egion will depend both on the potential applied t the eletode, and on the onentation of ions pesent in the eletolyte solution. Gouyhapman model of diffuse double laye. Eletode Exess positive Exess negative hage density Solution OP L a. nm In the Gouyhapman model of the intefae egion, it is assumed that the exess hage density on the solution side of the intefae an be epesented in tems of a thee dimensional spae hage. s x iffuse laye thikness 6
7 At potential of Zeo hage:, L Gouyhapman model of diffuse double laye. Eletode Exess positive M Exess negative hage density Solution 4ze L sinh hage density in diffuse laye d ze L osh L kbt osh osh L, osh Valid when, is L small. L a. nm osh s x L = ebye Length. Measues diffuse laye thikness. iffuse laye thikness. The diffuse laye thikness is alled the ebye Length and is given the symbol L. In many books this is denoted as /. Fo a z,z eletolyte the ebye length is given by the expession aoss. / kbt RT Evaluation of the onstants gives a L useful expession fo omputation. z e z F / L 6.3 z T e8 L / m mol m 3 o mm Note that the ebye Length ineases as the ioni onentation deeases. The diffuse laye thikness will be geatest fo the most dilute solutions. L /m e9 e /mol m 3 (,) eletolyte, wate = 78, T = 98K 7
8 Sten model of the intefae egion. Neithe the elmholtz ompat laye model no the Gouy hapman diffuse laye model is totally satisfatoy. In the G model the solvated ions ae modelled as point hages. This neglet of ion size is unealisti. In eality the solvated ion an only appoah the eletode sufae to a distane equal to its solvated adius a. ene a moe logial appoah is to ombine the featues of the elmholtz and Gouy hapman models. This was done by Sten. The Sten model is as follows. Next to the eletode we have a egion of high eleti field and low dieleti onstant ( value a. 6) with a ow of fimly held ounte ions. Beyond that thee is an ioni atmosphee (the diffuse laye) whee thee is a balane between the odeing eletostati foe and disodeing themal motions. The dieleti onstant ineases apidly with distane in this egion. The eletial potential vaies linealy with distane (a. hydated ion adius) within the inne ompat laye and deeases in an appoximate exponential manne with distane within the diffuse laye, deaying to zeo in the bulk solution. Sten model of solid/solution intephase egion. L L Seies aangement of apaitos. The smalle of the two apaitanes will detemine the oveall apaitane. If and ae of vey diffeent size then the tem ontaining the lage one may be negleted. The diffuse laye apaitane will pedominate when the solution onentation is low. 8
9 A easonable model of the eletode/solution intefae. Linea potential Vaiation with distane The eletode/solution intefae is modelled as a seies aangement of two apaitos. This is an equivalent iuit epesentation of the intefae. Exponential Vaiation of Potential with distane x Fom basi physis : L ompat laye L Seies aangement of apaitos. Total apaitane L iffuse laye The PoissonBoltzmann equation (I). Fom lassial eletostatis we use the Poisson equation whih elates the dieleti displaement veto and the loal volume density of hage (the numbe of hages pe unit volume). E ivegene opeato E pemittivity pemittivity of vauum eleti field veto dieleti onstant 8.854x Eleti field veto E an be elated to the eletostati potential using basi physis. E This is the fom of the Poisson Gadient opeato equation whih elates the and the eletostati potential. Fm 9
10 The PoissonBoltzmann equation (II). We now need to evaluate the. The volume density of hage is obtained by adding togethe the podut of the hage q and onentation of eah ioni speies in the solution next to the eletode sufae. Ion valene q z e fundamental hage We use the Boltzmann equation of statistial mehanis to obtain a elationship between the loal ounteion onentation and the bulk onentation. To do this we need to evaluate the wok w done in binging the ion fom a efeene point at infinity, up to a point distane fom the eletode sufae. We assume that this wok is puely eletial in natue. w ) q z e( ) ( w exp kbt z e( ) exp kbt The PoissonBoltzmann equation (III). We ae now in a position to wite down the PB equation. This is a faily ompliated equation to solve fom fist piniples. The exat fom of the diffeential equation depends on the geomety. We shall assume a z,z eletolyte suh as Kl o Nal. The geomety detemines the fom that the opeato takes. A plana geomety is used fo maoeletodes, wheeas a spheial geomety is adopted fo ultamioeletodes. d x Plana geomety dx d dx d d d d Spheial geomety d d ze d ze d ze sinh kbt ( ) z L z z e z z e ( ) z e exp kbt z,z valent eletolyte ze ze ze exp exp kbt kbt ze ze sinh kbt ze ze exp exp kbt kbt kbt ze ebye Length, z,z eletolyte The PB equation is solved fo.
11 The Poisson Boltzmann equation (IV). The PB equation fully desibes the petinent eletial popeties of the diffuse laye. oweve it an only be solved analytially fo a few speial situations. Fo the most geneal ases a numeial solution has to be adopted. The PB equation fo flat plana sufaes an be igoously solved fo z,z eletolytes. This annot be done igoously fo spheial sufaes. Appoximate solutions of vaying degees of auay have been podued. A easonable appoah valid both fo plana and spheial intefaes involves the ebyeukel appoximation, whih esults in the tansfomation of the PB equation into a linea fom as indiated aoss. This appoximation will be valid povided that the potential at the sufae of the eletode is not too lage. Appoximate fom appoximation Of potential distibution Plana geomety d dx ze k T B x x exp exp L Poisson/Boltzmann equation : plana sufae. d sinh d Nomalised potential ze kbt ze k T B Vaiation of eletostati potential with distane in the diffuse laye egion. The potential is effetively exponentially deaying with distane fom solid sufae. Neglet ompat laye x x L Nomalised distane exp Small sufae potential ebyeukel appoximation Lage sufae potential tanh exp tanh tanh Full solution of PB equation Thikness of iffuse laye exp
12 hage density in diffuse laye egion. x dx d If the potential applied to the metal sufae is small we an assume a linea appoximation and Fo a z,z eletolyte the loal hage density is given by ze exp ze sinh exp ze sinh d ze sinh d sinh 4ze 4ze ebye Length d sinh d d d sinh osh sinh sinh z e k T 4ze 4ze ze kbt Note that the sign of is opposite to that of. ene a positive implies that negative ions ae attated to intefae and vie vesa. B iffuse Laye apaitane We define the diffeential apaitane of the diffuse laye as L d d d M d BT d 4ze ze k hage density within diffuse laye is d sinh We eall that ebye Length (diffuse laye thikness is kbt ze z e k T B z e k T z e k T z e k T ze osh kbt osh B B B d sinh d osh ze osh kbt depends on potential. It is not onstant.
13 Vaiation of diffuse laye apaitane with potential. ow good is the diffuse laye Theoy in patie? iffuse laye model also applies fo olloidal patile/solution intefae. ouble laye modelling still being Pefomed at eseah level to vaious degees of sophistiation. 3
14 Eletoneutality beakdown in diffuse laye egion : Plana sufae. ounte ion exess tanh oth f f f lntanh f ln tanh ounteion onentation ineases lose to haged solid sufae and oion onentation deeases lose to haged sufae. oion depletion Typial vaiation of L with applied potential. g/aqueous Kl intefae. apaitane maximum Moden models inopoating speifi adsoption of ions in the inne ompat laye, allied with a model fo the wate stutue in the inne laye explain the apaitane maximum onstant apaity Region. Explained by elmholtz model apaitane minimum Explained by Gouyhapman model 4
15 The BM tiple laye Model of the metal/solution Intefae. One uently aepted model is the BM model. It epesents the situation on the solution side of the intefae in tems of a tiple laye. an inne helmholtz plane (IP) is intodued and is egaded as the lous of the eletial entes of speifially adsobed ions. These anions ae stongly adsobed onto the eletode sufae and ae patially desolvated. This fat auses a evesal of eletostati potential in the egion between the IP and the OP. x A pimay and seonday wate laye is intodued with diffeing dieleti popeties. x The pimay wate laye ( = 5) is loated immediately adaent to the metal eletode sufae. The seonday wate laye (e = 36) is loated as a hydation sphee aound a solvated ation and anion. 5
16 Polaizable and nonpolaizable intefaes. L Intefaial stutue Eletode/solution intefae No leakage of hage aoss M/S intefae Ideally Polaizable Intefae : R T Eletial equivalent iuit R T Ideally nonpolaizable Intefae: R T hage tansfe ous aoss M/S intefae Measues ET Aoss intefae Simple equivalent iuit epesentation of eletode/solution intefae egion. L haging uent i i L R S Resistane of solution Faadai uent i F i i i F R T Solution Eletode Evaluation of L (and hene i ) always neessay when making kineti measuements at shot timesales. ouble laye haging uent always pesent in addition to Faadai uent in eletohemial measuements. 6
17 Expeimental inteogation of eletode/solution intefaes. onventional eletohemial tehniques. Based on measuement of uent, potential and hage. V, RV, RRV, PSA, IS et. Applied both to maosized and mioeletodes. Theoy, instumentation, and patie well developed. No diet infomation on miosopi stutue of eletode/solution intefae. Spetosopi tehniques. Povides useful hemial infomation anout speies at intefaes. FTIR, Raman, UV/VIS, XPS, EXAFS, Ellipsomety, E/NMR (new tehnique, vey speialised, limited appliation at pesent). Sanning pobe miosopy. igh esolution topogaphial imaging of eletode sufaes with atomi esolution. Sufae eativity also pobed with high spatial esolution. STM, AFM, SEM. Refe to: P.A. histensen, A. amnett, Tehniques and Mehanisms in Eletohemisty, hapman and all, UK, 994 fo details onening Spetosopi tehniques. 7
Electrochemistry 2013
Physial and Interfaial Eletrohemistry 13 Leture 5 Eletrode Exess negative harge density + + + Solution Potential Helmholtz Layer Eletrode/solution Interfae + + s Exess positive harge density x H a..6 nm
More informationmatschek (ccm2548) Ch17-h3 chiu (57890) 1
matshek m2548) Ch17-h3 hiu 5789) 1 This pint-out should have 16 questions. Multiple-hoie questions may ontinue on the next olumn o page find all hoies efoe answeing. 1 1. points A student said, The eleti
More informationExperiment 1 Electric field and electric potential
Expeiment 1 Eleti field and eleti potential Pupose Map eleti equipotential lines and eleti field lines fo two-dimensional hage onfiguations. Equipment Thee sheets of ondutive papes with ondutive-ink eletodes,
More informationExtra Examples for Chapter 1
Exta Examples fo Chapte 1 Example 1: Conenti ylinde visomete is a devie used to measue the visosity of liquids. A liquid of unknown visosity is filling the small gap between two onenti ylindes, one is
More information8.022 (E&M) Lecture 13. What we learned about magnetism so far
8.0 (E&M) Letue 13 Topis: B s ole in Mawell s equations Veto potential Biot-Savat law and its appliations What we leaned about magnetism so fa Magneti Field B Epeiments: uents in s geneate foes on hages
More informationAVS fiziks. Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
ELECTROMAGNETIC THEORY SOLUTIONS GATE- Q. An insulating sphee of adius a aies a hage density a os ; a. The leading ode tem fo the eleti field at a distane d, fa away fom the hage distibution, is popotional
More informationPhysics 218, Spring March 2004
Today in Physis 8: eleti dipole adiation II The fa field Veto potential fo an osillating eleti dipole Radiated fields and intensity fo an osillating eleti dipole Total satteing oss setion of a dieleti
More information1 Fundamental Solutions to the Wave Equation
1 Fundamental Solutions to the Wave Equation Physial insight in the sound geneation mehanism an be gained by onsideing simple analytial solutions to the wave equation. One example is to onside aousti adiation
More informationMass Transfer (Stoffaustausch)
Mass Tansfe (Stoffaustaush) Examination 3. August 3 Name: Legi-N.: Edition Diffusion by E. L. Cussle: none nd 3 d Test Duation: minutes The following mateials ae not pemitted at you table and have to be
More informationReflectance spectra for Si
Refletane speta fo Si Notie R and ε i and ε show onsideable stutues in the fom of peas and shouldes. These stutues aise fom the optial tansitions between alene bands to the ondution bands. 16 Miosopi Theoy:
More informationPHYS 110B - HW #7 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS B - HW #7 Fall 5, Solutions by David Pae Equations efeened as Eq. # ae fom Giffiths Poblem statements ae paaphased [.] Poblem.4 fom Giffiths Show that Eq..4, V, t an be witten as Eq..44, V, t q t
More informationnot to be republished NCERT ELECTROMAGNETIC WAVES Chapter Eight MCQ I
Chapte Eight ELECTROMAGNETIC WAVES MCQ I 8 One equies ev of enegy to dissoiate a abon monoxide moleule into abon and oxygen atoms The minimum fequeny of the appopiate eletomagneti adiation to ahieve the
More informationLaplace Potential Distribution and Earnshaw s Theorem
Laplae Potential Distibution and Eanshaw s Theoem Fits F.M. de Mul Laplae and Eanshaw Pesentations: Eletomagnetism: Histoy Eletomagnetism: Elet. topis Eletomagnetism: Magn. topis Eletomagnetism: Waves
More informationCOMPARING MORE THAN TWO POPULATION MEANS: AN ANALYSIS OF VARIANCE
COMPARING MORE THAN TWO POPULATION MEANS: AN ANALYSIS OF VARIANCE To see how the piniple behind the analysis of vaiane method woks, let us onside the following simple expeiment. The means ( 1 and ) of
More informationMass Transfer. Dispersion. Lecture 13, , Dr. K. Wegner
Mass Tansfe Dispesion Letue 3, 3..7, D. K. Wegne Dispesion to dispese to spead widely. Dispesion is the at o poess of dispesing, of distibuting one substane (small volume fation) in anothe (lage volume
More informationMolecular Energy Changes During a Reaction
Reation Kinetis Moleula Enegy Changes Duing a Reation Chemial Enegy of Speies E xn E* +BP E* P+B Moleules above this enegy level (defined somewhat abitaily) ae alled ativated omplexes Poduts Reatants Pogession
More information1 Fundamental Solutions to the Wave Equation
1 Fundamental Solutions to the Wave Equation Physial insight in the sound geneation mehanism an be gained by onsideing simple analytial solutions to the wave equation One example is to onside aousti adiation
More informationAnswers to Coursebook questions Chapter 2.11
Answes to Couseook questions Chapte 11 1 he net foe on the satellite is F = G Mm and this plays the ole of the entipetal foe on the satellite, ie mv mv Equating the two gives π Fo iula motion we have that
More informationPROPAGATION OF PHOTON IN RESTING AND MOVING MEDIUM. J. Zaleśny. Institute of Physics, Technical University of Szczecin, A b s t r a c t
PROPGTION OF PHOTON IN RESTING ND MOVING MEDIUM J Zaleśny Institute of Physis, Tehnial Univesity of Szzein, l 30 70 Piastów 48, Szzein, Poland b s t a t The popagation of photon in a dieleti may be desibed
More informationDissolution of Solid Particles in Liquids: A Shrinking Core Model
Wold Aademy of Siene, Engineeing and Tehnology 5 9 Dissolution of Solid Patiles in Liquids: A Shining oe Model Wei-Lun Hsu, Mon-Jyh Lin, and Jyh-Ping Hsu Astat The dissolution of spheial patiles in liquids
More informationRevised Newtonian Formula of Gravity and Equation of Cosmology in Flat Space-Time Transformed from Schwarzschild Solution
Intenational Jounal of Astonomy and Astophysis,,, 6-8 http://dx.doi.og/.46/ijaa.. Published Online Mah (http://www.sip.og/jounal/ijaa) evised Newtonian Fomula of Gavity and Equation of Cosmology in Flat
More informationRed Shift and Blue Shift: A realistic approach
Red Shift and Blue Shift: A ealisti appoah Benhad Rothenstein Politehnia Uniesity of Timisoaa, Physis Dept., Timisoaa, Romania E-mail: benhad_othenstein@yahoo.om Coina Nafonita Politehnia Uniesity of Timisoaa,
More informationElectromagnetism. Christopher R Prior. ASTeC Intense Beams Group Rutherford Appleton Laboratory
letomagnetism Chistophe R Pio Fellow and Tuto in Mathematis Tinity College Oxfod ASTeC Intense Beams Goup Ruthefod Appleton Laboatoy Contents Maxwell s equations and Loentz Foe Law Motion of a haged patile
More informationClassical Approach to the Theory of Elementary Particles
Classial Appoah to the Theoy of Elementay Patiles By Yui N. Keilman Abstat: Pesented hee is an attempt to modify /extend lassial eletodynamis (CED) in ode to enable the lassial appoah (the appoah based
More informationOBSTACLE DETECTION USING RING BEAM SYSTEM
OBSTACLE DETECTION USING RING BEAM SYSTEM M. Hiaki, K. Takamasu and S. Ozono Depatment of Peision Engineeing, The Univesity of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan Abstat: In this pape, we popose
More informationPhotographing a time interval
Potogaping a time inteval Benad Rotenstein and Ioan Damian Politennia Univesity of imisoaa Depatment of Pysis imisoaa Romania benad_otenstein@yaoo.om ijdamian@yaoo.om Abstat A metod of measuing time intevals
More informationOn the indirect e ect in the Stokes±Helmert method of geoid determination
Jounal of Geodesy (1999) 7: 87±9 On the indiet e et in the Stokes±Helmet method of geoid detemination L. E. SjoÈ beg, H. Nahavandhi oyal Institute of Tehnology, Depatment of Geodesy and Photogammety, S-100
More information2. Electrostatics. 2.1 What is electrostatics and electrostatic induction? 2.2 Explain coulomb s law of electrostatics
. Eletostatis. What is eletostatis an eletostati inution? Eletostatis is a banh of siene ealing with eletiity at est i.e. stati eletiity. Nomally evey atom has an ual numbe of potons (ve hage) an eletons
More informationAn analytic calculation method on air gap flux in permanent magnet. brushless DC motor with ironless rotor
Intenational Confeene on Enegy and Envionmental Potetion ICEEP 6 An analyti alulation method on ai gap flux in pemanent magnet bushless DC moto with ionless oto Xinghua Wang,Yaolong Sheng andshugang Zhao,,
More informationdp p v= = ON SHOCK WAVES AT LARGE DISTANCES FROM THE PLACE OF THEIR ORIGIN By Lev D. Landau J. Phys. U.S.S.R. 9, 496 (1945).
ON SHOCK WAVES AT LARGE DISTANCES FROM THE PLACE OF THEIR ORIGIN By Lev D. Landau J. Phys. U.S.S.R. 9, 496 (1945). It is shown that at lage distanes fom the body, moving with a. veloity exeeding that of
More informationElectric Anisotropy, Magnetic Anisotropy, Uniaxial and Biaxial Materials, Bianisotropic Media (Definitions)
leti nisotop agneti nisotop Uniaial and iaial ateials ianisotopi edia efinitions medium is alled eletiall anisotopi if tenso Note that and ae no longe paallel medium is magnetiall anisotopi if tenso Note
More informationKhmelnik S.I. Mathematical Model of Dust Whirl
Khmelnik S.I. Mathematial Model of Dust Whil Abstat The question of the soue of enegy in a dust whil is onsideed. Atmosphei onditions annot be the sole soue of enegy, as suh dust whils exist on Mas, whee
More informationspherical dp dr The gravitational force acting inwards on the same element of gas is,
Rik s Cosmology Tutoial: Chapte Stella Stutue Pat Chapte Stella Stutue Pat : Pessue, Density and Tempeatue Distibutions in Spheially Symmeti, Main Sequene Stas The Clayton Model Last Update: July 006.
More informationInvestigation of Magnitude and Phase Errors in Waveguide Samples for the Nicolson-Ross-Weir Permittivity Technique
Univesity of New Hampshie Univesity of New Hampshie Sholas' Repositoy Maste's Theses and Capstones Student Sholaship Winte 016 Investigation of Magnitude and Phase Eos in Waveguide Samples fo the Niolson-Ross-Wei
More informationIn electrostatics, the electric field E and its sources (charges) are related by Gauss s law: Surface
Ampee s law n eletostatis, the eleti field E and its soues (hages) ae elated by Gauss s law: EdA i 4πQenl Sufae Why useful? When symmety applies, E an be easily omputed Similaly, in magnetism the magneti
More informationFrom E.G. Haug Escape Velocity To the Golden Ratio at the Black Hole. Branko Zivlak, Novi Sad, May 2018
Fom E.G. Haug Esape eloity To the Golden Ratio at the Blak Hole Banko Zivlak, bzivlak@gmail.om Novi Sad, May 018 Abstat Esape veloity fom the E.G. Haug has been heked. It is ompaed with obital veloity
More informationE(r,t) = e 3. r 3. (b) Show that the transverse current, J t,is 3n(n e 3 ) e 3
Polem Set 3 (Jakson 6.20).. An example of the pesevation of ausality and finite speed of popagation in spite of the use of the Coulomg gauge is affoded y a unit stength dipole soue that is flashed on and
More informationIon-sound waves (electrostatic low frequency waves)
OTHER TYPES of WAVES Ion-sound waves (eletostati low fequeny waves) ae longitudinal waves simila lassial sound in gas s kt k M B plasma sound is slow fo eletons, but fast fo ions Eleton density is in eah
More informationGeneralized Vapor Pressure Prediction Consistent with Cubic Equations of State
Genealized Vapo Pessue Pedition Consistent with Cubi Equations of State Laua L. Petasky and Mihael J. Misovih, Hope College, Holland, MI Intodution Equations of state may be used to alulate pue omponent
More informationCorrespondence Analysis & Related Methods
Coespondene Analysis & Related Methods Oveview of CA and basi geometi onepts espondents, all eades of a etain newspape, osstabulated aoding to thei eduation goup and level of eading of the newspape Mihael
More informationSAMPLE LABORATORY SESSION FOR JAVA MODULE B. Calculations for Sample Cross-Section 2
SAMPLE LABORATORY SESSION FOR JAVA MODULE B Calulations fo Sample Coss-Setion. Use Input. Setion Popeties The popeties of Sample Coss-Setion ae shown in Figue and ae summaized below. Figue : Popeties of
More informationTime Dilation in Gravity Wells
Time Dilation in Gavity Wells By Rihad R. Shiffman Digital Gaphis Asso. 038 Dunkik Ave. L.A., Ca. 9005 s@isi.edu This doument disusses the geneal elativisti effet of time dilation aused by a spheially
More informationNumerical solution of diffusion mass transfer model in adsorption systems. Prof. Nina Paula Gonçalves Salau, D.Sc.
Numeical solution of diffusion mass tansfe model in adsoption systems Pof., D.Sc. Agenda Mass Tansfe Mechanisms Diffusion Mass Tansfe Models Solving Diffusion Mass Tansfe Models Paamete Estimation 2 Mass
More informationSuppose you have a bank account that earns interest at rate r, and you have made an initial deposit of X 0
IOECONOMIC MODEL OF A FISHERY (ontinued) Dynami Maximum Eonomi Yield In ou deivation of maximum eonomi yield (MEY) we examined a system at equilibium and ou analysis made no distintion between pofits in
More informationPulse Neutron Neutron (PNN) tool logging for porosity Some theoretical aspects
Pulse Neuton Neuton (PNN) tool logging fo poosity Some theoetical aspects Intoduction Pehaps the most citicism of Pulse Neuton Neuon (PNN) logging methods has been chage that PNN is to sensitive to the
More informationNon-Ideal Gas Behavior P.V.T Relationships for Liquid and Solid:
hemodynamis Non-Ideal Gas Behavio.. Relationships fo Liquid and Solid: An equation of state may be solved fo any one of the thee quantities, o as a funtion of the othe two. If is onsideed a funtion of
More informationIn the previous section we considered problems where the
5.4 Hydodynamically Fully Developed and Themally Developing Lamina Flow In the pevious section we consideed poblems whee the velocity and tempeatue pofile wee fully developed, so that the heat tansfe coefficient
More informationG. S. Seth, R. Sharma and S. Sarkar
Jounal of Applied Fluid Mehanis Vol. 8 No. pp. 7-0 05. Available online at www.jafmonline.net ISSN 735-357 EISSN 735-3645. Natual Convetion Heat and Mass Tansfe Flow with Hall Cuent Rotation Radiation
More information2. Equation of generalized Dynamics. Let rectangular right hand coordinate triple is fixed in three-dimensional Euclidian space.
Genealized Dynamis about Foes Ating on Chage Moving in Capaito and Solenoid. J.G. Klyushin, Ph. D. Aademy of Civil Aviation, hai of applied mathematis; e-mail: klyushin@shaping.og; mail: Intenational Club
More information(conservation of momentum)
Dynamis of Binay Collisions Assumptions fo elasti ollisions: a) Eletially neutal moleules fo whih the foe between moleules depends only on the distane between thei entes. b) No intehange between tanslational
More informationReview: Electrostatics and Magnetostatics
Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion
More informationVibrational Modes and Instabilities of a Dust Particle Pair in a Complex Plasma
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < Vibational Modes and Instabilities of a Dust Patile Pai in a Comple Plasma K. Qiao, L. S. Matthews, and T. W. Hyde,
More informationComputational chemistry
6 Computational hemisty The ental hallenge 6. The Hatee Fok fomalism 6. The Roothaan equations 6.3 Basis sets The fist appoah: semiempiial methods 6.4 The Hükel method evisited 6.5 Diffeential ovelap The
More informationDARK MATTER AND THE DYNAMICS OF GALAXIES: A NEWTONIAN APPROACH 1. INTRODUCTION
DARK MATTER AND THE DYNAMICS OF GALAXIES: A NEWTONIAN APPROACH Mugu B. RĂUŢ Coesponding autho: Mugu RĂUŢ, E-mail: m_b_aut@yahoo.om Abstat In this pape I popose a oetion to the well-known Newtonian gavitational
More informationStudy of the Endface Friction of the Revolving Vane Mechanism
Pudue Univesity Pudue e-pubs Intenational Compesso Engineeing Confeene Shool of Mehanial Engineeing 010 Study of the Endfae Fition of the Revolving Vane Mehanism Alison Subiantoo Shool of Mehanial and
More informationAN ELECTROMAGNETIC LAUNCH SYSTEM FOR UAVs
Tehnial Sienes and Applied athematis AN ELECTROAGNETIC LAUNCH SYSTE FOR UAVs Lauian GHERAN Depatment of Eletonis and Infomatis, Faulty of Aeonautial anagement, Heni Coandă Ai Foe Aademy, Basov, Romania
More information2 Governing Equations
2 Govening Equations This chapte develops the govening equations of motion fo a homogeneous isotopic elastic solid, using the linea thee-dimensional theoy of elasticity in cylindical coodinates. At fist,
More informationMath 2263 Solutions for Spring 2003 Final Exam
Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate
More informationarxiv: v4 [physics.class-ph] 14 Jul 2018
Noname manusipt No. will be inseted by the edito Long-Range Longitudinal Eleti Wave in Vauum Radiated by Eleti Dipole: Pat I Altay Zhakatayev, Leila Tlebaldiyeva axiv:7.v4 [physis.lass-ph] 4 Jul 8 Reeived:
More informationContact impedance of grounded and capacitive electrodes
Abstact Contact impedance of gounded and capacitive electodes Andeas Hödt Institut fü Geophysik und extateestische Physik, TU Baunschweig The contact impedance of electodes detemines how much cuent can
More informationPhysics 221 Lecture 41 Nonlinear Absorption and Refraction
Physics 221 Lectue 41 Nonlinea Absoption and Refaction Refeences Meye-Aendt, pp. 97-98. Boyd, Nonlinea Optics, 1.4 Yaiv, Optical Waves in Cystals, p. 22 (Table of cystal symmeties) 1. Intoductoy Remaks.
More informatione sin cos i sin sin j cos k [2 POINTS] (c) Hence, determine expressions for sin sin i sin cos j sin e
EN: Continuum Mehanis Homewok : Kinematis Due : noon Fiday Febuay 4th Shool of Engineeing Bown Univesity. To analyze the defomation of a onial membane, it is poposed to use a two-dimensional onial-pola
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Seies UG Examination 2015 16 FLUID DYNAMICS WITH ADVANCED TOPICS MTH-MD59 Time allowed: 3 Hous Attempt QUESTIONS 1 and 2, and THREE othe questions.
More informationElectrostatics (Electric Charges and Field) #2 2010
Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when
More informationDiffusion and Transport. 10. Friction and the Langevin Equation. Langevin Equation. f d. f ext. f () t f () t. Then Newton s second law is ma f f f t.
Diffusion and Tanspot 10. Fiction and the Langevin Equation Now let s elate the phenomena of ownian motion and diffusion to the concept of fiction, i.e., the esistance to movement that the paticle in the
More informationMass- and light-horizons, black holes' radii, the Schwartzschild metric and the Kerr metric
006-010 Thiey De Mees Mass- and light-hoizons, blak holes' adii, the Shwatzshild meti and the Ke meti mpoved alulus. (using gavitomagnetism) T. De Mees - thieydm@pandoa.be Abstat Blak holes geneally ae
More informationTransmission Line Analysis of Beam Deflection in a BPM Stripline Kicker
UCR-JC-126073 PREPRINT Tansmission ine Analysis of Beam Defletion in a BPM Stipline Kike Geoge J. Capoaso Yu Ju Chen Bian Poole This pape was pepaed fo submittal to the 1997 Patile Aeleato Confeene Vanouve,
More informationSpecial relativity with clock synchronization
Speial elativity with lok synhonization Benhad Rothenstein ), Stefan Popesu ) and Geoge J. Spi 3) ) Politehnia Univesity of Timisoaa, Physis Depatment, Timisoaa, Romania, benhad_othenstein@yahoo.om ) Siemens
More information17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other
Electic Potential Enegy, PE Units: Joules Electic Potential, Units: olts 17.1 Electic Potential Enegy Electic foce is a consevative foce and so we can assign an electic potential enegy (PE) to the system
More informationTHE A-TEMPORAL COSMIC SPACE AND A GENERALIZATION OF THE DIRAC EQUATION
THE -TEMPORL COMC PCE N GENERLZTON OF THE RC EQUTON avide Fisaletti aelife nstitute an Loenzo in Camo PU taly fisalettidavide@libeo.it bstat model desibing an a-temoal sae-gavity endowed with a quantum
More informationTHEORETICAL AND EXPERIMENTAL STUDY ON DROPWISE CONDENSATION IN PLATE HEAT EXCHANGERS
Abstat THEORETICAL AND EXPERIMENTAL STUDY ON DROPWISE CONDENSATION IN PLATE HEAT EXCHANGERS V. Bendt, S. Zunft and H. Mülle-Steinhagen Geman Aeospae Cente (DLR), Stuttgat, Gemany This pape desibes the
More information2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum
2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known
More informationIMPLEMENTATION OF MUR S ABSORBING BOUNDARIES WITH PERIODIC STRUCTURES TO SPEED UP THE DESIGN PROCESS USING FINITE-DIFFERENCE TIME-DOMAIN METHOD
Pogess In Eletomagnetis Reseah, PIER 58, 101 114, 006 IMPLEMENTATION OF MUR S ABSORBING BOUNDARIES WITH PERIODIC STRUCTURES TO SPEED UP THE DESIGN PROCESS USING FINITE-DIFFERENCE TIME-DOMAIN METHOD G.
More informationPHYS 301 HOMEWORK #10 (Optional HW)
PHYS 301 HOMEWORK #10 (Optional HW) 1. Conside the Legende diffeential equation : 1 - x 2 y'' - 2xy' + m m + 1 y = 0 Make the substitution x = cos q and show the Legende equation tansfoms into d 2 y 2
More informationChapter 4. Sampling of Continuous-Time Signals
Chapte 4 Sampling of Continuous-Time Signals 1 Intodution Disete-time signals most ommonly ou as epesentations of sampled ontinuous-time signals. Unde easonable onstaints, a ontinuous-time signal an be
More information$ i. !((( dv vol. Physics 8.02 Quiz One Equations Fall q 1 q 2 r 2 C = 2 C! V 2 = Q 2 2C F = 4!" or. r ˆ = points from source q to observer
Physics 8.0 Quiz One Equations Fall 006 F = 1 4" o q 1 q = q q ˆ 3 4" o = E 4" o ˆ = points fom souce q to obseve 1 dq E = # ˆ 4" 0 V "## E "d A = Q inside closed suface o d A points fom inside to V =
More informationNumerical Modeling in Biomedical Systems
Numeial Modeling in Biomedial Systems BME 15:35 Letue 7 9/6/17 Nonlinea Systems Dunn Chapte 5 Nonlinea equations Root finding Baketing methods Open methods Gaphial Bisetion False Position Newton s method
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationA Theory of the Podkletnov Effect based on General Relativity: Anti-Gravity Force due to the Perturbed Non-Holonomic Background of Space
July, 007 PROGRESS IN PHYSICS Volume 3 SPECIAL REPORT A Theoy of the Podkletnov Effet based on Geneal Relativity: Anti-Gavity Foe due to the Petubed Non-Holonomi Bakgound of Spae Dmiti Rabounski and Laissa
More informationFlux. Area Vector. Flux of Electric Field. Gauss s Law
Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is
More informationPHYS 1444 Section 501 Lecture #7
PHYS 1444 Section 51 Lectue #7 Wednesday, Feb. 8, 26 Equi-potential Lines and Sufaces Electic Potential Due to Electic Dipole E detemined fom V Electostatic Potential Enegy of a System of Chages Capacitos
More informationConfidence Intervals for the Squared Multiple Semipartial Correlation Coefficient. James Algina. University of Florida. H. J.
Eet Size Conidene Inteval 1 Conidene Intevals o the Squaed Multiple Semipatial Coelation Coeiient by James Algina Univesity o Floida H. J. Keselman Univesity o Manitoba all D. Penield Univesity o Miami
More informationThe Kerr-metric, mass- and light-horizons, and black holes' radii.
006 Thiey De Mees The Ke-meti, mass- and light-hoizons, and blak holes' adii. (using the Analogue Maxwell theoy) T. De Mees - thieydm @ pandoa.be Abstat Blak holes an geneally be defined as stella objets
More informationCurrent, Resistance and
Cuent, Resistance and Electomotive Foce Chapte 25 Octobe 2, 2012 Octobe 2, 2012 Physics 208 1 Leaning Goals The meaning of electic cuent, and how chages move in a conducto. What is meant by esistivity
More informationInvestigation of the Centrifugal Force Effect to a Revolving Vane (RV) Machine
Pudue Univesity Pudue e-pubs Intenational Compesso Engineeing Confeene Shool of Mehanial Engineeing 01 Investigation of the Centifugal oe Effet to a Revolving Vane (RV) Mahine Alison Subiantoo alis0004@ntu.edu.sg
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More informationFresnel Diffraction. monchromatic light source
Fesnel Diffaction Equipment Helium-Neon lase (632.8 nm) on 2 axis tanslation stage, Concave lens (focal length 3.80 cm) mounted on slide holde, iis mounted on slide holde, m optical bench, micoscope slide
More information( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is
Mon., 3/23 Wed., 3/25 Thus., 3/26 Fi., 3/27 Mon., 3/30 Tues., 3/31 21.4-6 Using Gauss s & nto to Ampee s 21.7-9 Maxwell s, Gauss s, and Ampee s Quiz Ch 21, Lab 9 Ampee s Law (wite up) 22.1-2,10 nto to
More informationDiffraction from Crystals Shape Factor
Letue 3 Diffation fom Cystals Shape Fato - Fultz & Howe, Chap. 5 - Williams & Cate, Chap. 7 - Reime, Chap. 7 Satteed wave: Shape Fato Cystal size effet k F k ep i k Shape fato: S k ep ik z Unit ell (/w
More informationCapacitors and Capacitance
Capacitos and Capacitance Capacitos ae devices that can stoe a chage Q at some voltage V. The geate the capacitance, the moe chage that can be stoed. The equation fo capacitance, C, is vey simple: C Q
More informationConcepts of radial and angular kinetic energies
ownloaded fom obit.dtu.dk on: Ot 3, 08 Conepts of adial angula kineti enegies ahl, Jens Pede; Shleih, W.P. Published in: Physial Review A Link to atile, OI: 0.03/PhysRevA.65.009 Publiation date: 00 oument
More informationThe Radii of Baryons
Jounal Heading Yea; Vol. (No.): page ange DOI: 0.592/j.xxx.xxxxxxxx.xx The Radii of Bayons Maio Evealdo de Souza Depatmento de Físia, Univesidade Fedeal de Segipe, São Cistovão, 4900-000, Bazil Astat Consideing
More informationThe Research of AQI Index Changing Regularity Mainly in Tianjin Ziyu Guo
nd Intenational Confeene on Eduation Tehnology, Management and Humanities Siene (ETMHS 06 The Reseah of AQI Index Changing Regulaity Mainly in Tianjin Ziyu Guo Shool of Institute of Eletial and Eletoni
More informationIntroduction to Nuclear Forces
Intoduction to Nuclea Foces One of the main poblems of nuclea physics is to find out the natue of nuclea foces. Nuclea foces diffe fom all othe known types of foces. They cannot be of electical oigin since
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More information1 Similarity Analysis
ME43A/538A/538B Axisymmetic Tubulent Jet 9 Novembe 28 Similaity Analysis. Intoduction Conside the sketch of an axisymmetic, tubulent jet in Figue. Assume that measuements of the downsteam aveage axial
More informationDark matter, dark energy and gravitational proprieties of antimatter
Dak matte, dak enegy and gavitational popieties of antimatte Dagan Slavkov Hajdukovi PH Division CEN CH-111 Geneva dagan.hajdukovi@en.h Abstat We suggest that the eventual gavitational epulsion between
More informationOn the Sun s Electric-Field
On the Sun s Electic-Field D. E. Scott, Ph.D. (EE) Intoduction Most investigatos who ae sympathetic to the Electic Sun Model have come to agee that the Sun is a body that acts much like a esisto with a
More informationOn the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
More informationRight-handed screw dislocation in an isotropic solid
Dislocation Mechanics Elastic Popeties of Isolated Dislocations Ou study of dislocations to this point has focused on thei geomety and thei ole in accommodating plastic defomation though thei motion. We
More information