Elements of Kinetic Theory
|
|
- Hugh McDowell
- 5 years ago
- Views:
Transcription
1 Eements of Kinetic Theory Statistica mechanics Genera description computation of macroscopic quantities Equiibrium: Detaied Baance/ Equipartition Fuctuations Diffusion Mean free path Brownian motion Diffusion against a density gradient Drift in a fied Einstein equation Baance between diffusion and drift Einstein reation Constancy of chemica potentia Other transport phenomena Heat transport Momentum transport=viscosity Johnson noise Mosty in Kitte and Kroemer Chap. 14 Phys 11 (S006) 9 Kinetic theory 1 B. Sadouet
2 Thermodynamic quantities Pressure cf. Kitte and Kroemer Chapter 14 p. 391 If the partices have specuar refection by the wa, the momentum transfer for a partice arriving at ange θ is pcos P = Force da Integration on anges gives p d = t da = # d 1 d cos p cos v cos n p pv n( p) p dp 3 that we woud ike to compare with the energy density u = n( p)d 3 # p = 4 # n( p) p dp 0 # 0 0 ( ) p dp non reativistic pv = pressure P = u (energy density) 3 u = 3 N V # P = N # # = same pressure as thermodynamic definition = V V U,N utra reativistic pv = P = 1 3 u Phys 11 (S006) 9 Kinetic theory B. Sadouet
3 Detaied Baance Consider boxes fied with gas in communication through a sma aperture Each supposed in therma equiibrium Temperature T 1, T Concentration n 1, n What is gained by one box= what is ost by the other one Detaied baance argument At equiibrium: No net fux=> gains of box 1= osses of box 1 In particuar: Outgoing fux of partices = Incoming fux of partices ( ) = ( n, ) n 1, 1 For idea gas Outgoing energy fux = Incoming energy fux J( n 1, 1 ) = J( n, ) n n Ony way for idea gas 1, ( p)d 3 p = 1, Vn Q 1, n 1 = n 1 = ( ) ( ) exp # p 1, ) d 3 p ( h 3 => equaity of temperature and chemica potentia (and pressure) Equipartition of energy can be thought as equiibrium between 3 degrees of freedom each having Phys 11 (S006) 9 Kinetic theory 3 B. Sadouet 1 d.f. = 1
4 Fuctuations Microscopic exchanges In a system in equiibrium, exchanges sti go on at the microscopic eve They are just baanced Macroscopic quantities= sum of microscopic quantities N x mean e # U = = s p s = s s s Z But with a finite system, reative fuctuations on such sum is of the order 1/ N e.g., fuctuation on tota energy ( ) = E E = # s e E = E E Variance not entropy µ,v Phys 11 (S006) 9 Kinetic theory 4 B. Sadouet s # s s Z ( # e ( s ( s Z # s ) * U = E = og Z Computation with partition function; By substitution we can see that E = E E = # U # # og Z = # # # Simiary when there is exchange of partices N = ogz # #ogz N = N N µ = = # N µ µ
5 How do systems come in equiibrium? Energy transfer If gases at different temperatures are put in contact, moecues of the hotter gas have in average higher energy and transfer net energy to the ower temperature gas => temperature equiibrium. Energy transport by diffusion. Not instantaneous => therma conductivity Heat transfer equation Simiary, transfer between wa of gas encosure and gas. =>e.g., back body radiation: equiibrium between was and photons inside cavity Momentum transfer if shear between fuid voumes reated to viscosity (see ater) Partice transfer If a gas system 1 is put in contact with another gas system where the concentration of gas moecues is ower, the higher density in system 1 wi favor diffusion of moecues to system => concentration equiibrium => Diffusion equation Phys 11 (S006) 9 Kinetic theory 5 B. Sadouet
6 Scattering Mean Free Path Interaction cross section dz Mean free path Consider a beam of partices incident on a target Probabiity of interaction in a sab of thickness dz = Exampe: hard spheres Probabiity of interaction in interva dz Surviva probabiity = 1 n Partice enters medium at z=0. is the attenuation ength ( ) = N ( z) 1 dz N z dz # P( z dz) = P( z) 1 dz Probabiity of interaction between z, zdz # d n dz Cross section : dimension = area Phys 11 (S006) 9 Kinetic theory 6 B. Sadouet dz d = d ( The surviva probabiity varies as: ( ) ) dp dz = P z Prob ( z )dz = exp z # dz ) P z ( ) = exp z #
7 Diffusion: No Concentration Gradient Brownian motion Succession of scatters:consider a partice of speed v Assume isotropic scattering, no concentration gradient => Average dispacement between two scatters aong z axis z z between scatters = scos e # s ds d cos d = 0 constant concentration => does not depend on θ Average dispacement squared between two scatters aong z axis = Variance z between scatters = ( scos ) e #s ds d cos d = 3 dn Number of scatters for partices of speed v per unit time scatters dt d z => Evoution of variance with time fixed v = v Average on distribution of veocities dt 3 d z = v dt 3 = vf ( v)dv = D = d x = d y 3 f ( v)dv dt dt D = v 3 = Diffusion coefficient Phys 11 (S006) 9 Kinetic theory 7 B. Sadouet = v
8 Diffusion: Concentration Gradient Suppose that we have concentration gradient aong the z axis s to first order in s dn n 0 dz 1 = n 1 = 1 # 1 1 dn o o n o dz z ( = 1 # 1 1 dn o o n o dz scos) ( z => depends on θ => Probabiity of surviva aong direction θ is such that P surviva ( s ds) = P surviva ( s) 1 ds dp # surviva ( s) = P surviva ( s) # 1 1 dn o o n o dz scos ( ds P surviva s,cos ( ) = exp # 1 o s s 1 dn o n o dz cos ( ) ) ( * e# s o 1# s 1 dn o o n o dz cos ( ) => Probabiity of interaction between s and sds P interact = P surviva ( s) ds = ds s # e o 1 s 1 dn o o o n o dz cos # ( 1 1 dn o n o dz scos ( The mean dispacement aong the z axis between coisions (keeping ony first order in reative gradient) ds s z between coisions = # scos e o 1 s s ( 1 dn o * o o ) n o dz cos ( * d cos d ), Phys 11 (S006) 9 Kinetic theory 8 B. Sadouet
9 Diffusion: Concentration Gradient taking into account that we get s m e Each partice undergoes v/ scatters per unit time. Hence, the mean transport veocity aong the concentration gradient is Averaging on veocity distribution 0 # s o ds o = m o m z between coisions = 1 3 o ( 3 o ) 1 dn o n o dz dt d z dt = w z = v o 3 dz = o 3 v = z between coisions = o v o 3 1 n o dn o dz = D 1 dn o n o dz 1 dn o n o dz 1 dn o n o dz Phys 11 (S006) 9 Kinetic theory 9 B. Sadouet
10 Transport Diffusive transport => Partice fux J z = n o w z = D dn or more generay dz Fisk s aw: Opposite to gradient This is one exampe of transport: In addition to random veocity v there is a coherent transport (or drift) veocity w and net fuxes of partices Simiar transport of charged partices expains eectric mobiity, of energy expains heat conduction, of momentum expains viscosity. Conservation of the number of partices n Consider a voume V. The decrease of the number of partices inside voume has to be equa to the tota partice fux through the surface. n d 3 x = # t J d S V S but by the Divergence Theorem J d S = # J d 3 x S V This has to be fufied whatever the voume. d S Hence the partice conservation equation: Diffusion equation Repacing J by its vaue we get J = D n n t = D n wave equation n t # J = 0 J = D n 1 c A t = A Phys 11 (S006) 9 Kinetic theory 10 B. Sadouet
11 Drift in an Eectric Fied Charged Partices (eectrons, hoes, ions) Drift Veocity Consider an eectric fied aong the z axis. In addition to its random veocity, each dz dt E partice wi acquire a net veocity in z direction from acceeration between coisions It is advantageous therefore to work with the time δt before the next coision instead of s. If the coision time τ c =/v is constant, the probabiity of coision between δt and δt dδt is Mean dispacement between coisions: z co. = e Each partice undergoes v/ coisions per unit time => = z between coisions v = qe m => Averaging on random veocities t # c v z = qe m t z = v cos#t 1 dt d cos d ( # c v cost 1 ) v w z = qe m qe m t e t / # c # # dt # c qe m t * = qe m # c = qe mv surviva probabiity v = qe m c =acceeration x τ c Note: this stricty appies to case where coision time τ c =/v is constant. Otherwise w = qe z m 3 v 1 = qe # 3 v m ( c eff Phys 11 (S006) 9 Kinetic theory 11 B. Sadouet
12 Mobiity Drift in an Eectric Fied () w = µ E Constant τ c =/v µ = q m c Ohm s aw Consider a wire of ength L and sectiona area A. If the wire is thin enough L A E = V where V is the appied votage L I = naqw = na q m V c L V = L 1 A nq m c # conductivity # c I Phys 11 (S006) 9 Kinetic theory 1 B. Sadouet
13 Einstein Reation Constant coision time Consider the ratio for constant τ c =/v qd µ = m v c 3 = m v c c 3 # for constant c = /v D = c m = k Tµ B q = 1 / mv 3 = = k B T Genera case If the fied is ow enough for the partice to remain in therma equiibrium at temperature τ = k B T, it can be easiy shown by integrating by part the integra giving <v> that D = m# 3 v 1 3 d dv = c eff m = µ q We then sti have D = c eff m = k BT µ q Phys 11 (S006) 9 Kinetic theory 13 B. Sadouet
14 Constancy of the tota chemica potentia Baance between Eectric Potentia and Density Gradient Consider a charged partice in an eectric fied aong Oz=> For constant τ c =/v, this induces a drift veocity which wi increase the concentration aong w z Any density gradient wi induce a diffusion such that Inversey a gradient of charged partices wi induce an eectric fied which wi create a drift veocity These two contributions wi baance when these two veocities are opposite => at equiibrium w z = D 1 n integrating to have the potentia w z = µ E dn dz = µ q µ E = µ 1 q n qe 1 n we get # n 1 Remembering that the interna chemica potentia is we concude that the tota chemica potentia is constant dn dz 1 n dn dz = 0 dn dz z V(z) = Edz z o qv(z) og n( z) = constant µ int z ( ) = og n( z) # n Q qv (z) µ int ( z) = constant Phys 11 (S006) 9 Kinetic theory 14 B. Sadouet
15 Energy and Momentum Transfer Average on random directions of scatter. Energy transfer 1> E = 1 m v v = 1 m v 1 v p = m Consequence : heat conduction viscosity Momentum transfer v v v = m 1 v Phys 11 (S006) 9 Kinetic theory 15 B. Sadouet
16 Therma Conductivity Consider a medium in oca therma equiibrium but with a therma gradient aong z. Diffusion wi transport energy from hotter region to cooer regions: Consider a partice 1 which just has been scattered: its initia veocity is v 1 and ange θ,ϕ. At the next coision with partice after path s, it wi transfer in average 1 m v 1 v # if partice 1comes from region of temperature T 1 and partice comes from a region of temperature T. The mean energy transport aong z per coision is 3 < Average energy transfer z >= k T 1 # T # B( ) s scos e ds with T 1 T = T T #z = z z scos < Average energy transfer #z >= 3 k T B ( z scos ) s e ds d cos v Taking into account the tota number of coisions per unit time we obtain the energy fux aong z (averaged over v) is J Qz = 3 nk T B z therma conductance = 3 nk B v 3 = 3 k T 1 T B # or J Q = # T v 3 = C v 3 = CD d cos d( ) = 3 k B where C is the heat capacity per unit voume Phys 11 (S006) 9 Kinetic theory 16 B. Sadouet d T z 3
17 Therma Conductivity () Heat equation CT = u where u is the energy density The oca increase of temperature with time is By same argument of energy conservation => T t = 1 C # T = D# T T t = 1 u C t u t # J Q = 0 This is the diffusion equation again Phys 11 (S006) 9 Kinetic theory 17 B. Sadouet
18 Reation to Brownian motion Exampe of current fuctuation across a resistor Let us consider a charge moving between pates whose votages differ by V. L E Eectron transfer If the charge is moving randomy i = q L v x and i = 0 i = q # L v x at a given time t Power spectrum However for cacuation of noise through a circuit which has some frequency dependence, we need to compute the noise as a function of the frequency. The conservation of energy impies qedx = Vdq or dq dt = q E dx V dt = q L v x Note: v is the veocity the frequency < i ( ) > dv Caed the power spectrum or the spectra density of the noise. Phys 11 (S006) 9 Kinetic theory 18 B. Sadouet
19 Johnson Noise area A As a mode of a resistor we consider the same system as ast side with N eectrons N = nla L q E One eectron moving a ength s aong the direction produces a square puse of ength t = s v and ampitude i = qv L cos Eectron transfer Fourier transform f ( t) For each fight path between interactions sin (#t) f ( v) = e i#t f ( t)dt = e i# (t t /) i e i# (t t /) ti for sma # # The moduus of f v ( ) at ow frequency is ti and its phase is random where the factor comes from the fact for the spectra density we combine positive and negative frequencies Fourier transform i ( ) int = 0 ( ) = f v int ( ) int i = t i int Phys 11 (S006) 9 Kinetic theory 19 B. Sadouet t i tδτ
Elements of Kinetic Theory
Eements of Kinetic Theory Diffusion Mean free path rownian motion Diffusion against a density gradient Drift in a fied Einstein equation aance between diffusion and drift Einstein reation Constancy of
More informationElements of Kinetic Theory
Eements of Kinetic Theory Statistica mechanics Genera description computation of macroscopic quantities Equiibrium: Detaied Baance/ Equipartition Fuctuations Diffusion Mean free path Brownian motion Diffusion
More informationPhysics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions
Physics 27c: Statistica Mechanics Fermi Liquid Theory: Coective Modes Botzmann Equation The quasipartice energy incuding interactions ε p,σ = ε p + f(p, p ; σ, σ )δn p,σ, () p,σ with ε p ε F + v F (p p
More informationProblem Set 6: Solutions
University of Aabama Department of Physics and Astronomy PH 102 / LeCair Summer II 2010 Probem Set 6: Soutions 1. A conducting rectanguar oop of mass M, resistance R, and dimensions w by fas from rest
More informationGauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law
Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s
More informationCandidate Number. General Certificate of Education Advanced Level Examination January 2012
entre Number andidate Number Surname Other Names andidate Signature Genera ertificate of Education dvanced Leve Examination January 212 Physics PHY4/1 Unit 4 Fieds and Further Mechanics Section Tuesday
More informationElectromagnetism Spring 2018, NYU
Eectromagnetism Spring 08, NYU March 6, 08 Time-dependent fieds We now consider the two phenomena missing from the static fied case: Faraday s Law of induction and Maxwe s dispacement current. Faraday
More information$, (2.1) n="# #. (2.2)
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More informationElectromagnetic Waves
Eectromagnetic Waves Dispacement Current- It is that current that comes into existence (in addition to conduction current) whenever the eectric fied and hence the eectric fux changes with time. It is equa
More informationMass Transport 2: Fluids Outline
ass Transport : Fuids Outine Diffusivity in soids, iquids, gases Fick s 1st aw in fuid systems Diffusion through a stagnant gas fim Fick s nd aw Diffusion in porous media Knudsen diffusion ass Transfer
More information6.1 Introduction to Scaling Scaling theory is a value guide to what may work and what may not work when we start to design the world of micro.
Chapter 6 Scaing Laws in Miniaturization 6. Introduction to Scaing Scaing theory is a vaue guide to what may work and what may not work when we start to design the word of micro. Three genera scae sizes:
More informationVersion 2.2 NE03 - Faraday's Law of Induction
Definition Version. Laboratory Manua Department of Physics he University of Hong Kong Aims o demonstrate various properties of Faraday s Law such as: 1. Verify the aw.. Demonstrate the ighty damped osciation
More informationPhysics 111. Thursday, Dec. 9, 3-5pm and 7-9pm. Announcements. Tuesday, December 7, Stress Strain. For the rest of the semester
ics day, ember 7, 004 Ch 17: Kinetic Theory Stress Strain Ch 18: 1st Law of Thermodynamics nd Law of Thermodynamics or the rest of the semester Thursday,. 9, 3-5pm and 7-9pm Monday,. 13, 004 10:30 am 1:30
More informationDemonstration of Ohm s Law Electromotive force (EMF), internal resistance and potential difference Power and Energy Applications of Ohm s Law
Lesson 4 Demonstration of Ohm s Law Eectromotive force (EMF), interna resistance and potentia difference Power and Energy Appications of Ohm s Law esistors in Series and Parae Ces in series and Parae Kirchhoff
More informationPhysics 235 Chapter 8. Chapter 8 Central-Force Motion
Physics 35 Chapter 8 Chapter 8 Centra-Force Motion In this Chapter we wi use the theory we have discussed in Chapter 6 and 7 and appy it to very important probems in physics, in which we study the motion
More informationMP203 Statistical and Thermal Physics. Solutions to Problem Set 3
MP03 Statistica and Therma Physics Soutions to Probem Set 3 1. Consider a cyinder containing 1 mo of pure moecuar nitrogen (N, seaed off withamovabepiston,sothevoumemayvary. Thecyinderiskeptatatmospheric
More informationMONTE CARLO SIMULATIONS
MONTE CARLO SIMULATIONS Current physics research 1) Theoretica 2) Experimenta 3) Computationa Monte Caro (MC) Method (1953) used to study 1) Discrete spin systems 2) Fuids 3) Poymers, membranes, soft matter
More informationQuantum Mechanical Models of Vibration and Rotation of Molecules Chapter 18
Quantum Mechanica Modes of Vibration and Rotation of Moecues Chapter 18 Moecuar Energy Transationa Vibrationa Rotationa Eectronic Moecuar Motions Vibrations of Moecues: Mode approximates moecues to atoms
More informationBohr s atomic model. 1 Ze 2 = mv2. n 2 Z
Bohr s atomic mode Another interesting success of the so-caed od quantum theory is expaining atomic spectra of hydrogen and hydrogen-ike atoms. The eectromagnetic radiation emitted by free atoms is concentrated
More informationPreamble. Flow and Fluid Velocity. In this section of my lectures we will be. To do this we will use as an analogy
Preambe Resistance Physics, 8 th Edition Custom Edition Cutne & Johnson Chapter 20.3 Pages 602-605 In this section of my ectures we wi be deveoping the concept of resistance. To do this we wi use as an
More informationForces of Friction. through a viscous medium, there will be a resistance to the motion. and its environment
Forces of Friction When an object is in motion on a surface or through a viscous medium, there wi be a resistance to the motion This is due to the interactions between the object and its environment This
More informationStrain Energy in Linear Elastic Solids
Strain Energ in Linear Eastic Soids CEE L. Uncertaint, Design, and Optimiation Department of Civi and Environmenta Engineering Duke Universit Henri P. Gavin Spring, 5 Consider a force, F i, appied gradua
More informationSection 6: Magnetostatics
agnetic fieds in matter Section 6: agnetostatics In the previous sections we assumed that the current density J is a known function of coordinates. In the presence of matter this is not aways true. The
More informationRelated Topics Maxwell s equations, electrical eddy field, magnetic field of coils, coil, magnetic flux, induced voltage
Magnetic induction TEP Reated Topics Maxwe s equations, eectrica eddy fied, magnetic fied of cois, coi, magnetic fux, induced votage Principe A magnetic fied of variabe frequency and varying strength is
More informationChemical Kinetics Part 2. Chapter 16
Chemica Kinetics Part 2 Chapter 16 Integrated Rate Laws The rate aw we have discussed thus far is the differentia rate aw. Let us consider the very simpe reaction: a A à products The differentia rate reates
More informationInduction and Inductance
Induction and Inductance How we generate E by B, and the passive component inductor in a circuit. 1. A review of emf and the magnetic fux. 2. Faraday s Law of Induction 3. Lentz Law 4. Inductance and inductor
More informationParallel-Axis Theorem
Parae-Axis Theorem In the previous exampes, the axis of rotation coincided with the axis of symmetry of the object For an arbitrary axis, the paraeaxis theorem often simpifies cacuations The theorem states
More information12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes
Maima and Minima 1. Introduction In this Section we anayse curves in the oca neighbourhood of a stationary point and, from this anaysis, deduce necessary conditions satisfied by oca maima and oca minima.
More informationNotes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n.
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More informationarxiv:quant-ph/ v3 6 Jan 1995
arxiv:quant-ph/9501001v3 6 Jan 1995 Critique of proposed imit to space time measurement, based on Wigner s cocks and mirrors L. Diósi and B. Lukács KFKI Research Institute for Partice and Nucear Physics
More information1.2 Partial Wave Analysis
February, 205 Lecture X.2 Partia Wave Anaysis We have described scattering in terms of an incoming pane wave, a momentum eigenet, and and outgoing spherica wave, aso with definite momentum. We now consider
More information22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 2: The Moment Equations
.615, MHD Theory of Fusion ystems Prof. Freidberg Lecture : The Moment Equations Botzmann-Maxwe Equations 1. Reca that the genera couped Botzmann-Maxwe equations can be written as f q + v + E + v B f =
More informationXI PHYSICS. M. Affan Khan LECTURER PHYSICS, AKHSS, K. https://promotephysics.wordpress.com
XI PHYSICS M. Affan Khan LECTURER PHYSICS, AKHSS, K affan_414@ive.com https://promotephysics.wordpress.com [TORQUE, ANGULAR MOMENTUM & EQUILIBRIUM] CHAPTER NO. 5 Okay here we are going to discuss Rotationa
More informationInterim Exam 1 5AIB0 Sensing, Computing, Actuating , Location AUD 11
Interim Exam 1 5AIB0 Sensing, Computing, Actuating 3-5-2015, 14.00-15.00 Location AUD 11 Name: ID: This interim exam consists of 1 question for which you can score at most 30 points. The fina grade for
More information5. Systems in contact with a thermal bath
5. Systems in contact with a thermal bath So far, isolated systems (micro-canonical methods) 5.1 Constant number of particles:kittel&kroemer Chap. 3 Boltzmann factor Partition function (canonical methods)
More informationEffect of Oxygen Injection into Argon Induction Plasmas on Chemically Non-Equilibrium Conditions
Proceedings of 17th Internationa Symposium on Pasma Chemistry, Toronto, Canada, August 7-12, 25 Effect of Oxygen Injection into Argon Induction Pasmas on Chemicay Non-Equiibrium Conditions Nobuhiko Atsuchi
More informationSEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l
Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed
More information1. Measurements and error calculus
EV 1 Measurements and error cacuus 11 Introduction The goa of this aboratory course is to introduce the notions of carrying out an experiment, acquiring and writing up the data, and finay anayzing the
More informationChemical Kinetics Part 2
Integrated Rate Laws Chemica Kinetics Part 2 The rate aw we have discussed thus far is the differentia rate aw. Let us consider the very simpe reaction: a A à products The differentia rate reates the rate
More information14-6 The Equation of Continuity
14-6 The Equation of Continuity 14-6 The Equation of Continuity Motion of rea fuids is compicated and poory understood (e.g., turbuence) We discuss motion of an idea fuid 1. Steady fow: Laminar fow, the
More informationQuantum Electrodynamical Basis for Wave. Propagation through Photonic Crystal
Adv. Studies Theor. Phys., Vo. 6, 01, no. 3, 19-133 Quantum Eectrodynamica Basis for Wave Propagation through Photonic Crysta 1 N. Chandrasekar and Har Narayan Upadhyay Schoo of Eectrica and Eectronics
More informationSeparation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More informationChapter 4: Electrostatic Fields in Matter
Chapter 4: Eectrostatic Fieds in Matter 4. Poarization 4. The Fied of a Poarized Oject 4.3 The Eectric Dispacement 4.4 Sef-Consistance of Eectric Fied and Poarization; Linear Dieectrics 4. Poarization
More informationA sta6s6cal view of entropy
A sta6s6ca view of entropy 20-4 A Sta&s&ca View of Entropy The entropy of a system can be defined in terms of the possibe distribu&ons of its moecues. For iden&ca moecues, each possibe distribu&on of moecues
More informationModule 22: Simple Harmonic Oscillation and Torque
Modue : Simpe Harmonic Osciation and Torque.1 Introduction We have aready used Newton s Second Law or Conservation of Energy to anayze systems ike the boc-spring system that osciate. We sha now use torque
More informationHYDROGEN ATOM SELECTION RULES TRANSITION RATES
DOING PHYSICS WITH MATLAB QUANTUM PHYSICS Ian Cooper Schoo of Physics, University of Sydney ian.cooper@sydney.edu.au HYDROGEN ATOM SELECTION RULES TRANSITION RATES DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS
More informationApplied Nuclear Physics (Fall 2006) Lecture 7 (10/2/06) Overview of Cross Section Calculation
22.101 Appied Nucear Physics (Fa 2006) Lecture 7 (10/2/06) Overview of Cross Section Cacuation References P. Roman, Advanced Quantum Theory (Addison-Wesey, Reading, 1965), Chap 3. A. Foderaro, The Eements
More informationPhysics 566: Quantum Optics Quantization of the Electromagnetic Field
Physics 566: Quantum Optics Quantization of the Eectromagnetic Fied Maxwe's Equations and Gauge invariance In ecture we earned how to quantize a one dimensiona scaar fied corresponding to vibrations on
More informationTerm Test AER301F. Dynamics. 5 November The value of each question is indicated in the table opposite.
U N I V E R S I T Y O F T O R O N T O Facuty of Appied Science and Engineering Term Test AER31F Dynamics 5 November 212 Student Name: Last Name First Names Student Number: Instructions: 1. Attempt a questions.
More informationSure Shot 2016 Electric Current By M K Ezaz
Sure Shot 06 Eectric Current B M K Ezaz. A 0 V batter of negigibe interna resistance is connected across a 00 V batter and a resistance of 38 Ω. Find the vaue of the current in circuit. () E 00 0 A: I
More informationOur introduction to the concept of entropy will be based on the canonical distribution,
Chapter 6 Entropy Our introduction to the concept of entropy wi be based on the canonica distribution, ρ(h) = e βh Z. (6.1) Cassicay, we can define the mean voume of phase space occupied by ρ(ē) q p =
More informationOSCILLATIONS. dt x = (1) Where = k m
OSCILLATIONS Periodic Motion. Any otion, which repeats itsef at reguar interva of tie, is caed a periodic otion. Eg: 1) Rotation of earth around sun. 2) Vibrations of a sipe penduu. 3) Rotation of eectron
More informationSelf Inductance of a Solenoid with a Permanent-Magnet Core
1 Probem Sef Inductance of a Soenoid with a Permanent-Magnet Core Kirk T. McDonad Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (March 3, 2013; updated October 19, 2018) Deduce the
More informationThe basic equation for the production of turbulent kinetic energy in clouds is. dz + g w
Turbuence in couds The basic equation for the production of turbuent kinetic energy in couds is de TKE dt = u 0 w 0 du v 0 w 0 dv + g w q 0 q 0 e The first two terms on the RHS are associated with shear
More informationELASTICITY PREVIOUS EAMCET QUESTIONS ENGINEERING
ELASTICITY PREVIOUS EAMCET QUESTIONS ENGINEERING. If the ratio of engths, radii and young s modui of stee and brass wires shown in the figure are a, b and c respectivey, the ratio between the increase
More informationCluster modelling. Collisions. Stellar Dynamics & Structure of Galaxies handout #2. Just as a self-gravitating collection of objects.
Stear Dynamics & Structure of Gaaxies handout # Custer modeing Just as a sef-gravitating coection of objects. Coisions Do we have to worry about coisions? Gobuar custers ook densest, so obtain a rough
More informationRELUCTANCE The resistance of a material to the flow of charge (current) is determined for electric circuits by the equation
INTRODUCTION Magnetism pays an integra part in amost every eectrica device used today in industry, research, or the home. Generators, motors, transformers, circuit breakers, teevisions, computers, tape
More informationTheoretical Cosmology
Theoretica Cosmoogy Ruth Durrer, Roy Maartens, Costas Skordis Geneva, Capetown, Nottingham Benasque, February 16 2011 Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque 2011 1 / 14 Theoretica
More informationSession : Electrodynamic Tethers
Session : Eectrodynaic Tethers Eectrodynaic tethers are ong, thin conductive wires depoyed in space that can be used to generate power by reoving kinetic energy fro their orbita otion, or to produce thrust
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f(q ) = exp ( βv( q )) 1, which is obtained by summing over
More informationPhysicsAndMathsTutor.com
. Two points A and B ie on a smooth horizonta tabe with AB = a. One end of a ight eastic spring, of natura ength a and moduus of easticity mg, is attached to A. The other end of the spring is attached
More informationCopyright information to be inserted by the Publishers. Unsplitting BGK-type Schemes for the Shallow. Water Equations KUN XU
Copyright information to be inserted by the Pubishers Unspitting BGK-type Schemes for the Shaow Water Equations KUN XU Mathematics Department, Hong Kong University of Science and Technoogy, Cear Water
More information(Refer Slide Time: 2:34) L C V
Microwave Integrated Circuits Professor Jayanta Mukherjee Department of Eectrica Engineering Indian Intitute of Technoogy Bombay Modue 1 Lecture No 2 Refection Coefficient, SWR, Smith Chart. Heo wecome
More informationMODELING OF A THREE-PHASE APPLICATION OF A MAGNETIC AMPLIFIER
TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES MODELING OF A THREE-PHASE APPLICATION OF A MAGNETIC AMPLIFIER L. Austrin*, G. Engdah** * Saab AB, Aerosystems, S-81 88 Linköping, Sweden, ** Roya
More information1) For a block of mass m to slide without friction up a rise of height h, the minimum initial speed of the block must be
v m 1) For a bock of mass m to side without friction up a rise of height h, the minimum initia speed of the bock must be a ) gh b ) gh d ) gh e ) gh c ) gh P h b 3 15 ft 3) A man pus a pound crate up a
More informationPhys 7654 (Basic Training in CMP- Module III)/ Physics 7636 (Solid State II) Homework 1 Solutions
Phys 7654 Basic Training in CMP- Modue III/ Physics 7636 Soid State II Homework 1 Soutions by Hitesh Changani adapted from soutions provided by Shivam Ghosh Apri 19, 011 Ex. 6.4.3 Phase sips in a wire
More information1D Heat Propagation Problems
Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2
More informationLecture 6 Povh Krane Enge Williams Properties of 2-nucleon potential
Lecture 6 Povh Krane Enge Wiiams Properties of -nuceon potentia 16.1 4.4 3.6 9.9 Meson Theory of Nucear potentia 4.5 3.11 9.10 I recommend Eisberg and Resnik notes as distributed Probems, Lecture 6 1 Consider
More informationCoupling of LWR and phase transition models at boundary
Couping of LW and phase transition modes at boundary Mauro Garaveo Dipartimento di Matematica e Appicazioni, Università di Miano Bicocca, via. Cozzi 53, 20125 Miano Itay. Benedetto Piccoi Department of
More informationCABLE SUPPORTED STRUCTURES
CABLE SUPPORTED STRUCTURES STATIC AND DYNAMIC ANALYSIS OF CABLES 3/22/2005 Prof. dr Stanko Brcic 1 Cabe Supported Structures Suspension bridges Cabe-Stayed Bridges Masts Roof structures etc 3/22/2005 Prof.
More informationParallel plate capacitor. Last time. Parallel plate capacitor. What is the potential difference? What is the capacitance? Quick Quiz "V = 1 C Q
Last time r r r V=V o Potentia an eectric fie Capacitors "V = 1 C Q Parae pate capacitor Charge Q move from right conuctor to eft conuctor Each pate has size Length With = Area = A outer Pate surfaces
More informationarxiv: v1 [hep-th] 10 Dec 2018
Casimir energy of an open string with ange-dependent boundary condition A. Jahan 1 and I. Brevik 2 1 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM, Maragha, Iran 2 Department of Energy
More informationEquations for thermal conduction
Equations for therma conduction Heat conduction and heat conductivity q - T q heat fux heat conductivity T temperature Incropera & De Witt 1990 Therma conductivities for some materias Materia / Part λ
More informationVTU-NPTEL-NMEICT Project
MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 VTU-NPTE-NMEICT Project Progress Report The Project on Deveopment of Remaining Three Quadrants to NPTE Phase-I under grant in aid
More informationIIT JEE, 2005 (MAINS) SOLUTIONS PHYSICS 1
IIT JEE, 5 (MINS) SOLUTIONS YSIS iscaimer: Tis booket contains te questions of IIT-JEE 5, Main Examination based on te memory reca of students aong wit soutions provided by te facuty of riiant Tutorias.
More informationPHYSICS LOCUS / / d dt. ( vi) mass, m moment of inertia, I. ( ix) linear momentum, p Angular momentum, l p mv l I
6 n terms of moment of inertia, equation (7.8) can be written as The vector form of the above equation is...(7.9 a)...(7.9 b) The anguar acceeration produced is aong the direction of appied externa torque.
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f( q ) = exp ( βv( q )), which is obtained by summing over
More informationLecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String
ecture Notes for Math 251: ODE and PDE. ecture 3: 1.7 Wave Equation and Vibrations of an Eastic String Shawn D. Ryan Spring 212 ast Time: We studied other Heat Equation probems with various other boundary
More informationNEW PROBLEMS. Bose Einstein condensation. Charles H. Holbrow, Editor
NEW PROBLEMS Chares H. Hobrow, Editor Cogate University, Hamiton, New York 3346 The New Probems department presents interesting, nove probems for use in undergraduate physics courses beyond the introductory
More informationProceedings of Meetings on Acoustics
Proceedings of Meetings on Acoustics Voume 9, 23 http://acousticasociety.org/ ICA 23 Montrea Montrea, Canada 2-7 June 23 Architectura Acoustics Session 4pAAa: Room Acoustics Computer Simuation II 4pAAa9.
More informationPhysics 505 Fall 2007 Homework Assignment #5 Solutions. Textbook problems: Ch. 3: 3.13, 3.17, 3.26, 3.27
Physics 55 Fa 7 Homework Assignment #5 Soutions Textook proems: Ch. 3: 3.3, 3.7, 3.6, 3.7 3.3 Sove for the potentia in Proem 3., using the appropriate Green function otained in the text, and verify that
More informationTheoretische Physik 2: Elektrodynamik (Prof. A-S. Smith) Tutorial 12
WiSe 2012 15.01.2013 Prof. Dr. A-S. Smith Dip.-Phys. Een Fischermeier Dip.-Phys. Matthias Saba am Lehrstuh für Theoretische Physik I Department für Physik Friedrich-Aexander-Universität Erangen-Nürnberg
More informationNuclear Size and Density
Nucear Size and Density How does the imited range of the nucear force affect the size and density of the nucei? Assume a Vecro ba mode, each having radius r, voume V = 4/3π r 3. Then the voume of the entire
More informationESCI 340 Physical Meteorology Radiation Lesson 5 Terrestrial Radiation and Radiation Balance Dr. DeCaria
ECI 30 Physica Meteoroogy Radiation Lesson 5 errestria Radiation and Radiation Baance Dr. DeCaria References: Atmospheric cience: An Introductory urvey, Waace and Hobbs An Introduction to Atmospheric Radiation,
More informationA sta6s6cal view of entropy
A sta6s6ca view of entropy 20-4 A Sta&s&ca View of Entropy The entropy of a system can be defined in terms of the possibe distribu&ons of its moecues. For iden&ca moecues, each possibe distribu&on of moecues
More information11 - KINETIC THEORY OF GASES Page 1. The constituent particles of the matter like atoms, molecules or ions are in continuous motion.
- KIETIC THEORY OF GASES Page Introduction The constituent partices of the atter ike atos, oecues or ions are in continuous otion. In soids, the partices are very cose and osciate about their ean positions.
More informationJackson 4.10 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 4.10 Homework Probem Soution Dr. Christopher S. Baird University of Massachusetts Lowe PROBLEM: Two concentric conducting spheres of inner and outer radii a and b, respectivey, carry charges ±.
More informationDislocations in the Spacetime Continuum: Framework for Quantum Physics
Issue 4 October PROGRESS IN PHYSICS Voume 11 15 Disocations in the Spacetime Continuum: Framework for Quantum Physics Pierre A. Miette PierreAMiette@aumni.uottawa.ca, Ottawa, Canada This paper provides
More informationc=lu Name Some Characteristics of Light So What Is Light? Overview
Chp 6: Atomic Structure 1. Eectromagnetic Radiation 2. Light Energy 3. Line Spectra & the Bohr Mode 4. Eectron & Wave-Partice Duaity 5. Quantum Chemistry & Wave Mechanics 6. Atomic Orbitas Overview Chemica
More information4. Systems in contact with a thermal bath
4. Systems in contact with a thermal bath So far, isolated systems microcanonical methods 4.1 Constant number of particles:kittelkroemer Chap. 3 Boltzmann factor Partition function canonical methods Ideal
More informationChapter 32 Inductance
Chapter 3 nductance 3. Sef-nduction and nductance Sef-nductance Φ BA na --> Φ The unit of the inductance is henry (H). Wb T H A A When the current in the circuit is changing, the agnetic fux is aso changing.
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationUniversity of California, Berkeley Physics 7A Spring 2009 (Yury Kolomensky) SOLUTIONS TO PRACTICE PROBLEMS FOR THE FINAL EXAM
1 University of Caifornia, Bereey Physics 7A Spring 009 (Yury Koomensy) SOLUIONS O PRACICE PROBLEMS FOR HE FINAL EXAM Maximum score: 00 points 1. (5 points) Ice in a Gass You are riding in an eevator hoding
More informationCharge Density from X-ray Diffraction. Methodology
Charge Density from X-ray Diffraction. Methodoogy Ignasi Mata imata@icmab.es Master on Crystaography and Crystaization, 2012 Outine I. Charge density in crystas II. The mutipoar refinement III. Methodoogy
More informationANISOTROPIES OF THE MICROWAVE BACKGROUND
ANISOTROPIES OF THE MICROWAVE BACKGROUND The Universe just before recombination is a very tighty couped fuid, due to the arge eectromagnetic Thomson cross section. Photons scatter off charged partices
More informationTHE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE
THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on
More informationPHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationMore Scattering: the Partial Wave Expansion
More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction
More information17 Lecture 17: Recombination and Dark Matter Production
PYS 652: Astrophysics 88 17 Lecture 17: Recombination and Dark Matter Production New ideas pass through three periods: It can t be done. It probaby can be done, but it s not worth doing. I knew it was
More information5. Systems in contact with a thermal bath
5. Systems in contact with a thermal bath So far, isolated systems (micro-canonical methods) 5.1 Constant number of particles:kittel&kroemer Chap. 3 Boltzmann factor Partition function (canonical methods)
More information