Classical Diamagnetism and the Satellite Paradox
|
|
- Phillip Newman
- 6 years ago
- Views:
Transcription
1 Classial Diamagnetism and the Satellite Paradox 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ (November 1, 008) In typial models of lassial diamagnetism (see, for example, se of [1] or se of []) the hange of the magneti moment of an atom is dedued when an external magneti field inreases slowly from 0 to B = B ẑ. The atom onsists of an eletron of harge e and mass m that is in a irular orbit of radius r 0 in the x-y plane about a fixed nuleus of harge e at the origin. It is taitly assumed that the radius of the orbit remains r 0 at all times, although this is not onsistent with the effet of perturbations on motion in a 1/r potential, even if radiation is ignored (as it must be in any lassial model of a stable atom). 1 Give a model of lassial diamagnetism that inludes the (small) effet of hanges in the radius of the orbit as external magneti field inreases. Compare the present ase to the so-alled satellite paradox [4, 5, 6]. The latter is that the effet of atmospheri drag on a satellite in a low orbit about the Earth is to inrease the speed of the satellite as it slowly spirals inwards towards the Earth s surfae. Solution Aspets of this problem have been disussed in [7]. We suppose that at time t = 0 an external magneti field, perpendiular to the (x-y) plane of the eletron s orbit, turns on aording to B(t) =Ḃt (t >0), (1) where Ḃ is a onstant. As a result, an external azimuthal eletri field, E ext (r, t) = rḃ ˆφ (t >0), () is generated aording to Faraday s law (in ylindrial oordinates (r, φ, z) withthe fixed nuleus at the origin, and in Gaussian units). Then, the external (Lorentz) fore on the eletron is ( F ext = e E ext + v ) B ext = erḃ ˆφ ev φb ˆr + eṙb ˆφ, (3) 1 Diamagnetism has reently been onsider in another type of lassial atom [3], onsisting of an eletron somehow onfined to the surfae of a sphere. Faraday s law atually only tells us that E φ rdφ = πr Ḃ/, sothat E φ = rḃ/ where the average is taken over the orbit. In general, the indued eletri field inludes another omponent in the x-y plane that is essentially uniform over the atom, whih has the effet of reating a small eletri dipole moment for the atom while the magneti field is hanging. We ignore this tiny effet, whih in any ase vanishes one the magneti field takes on a steady, nonzero value. 1
2 where v =ṙ ˆr + r φ ˆφ, andwewriter φ = v φ.asv φ B = v φ tḃ, the first and seond terms of eq. (3) are of similar order (and both are small ompared to the Coulomb fore ee ˆr/r ). We suppose that the hange in magneti field is so slow that the resulting radial veloity ṙ is negligible ompared to the azimuthal veloity v φ ; hene, the third term is small ompared to the first and seond terms of eq. (3). However, the third term plays a role in insuring that a magneti field does no work. Thus, during time dt the eletron moves distane and the work done on it by the external fields is ds = v dt =(ṙ ˆr + r φ ˆφ) dt (4) dw ext = F ds = e v E ext dt = er φḃ dt = erv φḃ The equation of motion for the eletron is ma = m ( r r φ ) ˆr + m ( r φ +ṙ φ ) ( ˆφ = e E tot + v ) B ext dt. (5) = ee erḃ ˆr + r ˆφ ˆr + eṙb ˆφ, (6) For slow hanges in the magneti field we neglet r ompared to r φ, so the radial equation of motion an be written as mr φ ee r + = mr3 0 φ 0 r +, (7) where r 0 and v 0 = r 0 φ0 are the radius and azimuthal veloity, respetively, of the irular orbit when B =0. The azimuthal equation of motion is m(r φ +ṙ φ) = erḃ + eṙb. (8) The terms r φ and ṙ φ are of omparable magnitude. Multiplying eq. (8) by r, we an integrate to find the (mehanial) angular momentum, Using eq. (9) for φ in eq. (7), we find L z = mr φ = mr 0 φ0 + er B. (9) r0 3 r r 0 3 r e rb. (10) 4m r 0 φ 0 Assuming a solution of the form r = r 0 (1 + Δ), we obtain r r 0 1 e B ( 4m φ = r 0 1 e r0b ). (11) 4m v 0 0
3 That the radius r of the perturbed orbit departs from r 0 by a term that is seond order in the magneti field B was mentioned in [7] without a detailed derivation. Returning to eq. (9), the angular veloity is, to seond order in B, φ φ 0 + eb m + e B m φ, (1) 0 and the azimuthal veloity is v φ v 0 + er 0B m + e r0 B, (13) m v 0 to the same order. The motion (11)-(13) does not have the harater of that in the satellite paradox. For positive v 0, the energy of the lassial atom is inreased by the perturbing magneti field. That is, W ext > 0 aording to eq. (5), while the radius of the orbit derease with time aording to eq. (11), and the azimuthal veloity inreases aording to eq. (13). This is the behavior for negative, rather than positive, Ẇ ext in the satellite paradox [4, 5]. To first order in the magneti field B, eq. (13) is the same as obtained by supposing that the azimuthal fore ee φ diretly affets the azimuthal veloity (i.e., thateq.(8)anbe approximated as v φ = er 0 Ḃ/m), in ontrast to the ase of the satellite paradox where the hange in azimuthal veloity is the negative of the naive expetation [4]. However, eq. (13) differs in seond order from the result of this assumption, as needed to explain the energy balane to this order (see Appendix A). Indeed, if one is willing to aept eq. (13) to first order as obvious, then the radial equation of motion an be written as mv φ r ee r + ev φb = mr 0v 0 r + ev φb whih is a quadrati equation in 1/r with the approximate solution, (14) 1 r 1 e r 0 B, (15) r 0 4m v0 whih leads to eq. (11) but with the opposite sign for the seond term. Coming at length to the issue of diamagnetism, we note that the orbital angular momentum is L = mrv φ ẑ, (16) The magneti moment of the system is, following Larmor, μ = e m L = erv φ ẑ erv 0 ẑ e r0 4m B+ e3 r0b 3 ẑ μ 8m 3 v 0 +μ diamagneti +O(B ), (17) 0 where μ 0 = erv 0 ẑ, and μ diamagneti = e r0 4m B (18) as in the usual treatments [1, ] whih simply assume that r = r 0 always. From eq. (11) we see that this is a very good assumption within the ontext of a lassial atom (so long as one ignores effets of radiation). 3
4 A Appendix: Energy Balane The energy of the lassial atom in a magneti field B = B(t) ẑ is 3 U = m(ṙ + v φ ) We ignore the ontribution of ṙ to the kineti energy, and write U mv φ mr 0v0 ( mv 0 1+ er 0B + e r0b ) mv r mv 0 4m v0 0 = mv 0 + er 0v 0 B to seond order in field strength B. Then, to this order, U er 0v 0 Ḃ realling eqs. (5) and (13). + e r 0BḂ 4m ee r. (19) ( 1+ e r 0B 4m v 0 + e r 0B 8m (0) = er 0v 0 Ḃ ( 1+ er ) 0B erv φḃ mv 0 ) = Ẇext, (1) B Appendix: Why Doesn t the Satellite Paradox Hold for Classial Diamagnetism? One answer is that magneti fields do no work. The phenomenon of lassial diamagnetism, as desribed by eqs. (11) and (13), ontradits the laim in [5] that the qualitative behavior seen in the satellite paradox (the veloity inreases as the radius dereases, and vie versa) holds for any perturbation of motion in a 1/r potential. The argument in [5] is based on the laim that all relevant behavior of the system an be related to its energy, and that hanges in the energy are due to the work done by the perturbing fores. However, sine magneti fields do no work, they an lead to perturbations whose effet is not aptured by arguments based on energy. For ompleteness, we analyze the satellite paradox in a manner similar to that given in se.. Suppose the external fore is purely a drag fore, F ext = Cv p v Cv p φ (ṙ ˆr + v φ ˆφ), () where p is a small non-negative number, perhaps 0 or 1, and C is the drag oeffiient. Unlike the ase of an atom in an external magneti field, it suffies to onsider only the ase that v φ is positive. We again neglet r ompared to r φ in the radial equation of motion, φ φ 0 r3 0 r 3 Cṙvp φ mr, or v φ v 0 r 0 Crṙvp φ r m, (3) 3 Note that the energy, μ B = ervb/ er 0 v 0 B/, of the magneti moment μ in the external magneti field B is already ontained in the mirosopi energy expression (19), whih is also the energy in the Darwin approximation (see, for example, eq. () of [8]). 4
5 where v 0 is the azimuthal veloity of the (nearly) irular orbit at some referene radius r 0. The effet of the drag fore is to derease r, sothatṙ is negative, and eq. (3) tells us that v φ >v 0 for all r<r 0. Thus, the radial equation of motion ontains the qualitative nature of the satellite paradox, that the drag fore inreases the azimuthal veloity, rather than reduing it. In the first approximation, we ignore the small radial omponent of the drag fore, so that φ φ 0r0 3 ( ) r0 1/, and v r 3 φ v 0. (4) r The azimuthal equation of motion is r φ +ṙ φ Cr1+p φ1+p =. (5) m An analyti solution is espeially simple for p = 0 (drag fore proportional to veloity). 4 As before, we use the derivative of the result (4) of the radial equation in the azimuthal equation (5) to find ṙ = C r, (6) m and hene r = r 0 e Ct/m. (7) Thus, using eq. (4), v φ = v 0 e Ct/m. (8) In ontrast, the radius (11) of the lassial atom in an external magneti field dereases as the field inreases whether the azimuthal fore applies a drag or a boost! Furthermore, when the azimuthal fore on the lassial atom is a drag the azimuthal veloity dereases in magnitude, and inrease when the fore is a boost. C Appendix: Adiabati Invariane A more formal approah to the problem of lassial diamagnetism an be based on the (nonrelativisti) Lagrangian, 5 L = mv ev A + ev = m(ṙ + r φ ) (t) + ee r, (9) for the eletron (of harge e) in the Coulomb potential V = e /r of the nuleus and the external magneti field B(t) =B ẑ = A where A = rb ˆφ/. As the Lagrangian (9) does not depend on oordinate φ, the anonial momentum p φ is onserved, and we identify this as the z omponent of the anonial angular momentum, l z = p φ = L φ = er mr φ B = L z er B = L 0 = mr 0 v 0, (30) 4 An analyti solution is also possible for the ase of p = 1, i.e., for a onstant drag fore [6]. 5 In the present example the term ev A/ an be rewritten as μ B, whereμ is the (orbital) magneti moment. 5
6 where L z = mr φ is the mehanial angular momentum about the z axis. Thus, φ = L 0 mr + eb m, (31) where eb(t)/m is the ylotron (Larmor) frequeny. The radial equation of motion is m r = mr φ ee r = L 0 mr 3 e rb 4m L 0 mr 0 r U eff r, (3) wherewenotethatee = L 0/mr 0 for the initial irular orbit. The effetive radial potential U eff is U eff (r, t)= L 0 mr + e r B L 0 8m mr 0 r. (33) However, the problem is not one of small osillations about a fixed minimum of the effetive potential. Rather, we suppose that the orbit at time t is nearly irular with radius r(t) suh that the effetive potential is minimum at this time. That is, we set eq. (3) to zero, whih leads again to eq. (10), to the solutions (11)-(13) and to the magneti moment (18). We have found one invariant in this problem, the anonial angular momentum l z of eq. (30). Beause this remains onstant during hanges in the magneti field, whether these are adiabati or not, we an ertainly identify the anonial angular momentum as an adiabati invariant. We ould define the anonial magneti moment μ C as μ C = e m l z = μ z + e r B m μ z + e r0b m, (34) whih is also an adiabati invariant. However, the ordinary magneti moment μ z is not an adiabati invariant in this problem. We an ompare the present ase of a lassial atom to that of an eletron in a uniform magneti field. For bound motion in the latter ase we annot begin with zero magneti field, but we suppose instead that B = B 0 ẑ at time t =0. Then, the Lagrangian is obtained from eq. (9) by setting e =0. The anonial angular momentum (30) is again (an adiabati) invariant, while the radial equation of motion simplifies to mr φ = er B/ for slow hanges in the magneti field B. Hene, the anonial angular momentum l z an be written as l z = mr er φ B and the anonial magneti moment is = er B = L z, (35) μ C = e m l z = μ z. (36) In this ase the ordinary magneti moment μ z is also an adiabati invariant. However, most disussions of the motion of harged partiles in slowly varying magneti fields do not emphasize that the ordinary magneti moment is (somewhat aidentally) twie the invariant anonial magneti moment, whih may leave a misimpression as to the (non)invariane of the ordinary magneti moment in other irumstanes. 6
7 Aknowledgment Thanks to Vladimir Onoohin for drawing the author s attention to this problem, and to Vladimr Hnizdo and Dmitry Peregoudov for e-disussions of it. Referenes [1] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Letures on Physis, Vol. (Addison-Wesley, Reading, MA, 1964). [] E.M. Purell, Eletriity and Magnetism, nd ed. (MGraw-Hill, New York, 1985). [3] On Non-zero Classial Diamagnetism: A Surprise, N. Kumar and K.V. Kumar (Nov. 0, 008), [4] B.D. Mills, Jr., Satellite Paradox, Am. J. Phys. 7, 115 (1959), [5] L. Blitzer, Satellite Orbit Paradox: A General View, Am. J. Phys. 39, 887 (1971), [6] F.P.J. Rimrott and F.A. Salustri, Open Orbits in Satellite Dynamis, Teh.Meh.1, 07 (001), [7] S.L. O Dell and R.K.P. Zia, Classial and semilassial diamagnetism: A ritique of treatment in elementary texts, Am. J. Phys. 54, 3 (1986), [8] K.T. MDonald, Darwin Energy Paradoxes (Ot. 9, 008), 7
Dr G. I. Ogilvie Lent Term 2005
Aretion Diss Mathematial Tripos, Part III Dr G. I. Ogilvie Lent Term 2005 1.4. Visous evolution of an aretion dis 1.4.1. Introdution The evolution of an aretion dis is regulated by two onservation laws:
More informationGreen s function for the wave equation
Green s funtion for the wave equation Non-relativisti ase January 2019 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials are (Notes 1 eqns 43 and 44): 1 2 A 2 2 2 A = µ 0
More informationRelativity in Classical Physics
Relativity in Classial Physis Main Points Introdution Galilean (Newtonian) Relativity Relativity & Eletromagnetism Mihelson-Morley Experiment Introdution The theory of relativity deals with the study of
More informationThe Concept of Mass as Interfering Photons, and the Originating Mechanism of Gravitation D.T. Froedge
The Conept of Mass as Interfering Photons, and the Originating Mehanism of Gravitation D.T. Froedge V04 Formerly Auburn University Phys-dtfroedge@glasgow-ky.om Abstrat For most purposes in physis the onept
More informationHamiltonian with z as the Independent Variable
Hamiltonian with z as the Independent Variable 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 Marh 19, 2011; updated June 19, 2015) Dedue the form of the Hamiltonian
More informationThe homopolar generator: an analytical example
The homopolar generator: an analytial example Hendrik van Hees August 7, 214 1 Introdution It is surprising that the homopolar generator, invented in one of Faraday s ingenious experiments in 1831, still
More informationQUANTUM MECHANICS II PHYS 517. Solutions to Problem Set # 1
QUANTUM MECHANICS II PHYS 57 Solutions to Problem Set #. The hamiltonian for a lassial harmoni osillator an be written in many different forms, suh as use ω = k/m H = p m + kx H = P + Q hω a. Find a anonial
More informationHidden Momentum in a Spinning Sphere
Hidden Momentum in a Spinning Sphere 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 8544 (August 16, 212; updated June 3, 217 A spinning sphere at rest has zero
More informationName Solutions to Test 1 September 23, 2016
Name Solutions to Test 1 September 3, 016 This test onsists of three parts. Please note that in parts II and III, you an skip one question of those offered. Possibly useful formulas: F qequb x xvt E Evpx
More informationTowards an Absolute Cosmic Distance Gauge by using Redshift Spectra from Light Fatigue.
Towards an Absolute Cosmi Distane Gauge by using Redshift Spetra from Light Fatigue. Desribed by using the Maxwell Analogy for Gravitation. T. De Mees - thierrydemees @ pandora.be Abstrat Light is an eletromagneti
More informationThe Thomas Precession Factor in Spin-Orbit Interaction
p. The Thomas Preession Fator in Spin-Orbit Interation Herbert Kroemer * Department of Eletrial and Computer Engineering, Uniersity of California, Santa Barbara, CA 9306 The origin of the Thomas fator
More informationAharonov-Bohm effect. Dan Solomon.
Aharonov-Bohm effet. Dan Solomon. In the figure the magneti field is onfined to a solenoid of radius r 0 and is direted in the z- diretion, out of the paper. The solenoid is surrounded by a barrier that
More informationPhysics 2D Lecture Slides Lecture 7: Jan 14th 2004
Quiz is This Friday Quiz will over Setions.-.6 (inlusive) Remaining material will be arried over to Quiz Bring Blue Book, hek alulator battery Write all answers in indelible ink else no grade! Write answers
More informationBrazilian Journal of Physics, vol. 29, no. 3, September, Classical and Quantum Mechanics of a Charged Particle
Brazilian Journal of Physis, vol. 9, no. 3, September, 1999 51 Classial and Quantum Mehanis of a Charged Partile in Osillating Eletri and Magneti Fields V.L.B. de Jesus, A.P. Guimar~aes, and I.S. Oliveira
More informationProblem 3 : Solution/marking scheme Large Hadron Collider (10 points)
Problem 3 : Solution/marking sheme Large Hadron Collider 10 points) Part A. LHC Aelerator 6 points) A1 0.7 pt) Find the exat expression for the final veloity v of the protons as a funtion of the aelerating
More informationTemperature-Gradient-Driven Tearing Modes
1 TH/S Temperature-Gradient-Driven Tearing Modes A. Botrugno 1), P. Buratti 1), B. Coppi ) 1) EURATOM-ENEA Fusion Assoiation, Frasati (RM), Italy ) Massahussets Institute of Tehnology, Cambridge (MA),
More informationElectromagnetic radiation of the travelling spin wave propagating in an antiferromagnetic plate. Exact solution.
arxiv:physis/99536v1 [physis.lass-ph] 15 May 1999 Eletromagneti radiation of the travelling spin wave propagating in an antiferromagneti plate. Exat solution. A.A.Zhmudsky November 19, 16 Abstrat The exat
More informationChapter 9. The excitation process
Chapter 9 The exitation proess qualitative explanation of the formation of negative ion states Ne and He in He-Ne ollisions an be given by using a state orrelation diagram. state orrelation diagram is
More informationMetal: a free electron gas model. Drude theory: simplest model for metals Sommerfeld theory: classical mechanics quantum mechanics
Metal: a free eletron gas model Drude theory: simplest model for metals Sommerfeld theory: lassial mehanis quantum mehanis Drude model in a nutshell Simplest model for metal Consider kinetis for eletrons
More informationTHEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE?
THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE? The stars are spheres of hot gas. Most of them shine beause they are fusing hydrogen into helium in their entral parts. In this problem we use onepts of
More informationNon-Markovian study of the relativistic magnetic-dipole spontaneous emission process of hydrogen-like atoms
NSTTUTE OF PHYSCS PUBLSHNG JOURNAL OF PHYSCS B: ATOMC, MOLECULAR AND OPTCAL PHYSCS J. Phys. B: At. Mol. Opt. Phys. 39 ) 7 85 doi:.88/953-75/39/8/ Non-Markovian study of the relativisti magneti-dipole spontaneous
More informationThe gravitational phenomena without the curved spacetime
The gravitational phenomena without the urved spaetime Mirosław J. Kubiak Abstrat: In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime,
More informationELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES.
ELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES. All systems with interation of some type have normal modes. One may desribe them as solutions in absene of soures; they are exitations of the system
More informationSimple Considerations on the Cosmological Redshift
Apeiron, Vol. 5, No. 3, July 8 35 Simple Considerations on the Cosmologial Redshift José Franiso Garía Juliá C/ Dr. Maro Mereniano, 65, 5. 465 Valenia (Spain) E-mail: jose.garia@dival.es Generally, the
More informationAnswers to Coursebook questions Chapter J2
Answers to Courseook questions Chapter J 1 a Partiles are produed in ollisions one example out of many is: a ollision of an eletron with a positron in a synhrotron. If we produe a pair of a partile and
More informationReview of classical thermodynamics
Review of lassial thermodynamis Fundamental Laws, roperties and roesses () First Law - Energy Balane hermodynami funtions of state Internal energy, heat and work ypes of paths (isobari, isohori, isothermal,
More informationRelativistic Dynamics
Chapter 7 Relativisti Dynamis 7.1 General Priniples of Dynamis 7.2 Relativisti Ation As stated in Setion A.2, all of dynamis is derived from the priniple of least ation. Thus it is our hore to find a suitable
More informationDepartment of Natural Sciences Clayton State University. Physics 3650 Quiz 1. c. Both kinetic and elastic potential energies can be negative.
Department of Natural Sienes Physis 3650 Quiz 1 August 5, 008 1. Whih one of the statements below is orret? a. Elasti potential energy an be negative but the kineti energy annot. b. Kineti energy an be
More informationAstr 5465 Mar. 29, 2018 Galactic Dynamics I: Disks
Galati Dynamis Overview Astr 5465 Mar. 29, 2018 Subjet is omplex but we will hit the highlights Our goal is to develop an appreiation of the subjet whih we an use to interpret observational data See Binney
More informationGyrokinetic calculations of the neoclassical radial electric field in stellarator plasmas
PHYSICS OF PLASMAS VOLUME 8, NUMBER 6 JUNE 2001 Gyrokineti alulations of the neolassial radial eletri field in stellarator plasmas J. L. V. Lewandowski Plasma Physis Laboratory, Prineton University, P.O.
More informationGravitation is a Gradient in the Velocity of Light ABSTRACT
1 Gravitation is a Gradient in the Veloity of Light D.T. Froedge V5115 @ http://www.arxdtf.org Formerly Auburn University Phys-dtfroedge@glasgow-ky.om ABSTRACT It has long been known that a photon entering
More informationInvestigation of the de Broglie-Einstein velocity equation s. universality in the context of the Davisson-Germer experiment. Yusuf Z.
Investigation of the de Broglie-instein veloity equation s universality in the ontext of the Davisson-Germer experiment Yusuf Z. UMUL Canaya University, letroni and Communiation Dept., Öğretmenler Cad.,
More informationThe Electromagnetic Radiation and Gravity
International Journal of Theoretial and Mathematial Physis 016, 6(3): 93-98 DOI: 10.593/j.ijtmp.0160603.01 The Eletromagneti Radiation and Gravity Bratianu Daniel Str. Teiului Nr. 16, Ploiesti, Romania
More informationChapter 26 Lecture Notes
Chapter 26 Leture Notes Physis 2424 - Strauss Formulas: t = t0 1 v L = L0 1 v m = m0 1 v E = m 0 2 + KE = m 2 KE = m 2 -m 0 2 mv 0 p= mv = 1 v E 2 = p 2 2 + m 2 0 4 v + u u = 2 1 + vu There were two revolutions
More informationAnnouncements. Lecture 5 Chapter. 2 Special Relativity. The Doppler Effect
Announements HW1: Ch.-0, 6, 36, 41, 46, 50, 51, 55, 58, 63, 65 *** Lab start-u meeting with TA yesterday; useful? *** Lab manual is osted on the ourse web *** Physis Colloquium (Today 3:40m anelled ***
More information+Ze. n = N/V = 6.02 x x (Z Z c ) m /A, (1.1) Avogadro s number
In 1897, J. J. Thomson disovered eletrons. In 1905, Einstein interpreted the photoeletri effet In 1911 - Rutherford proved that atoms are omposed of a point-like positively harged, massive nuleus surrounded
More informationThe Laws of Acceleration
The Laws of Aeleration The Relationships between Time, Veloity, and Rate of Aeleration Copyright 2001 Joseph A. Rybzyk Abstrat Presented is a theory in fundamental theoretial physis that establishes the
More informationDynamics of the Electromagnetic Fields
Chapter 3 Dynamis of the Eletromagneti Fields 3.1 Maxwell Displaement Current In the early 1860s (during the Amerian ivil war!) eletriity inluding indution was well established experimentally. A big row
More informationarxiv:physics/ v1 14 May 2002
arxiv:physis/0205041 v1 14 May 2002 REPLY TO CRITICISM OF NECESSITY OF SIMULTANEOUS CO-EXISTENCE OF INSTANTANEOUS AND RETARDED INTERACTIONS IN CLASSICAL ELECTRODYNAMICS by J.D.Jakson ANDREW E. CHUBYKALO
More informationELECTROMAGNETIC WAVES WITH NONLINEAR DISPERSION LAW. P. М. Меdnis
ELECTROMAGNETIC WAVES WITH NONLINEAR DISPERSION LAW P. М. Меdnis Novosibirs State Pedagogial University, Chair of the General and Theoretial Physis, Russia, 636, Novosibirs,Viljujsy, 8 e-mail: pmednis@inbox.ru
More informationConducting Sphere That Rotates in a Uniform Magnetic Field
1 Problem Conduting Sphere That Rotates in a Uniform Magneti Field Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (Mar. 13, 2002) A onduting sphere of radius a, relative
More informationThe Unified Geometrical Theory of Fields and Particles
Applied Mathematis, 014, 5, 347-351 Published Online February 014 (http://www.sirp.org/journal/am) http://dx.doi.org/10.436/am.014.53036 The Unified Geometrial Theory of Fields and Partiles Amagh Nduka
More informationarxiv:gr-qc/ v2 6 Feb 2004
Hubble Red Shift and the Anomalous Aeleration of Pioneer 0 and arxiv:gr-q/0402024v2 6 Feb 2004 Kostadin Trenčevski Faulty of Natural Sienes and Mathematis, P.O.Box 62, 000 Skopje, Maedonia Abstrat It this
More informationHidden Momentum in a Current Loop
1 Problem Hidden Momentum in a Current Loop Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (June 0, 2012; updated May 9, 2018) Disuss the momentum of a loop of eletrial
More informationLecture 15 (Nov. 1, 2017)
Leture 5 8.3 Quantum Theor I, Fall 07 74 Leture 5 (Nov., 07 5. Charged Partile in a Uniform Magneti Field Last time, we disussed the quantum mehanis of a harged partile moving in a uniform magneti field
More informationLecture #1: Quantum Mechanics Historical Background Photoelectric Effect. Compton Scattering
561 Fall 2017 Leture #1 page 1 Leture #1: Quantum Mehanis Historial Bakground Photoeletri Effet Compton Sattering Robert Field Experimental Spetrosopist = Quantum Mahinist TEXTBOOK: Quantum Chemistry,
More informationParticle-wave symmetry in Quantum Mechanics And Special Relativity Theory
Partile-wave symmetry in Quantum Mehanis And Speial Relativity Theory Author one: XiaoLin Li,Chongqing,China,hidebrain@hotmail.om Corresponding author: XiaoLin Li, Chongqing,China,hidebrain@hotmail.om
More informationVector Field Theory (E&M)
Physis 4 Leture 2 Vetor Field Theory (E&M) Leture 2 Physis 4 Classial Mehanis II Otober 22nd, 2007 We now move from first-order salar field Lagrange densities to the equivalent form for a vetor field.
More informationPh1c Analytic Quiz 2 Solution
Ph1 Analyti Quiz 2 olution Chefung Chan, pring 2007 Problem 1 (6 points total) A small loop of width w and height h falls with veloity v, under the influene of gravity, into a uniform magneti field B between
More informationWe consider the nonrelativistic regime so no pair production or annihilation.the hamiltonian for interaction of fields and sources is 1 (p
.. RADIATIVE TRANSITIONS Marh 3, 5 Leture XXIV Quantization of the E-M field. Radiative transitions We onsider the nonrelativisti regime so no pair prodution or annihilation.the hamiltonian for interation
More informationFour-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field
Four-dimensional equation of motion for visous ompressible substane with regard to the aeleration field, pressure field and dissipation field Sergey G. Fedosin PO box 6488, Sviazeva str. -79, Perm, Russia
More informationWavetech, LLC. Ultrafast Pulses and GVD. John O Hara Created: Dec. 6, 2013
Ultrafast Pulses and GVD John O Hara Created: De. 6, 3 Introdution This doument overs the basi onepts of group veloity dispersion (GVD) and ultrafast pulse propagation in an optial fiber. Neessarily, it
More informationThe Gravitational Potential for a Moving Observer, Mercury s Perihelion, Photon Deflection and Time Delay of a Solar Grazing Photon
Albuquerque, NM 0 POCEEDINGS of the NPA 457 The Gravitational Potential for a Moving Observer, Merury s Perihelion, Photon Defletion and Time Delay of a Solar Grazing Photon Curtis E. enshaw Tele-Consultants,
More informationClassical Trajectories in Rindler Space and Restricted Structure of Phase Space with PT-Symmetric Hamiltonian. Abstract
Classial Trajetories in Rindler Spae and Restrited Struture of Phase Spae with PT-Symmetri Hamiltonian Soma Mitra 1 and Somenath Chakrabarty 2 Department of Physis, Visva-Bharati, Santiniketan 731 235,
More informationAccelerator Physics Particle Acceleration. G. A. Krafft Old Dominion University Jefferson Lab Lecture 4
Aelerator Physis Partile Aeleration G. A. Krafft Old Dominion University Jefferson Lab Leture 4 Graduate Aelerator Physis Fall 15 Clarifiations from Last Time On Crest, RI 1 RI a 1 1 Pg RL Pg L V Pg RL
More informationPhysics 218, Spring February 2004
Physis 8 Spring 004 8 February 004 Today in Physis 8: dispersion Motion of bound eletrons in matter and the frequeny dependene of the dieletri onstant Dispersion relations Ordinary and anomalous dispersion
More informationPhysics 486. Classical Newton s laws Motion of bodies described in terms of initial conditions by specifying x(t), v(t).
Physis 486 Tony M. Liss Leture 1 Why quantum mehanis? Quantum vs. lassial mehanis: Classial Newton s laws Motion of bodies desribed in terms of initial onditions by speifying x(t), v(t). Hugely suessful
More informationIntroduction to Quantum Chemistry
Chem. 140B Dr. J.A. Mak Introdution to Quantum Chemistry Without Quantum Mehanis, how would you explain: Periodi trends in properties of the elements Struture of ompounds e.g. Tetrahedral arbon in ethane,
More informationOn the Quantum Theory of Radiation.
Physikalishe Zeitshrift, Band 18, Seite 121-128 1917) On the Quantum Theory of Radiation. Albert Einstein The formal similarity between the hromati distribution urve for thermal radiation and the Maxwell
More informationFinal Review. A Puzzle... Special Relativity. Direction of the Force. Moving at the Speed of Light
Final Review A Puzzle... Diretion of the Fore A point harge q is loated a fixed height h above an infinite horizontal onduting plane. Another point harge q is loated a height z (with z > h) above the plane.
More informationFeynman Cylinder Paradox
Feynman Cylinder Paradox John Belcher Division of Astrophysics, Massachussetts Institute of Technology, Cambridge, MA 02139 Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton,
More informationCritical Reflections on the Hafele and Keating Experiment
Critial Refletions on the Hafele and Keating Experiment W.Nawrot In 1971 Hafele and Keating performed their famous experiment whih onfirmed the time dilation predited by SRT by use of marosopi loks. As
More informationDerivation of Non-Einsteinian Relativistic Equations from Momentum Conservation Law
Asian Journal of Applied Siene and Engineering, Volue, No 1/13 ISSN 35-915X(p); 37-9584(e) Derivation of Non-Einsteinian Relativisti Equations fro Moentu Conservation Law M.O.G. Talukder Varendra University,
More informationNumerical Tests of Nucleation Theories for the Ising Models. Abstract
to be submitted to Physial Review E Numerial Tests of Nuleation Theories for the Ising Models Seunghwa Ryu 1 and Wei Cai 2 1 Department of Physis, Stanford University, Stanford, California 94305 2 Department
More informationCHAPTER 26 The Special Theory of Relativity
CHAPTER 6 The Speial Theory of Relativity Units Galilean-Newtonian Relativity Postulates of the Speial Theory of Relativity Simultaneity Time Dilation and the Twin Paradox Length Contration Four-Dimensional
More informationTENSOR FORM OF SPECIAL RELATIVITY
TENSOR FORM OF SPECIAL RELATIVITY We begin by realling that the fundamental priniple of Speial Relativity is that all physial laws must look the same to all inertial observers. This is easiest done by
More information8.333: Statistical Mechanics I Problem Set # 4 Due: 11/13/13 Non-interacting particles
8.333: Statistial Mehanis I Problem Set # 4 Due: 11/13/13 Non-interating partiles 1. Rotating gas: Consider a gas of N idential atoms onfined to a spherial harmoni trap in three dimensions, i.e. the partiles
More informationGravitomagnetic Effects in the Kerr-Newman Spacetime
Advaned Studies in Theoretial Physis Vol. 10, 2016, no. 2, 81-87 HIKARI Ltd, www.m-hikari.om http://dx.doi.org/10.12988/astp.2016.512114 Gravitomagneti Effets in the Kerr-Newman Spaetime A. Barros Centro
More informationNuclear Shell Structure Evolution Theory
Nulear Shell Struture Evolution Theory Zhengda Wang (1) Xiaobin Wang () Xiaodong Zhang () Xiaohun Wang () (1) Institute of Modern physis Chinese Aademy of SienesLan Zhou P. R. China 70000 () Seagate Tehnology
More informationPhysical Laws, Absolutes, Relative Absolutes and Relativistic Time Phenomena
Page 1 of 10 Physial Laws, Absolutes, Relative Absolutes and Relativisti Time Phenomena Antonio Ruggeri modexp@iafria.om Sine in the field of knowledge we deal with absolutes, there are absolute laws that
More informationThe Lorenz Transform
The Lorenz Transform Flameno Chuk Keyser Part I The Einstein/Bergmann deriation of the Lorentz Transform I follow the deriation of the Lorentz Transform, following Peter S Bergmann in Introdution to the
More informationEinstein s Three Mistakes in Special Relativity Revealed. Copyright Joseph A. Rybczyk
Einstein s Three Mistakes in Speial Relativity Revealed Copyright Joseph A. Rybzyk Abstrat When the evidene supported priniples of eletromagneti propagation are properly applied, the derived theory is
More informationElectromagnetic Field Angular Momentum of a Charge at Rest in a Uniform Magnetic Field
Eletromagneti Field Angular Momentum of a Charge at Rest in a Uniform Magneti Field 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 8544 (Deember 21, 214; updated
More information). In accordance with the Lorentz transformations for the space-time coordinates of the same event, the space coordinates become
Relativity and quantum mehanis: Jorgensen 1 revisited 1. Introdution Bernhard Rothenstein, Politehnia University of Timisoara, Physis Department, Timisoara, Romania. brothenstein@gmail.om Abstrat. We first
More informationBeams on Elastic Foundation
Professor Terje Haukaas University of British Columbia, Vanouver www.inrisk.ub.a Beams on Elasti Foundation Beams on elasti foundation, suh as that in Figure 1, appear in building foundations, floating
More informationRadiation processes and mechanisms in astrophysics 3. R Subrahmanyan Notes on ATA lectures at UWA, Perth 22 May 2009
Radiation proesses and mehanisms in astrophysis R Subrahmanyan Notes on ATA letures at UWA, Perth May 009 Synhrotron radiation - 1 Synhrotron radiation emerges from eletrons moving with relativisti speeds
More informationRecapitulate. Prof. Shiva Prasad, Department of Physics, IIT Bombay
18 1 Reapitulate We disussed how light an be thought of onsisting of partiles known as photons. Compton Effet demonstrated that they an be treated as a partile with zero rest mass having nonzero energy
More information20 Doppler shift and Doppler radars
20 Doppler shift and Doppler radars Doppler radars make a use of the Doppler shift phenomenon to detet the motion of EM wave refletors of interest e.g., a polie Doppler radar aims to identify the speed
More informationSpinning Charged Bodies and the Linearized Kerr Metric. Abstract
Spinning Charged Bodies and the Linearized Kerr Metri J. Franklin Department of Physis, Reed College, Portland, OR 97202, USA. Abstrat The physis of the Kerr metri of general relativity (GR) an be understood
More information4. (12) Write out an equation for Poynting s theorem in differential form. Explain in words what each term means physically.
Eletrodynamis I Exam 3 - Part A - Closed Book KSU 205/2/8 Name Eletrodynami Sore = 24 / 24 points Instrutions: Use SI units. Where appropriate, define all variables or symbols you use, in words. Try to
More informationarxiv: v1 [physics.plasm-ph] 5 Aug 2012
Classial mirosopi derivation of the relativisti hydrodynamis equations arxiv:1208.0998v1 [physis.plasm-ph] 5 Aug 2012 P. A. Andreev Department of General Physis, Physis Faulty, Mosow State University,
More informationGeneral Equilibrium. What happens to cause a reaction to come to equilibrium?
General Equilibrium Chemial Equilibrium Most hemial reations that are enountered are reversible. In other words, they go fairly easily in either the forward or reverse diretions. The thing to remember
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E')
22.54 Neutron Interations and Appliations (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E') Referenes -- J. R. Lamarsh, Introdution to Nulear Reator Theory (Addison-Wesley, Reading, 1966),
More informationarxiv:physics/ v1 [physics.class-ph] 8 Aug 2003
arxiv:physis/0308036v1 [physis.lass-ph] 8 Aug 003 On the meaning of Lorentz ovariane Lszl E. Szab Theoretial Physis Researh Group of the Hungarian Aademy of Sienes Department of History and Philosophy
More informationAn Effective Photon Momentum in a Dielectric Medium: A Relativistic Approach. Abstract
An Effetive Photon Momentum in a Dieletri Medium: A Relativisti Approah Bradley W. Carroll, Farhang Amiri, and J. Ronald Galli Department of Physis, Weber State University, Ogden, UT 84408 Dated: August
More informationPhys 561 Classical Electrodynamics. Midterm
Phys 56 Classial Eletrodynamis Midterm Taner Akgün Department of Astronomy and Spae Sienes Cornell University Otober 3, Problem An eletri dipole of dipole moment p, fixed in diretion, is loated at a position
More informationLine Radiative Transfer
http://www.v.nrao.edu/ourse/astr534/ineradxfer.html ine Radiative Transfer Einstein Coeffiients We used armor's equation to estimate the spontaneous emission oeffiients A U for À reombination lines. A
More informationCurrent density and forces for a current loop moving parallel over a thin conducting sheet
INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 5 (4) 655 666 EUROPEAN JOURNAL OF PHYSICS PII: S43-87(4)77753-3 Current density and fores for a urrent loop moving parallel over a thin onduting sheet BSPalmer
More informationChapter Outline The Relativity of Time and Time Dilation The Relativistic Addition of Velocities Relativistic Energy and E= mc 2
Chapter 9 Relativeity Chapter Outline 9-1 The Postulate t of Speial Relativity it 9- The Relativity of Time and Time Dilation 9-3 The Relativity of Length and Length Contration 9-4 The Relativisti Addition
More informationOn the internal motion of electrons. I.
Über die innere Bewegung des Elektrons. I, Zeit. Phys. (939), 5. On the internal motion of eletrons. I. By H. Hönl and A. Papapetrou With diagrams. (Reeived 3 Marh 939) Translated by D. H. Delphenih Upon
More informationTutorial 8: Solutions
Tutorial 8: Solutions 1. * (a) Light from the Sun arrives at the Earth, an average of 1.5 10 11 m away, at the rate 1.4 10 3 Watts/m of area perpendiular to the diretion of the light. Assume that sunlight
More informationENERGY AND MOMENTUM IN ELECTROMAGNETIC WAVES
MISN-0-211 z ENERGY AND MOMENTUM IN ELECTROMAGNETIC WAVES y È B` x ENERGY AND MOMENTUM IN ELECTROMAGNETIC WAVES by J. S. Kovas and P. Signell Mihigan State University 1. Desription................................................
More informationSTATISTICAL MECHANICS & THERMODYNAMICS
UVA PHYSICS DEPARTMENT PHD QUALIFYING EXAM PROBLEM FILE STATISTICAL MECHANICS & THERMODYNAMICS UPDATED: NOVEMBER 14, 212 1. a. Explain what is meant by the density of states, and give an expression for
More informationProbe-field reflection on a plasma surface driven by a strong electromagnetic field
J. Phys. B: At. Mol. Opt. Phys. 33 (2000) 2549 2558. Printed in the UK PII: S0953-4075(00)50056-X Probe-field refletion on a plasma surfae driven by a strong eletromagneti field Kazimierz Rz ażewski, Luis
More informationmax min z i i=1 x j k s.t. j=1 x j j:i T j
AM 221: Advaned Optimization Spring 2016 Prof. Yaron Singer Leture 22 April 18th 1 Overview In this leture, we will study the pipage rounding tehnique whih is a deterministi rounding proedure that an be
More informationAngular Distribution of Photoelectrons during Irradiation of Metal Surface by Electromagnetic Waves
Journal of Modern Physis, 0,, 780-786 doi:0436/jmp0809 Published Online August 0 (http://wwwsirporg/journal/jmp) Angular Distribution of Photoeletrons during Irradiation of Metal Surfae by letromagneti
More information(Newton s 2 nd Law for linear motion)
PHYSICS 3 Final Exaination ( Deeber Tie liit 3 hours Answer all 6 questions You and an assistant are holding the (opposite ends of a long plank when oops! the butterfingered assistant drops his end If
More informationOn the Geometrical Conditions to Determine the Flat Behaviour of the Rotational Curves in Galaxies
On the Geometrial Conditions to Determine the Flat Behaviour of the Rotational Curves in Galaxies Departamento de Físia, Universidade Estadual de Londrina, Londrina, PR, Brazil E-mail: andrenaves@gmail.om
More informationTWO WAYS TO DISTINGUISH ONE INERTIAL FRAME FROM ANOTHER
TWO WAYS TO DISTINGUISH ONE INERTIAL FRAME FROM ANOTHER (WHY IS THE SPEED OF LIGHT CONSTANT?) Dr. Tamas Lajtner Correspondene via web site: www.lajtnemahine.om. ABSTRACT... 2 2. SPACETIME CONTINUUM BY
More informationEnergy Gaps in a Spacetime Crystal
Energy Gaps in a Spaetime Crystal L.P. Horwitz a,b, and E.Z. Engelberg a Shool of Physis, Tel Aviv University, Ramat Aviv 69978, Israel b Department of Physis, Ariel University Center of Samaria, Ariel
More information