The Quantum Theory of Radiation


 Victor Reynolds
 2 years ago
 Views:
Transcription
1 A. Einstein, Physikalishe Zeitshrift 18, The Quantum Theory of Radiation A. Einstein (Reeived Marh, 1917) The formal similarity of the spetral distribution urve of temperature radiation to Maxwell s veloity distribution urve is too striking to have remained hidden very long. Indeed, in the important theoretial paper in whih Wien derived his displaement law ( ) ν ρ = ν 3 f (1) T he was led by this similarity to a farther orrespondene with the radiation formula. He disovered, as is known, the formula [Wien s radiation formula] ρ = αν 3 e hν kt (2) whih is reognized today as the orret limiting formula for large values of ν T. Today we known that no onsideration whih is based on lassial mehanis and eletrodynamis an lead to a useful radiation formula; rather that the lassial theory leads to the Rayleigh formula. ρ = kα h ν2 T (3) After Plank, in his ground breaking investigation, established his radiation formula ρ = αν 3 1 ekt hν (4) 1 1
2 on the assumption that there are disrete elements of energy, from whih quantum theory developed very rapidly, Wien s onsiderations, from whih formula (2) evolved, quite naturally were forgotten. A little while ago I obtained a derivation, related to Wien s original idea, of the Plank radiation formula whih is based on the fundamental assumption of quantum theory and whih makes use of the relationship of Maxwell s urve to the spetral distribution urve. This derivation deserves onsideration not only beause of its simpliity, but espeially beause it appears to larify the proesses of emission and absorption of radiation in matter, whih is still in suh darkness for us. In setting down ertain fundamental hypotheses onerning the absorption and emission of radiation by moleules that are losely related to quantum theory, I showed that moleules with a distribution of states in the quantum theoretial sense for temperature equilibrium are in dynamial equilibrium with the Plank radiation; in this way, the Plank formula (4) was obtained in a surprisingly simple and general way. It was obtained from the ondition that the quantum theoreti partition of states of the internal energy of the moleules is established only by the emission and absorption of radiation. If the assumed hypotheses about the interation of matter and radiation are orret, they will give us more than just the orret statistial partition or distribution of the internal energy of the moleules. During absorption and emission of radiation there is also present a transfer of momentum to the moleules; this means that just the interation of radiation and moleules leads to a veloity distribution of the latter. This must early be the same as the veloity distribution whih moleules aquire as the result of their mutual interation by ollisions, that is, it must oinide with the Maxwell distribution. we must require that the mean kineti energy whih a moleule (per degree of freedom) aquires in a Plank radiation field of temperature T be kt 2 ; this must be valid regardless of the nature of the moleules and independent of frequenies whih the moleules absorb and emit. In this paper we wish to verify that this far reahing requirement is, indeed, satisfied quite generally; as a result of this our simple hypotheses about the emission and absorption of radiation aquire new supports. In order to obtain this result, however, we must enlarge, in a definite way, the previous fundamental hypothesis whih were related entirely to the exhange of energy. We are faed with this question; Does the moleule suffer a push, when it absorbs or emits the energy ɛ? As an example we onsider, 2
3 from the lassial point of view, the emission of radiation. If a body emits the energy ɛ, it aquires a bakward thrust [impulse] ɛ if all the radiation ɛ is radiated in the same diretion. If, however, the radiation ours through a spatially symmetri proess, for example, spherial waves. there is then no reoil at all. This alternative also plays a role in the quantum theory of radiation. If a moleule, in going from one possible quantum theoreti state to another, absorbs or emits the energy ɛ in the form of radiation, suh an elementary proess an be looked upon as partly or fully direted in spae, or also as a symmetri (non direted) one. It turns out that we obtain a theory that is free of ontraditions only if we onsider the above elementary proesses as being fully direted events; herein lies the prinipal result of the onsiderations that follow. Fundamental Hypotheses of the Quantum Theory Canonial Distribution of State Aording to the quantum theory, a moleule of a definite kind may, aside from its orientation and its translational motion, be in one only a disrete set of states Z 1, Z 2,... Z n... whose (internal) energies are ɛ 1, ɛ 2,... ɛ n.... If the moleules of this kind belong to a gas of temperature T, then the relative abundane W n of the state Z n is given by the statistial mehanial anonial partition funtion for states W n = p n e ɛ n kt (5) In this formula k = R N is the wellknown Boltzmann onstant, p n a number that is independent of T and harateristi of the moleule and the state, whih we may all the statistial weight of the state. Formula (5) an be derived from the Boltzmann priniple or purely from thermodynamis. Equation (5) is the expression of the most far reahing generalization of the Maxwellian distribution of veloities. The latest important advanes in quantum theory deal with the theoretial determination of the quantum theoretial possible states Z n and their weights p n. For the prinipal part of the present investigation, it is not neessary to have a more detailed determination of the quantum states. 3
4 Hypotheses about the Energy Exhange Through Radiation Let Z n and Z m be two possible quantum theoretial states of a gas moleule whose energies ɛ n and ɛ m respetively, satisfy the inequality ɛ m > ɛ n Let the moleule be able to pass from the state Z n to the state Z m by absorbing the radiation energy ɛ m ɛ n, similarly let the transition from state Z n to the state Z m be possible through the emission of this amount of energy. Let the radiation emitted or absorbed by the moleule for the given index and ombination (m, n) have the harateristi frequeny ν. We now introdue ertain hypotheses about the laws whih are deisive for these transitions. These hypotheses are obtained by arrying over the known lassial relations for a Plank resonator to the unknown quantum theoretial relations. Emission A Plank resonator that is vibrating radiates energy aording to Hertz, in a known way independently of whether it is stimulated by an external field or not. In aordane with this, let a moleule be able to pass from the state Z m to the state Z n with the emission of radiant energy ɛ m ɛ n of frequeny ν without being exited by any external ause. Let the probability dw for this to happen in the time dt be dw = A n m dt (A) where A n m is a harateristi onstant for the given index ombination. The assumed statistial law orresponds to that of a radioative reation: that elementary proess of suh a reation in whih only γrays are emitted. We need not assume that this proess requires no time; this time need only be negligible ompared to the times whih the moleule spends in the states Z 1, and so on. Inident Radiation If a Plank resonator is in a radiation field, the energy of the resonator hanges beause the eletromagneti field of the radiation does work on the resonator; this work an be positive or negative depending on the phases of the resonator and the osillating field. In aordane with this, we introdue the following quantum theoretial hypothesis. Under the ation of the 4
5 radiation density ρ of the frequeny ν a moleule in state Z n an go over to state Z m absorbing the radiation energy ɛ m ɛ n in aordane with the probability law dw = B m n ρ dt (B) In the same way, let the transition Z m Z n under the ation of the radiation also be possible, whereby the radiation energy ɛ m ɛ n is emitted aording to the probability law dw = B n m ρ dt (B ) Bn m and Bm n are onstants. We all both proesses hanges of states through inident radiation. The question presents itself now as to the momentum that is transferred to the moleule in these hanges of state. We begin with the events assoiated with inident radiation. If a direted bundle of rays does work on a Plank resonator, then an equivalent amount of energy is removed from the bundle. this transfer of energy results, aording to the law of momentum, to a momentum transfer from the beam to the resonator. The latter therefore experienes a fore in the diretion of the ray of the radiation beam. If the energy transferred is negative, the fore ating on the resonator is opposite in diretion. In the ase of the quantum hypothesis, this learly means the following. If, as the result of inident radiation, the proess Z n Z m ours, then an amount of momentum ɛ m ɛ n is transferred to the moleule in the diretion of propagation of the bundle of radiation. If we have the proess Z m Z n for the ase of inident radiation, the magnitude of the transferred momentum is the same, but it is in the opposite diretion. If a moleule is simultaneously exposed to many bundles of radiation, we assume that the total energy Z m Z n is taken from or added to just one of these bundles, so that even in this ase the momentum ɛ m ɛ n is transferred to the moleule. In the ase of emission of energy by radiation by a Plank resonator, there is no net transfer of momentum to the resonator beause, aording to lassial theory, of emission ours as a spherial wave. However, we have already noted that we an arrive at a ontradition free quantum theory 5
6 only if we assume that the proess of emission is a direted one. Every elementary proess of emission (Z m Z n ) will then result in a transfer to the moleule of an amount of momentum ɛ m ɛ n. If the moleule is isotropi, we must take every diretion of emission as equally probable. If the moleule is not isotropi, we arrive at the same result if the orientation hanges in a random way in the ourse of time. We must, in any ase, make suh an assumption also for the statistial laws (B) and (B ) for inident radiation sine otherwise the onstants Bn m and Bm n would have to depend on diretion, whih we an avoid by assuming isotropy or pseudo isotropy (through setting up temporal mean values). Derivation of the Plank Radiation Law We now enquire about those effetive radiation densities ρ whih must prevail in order that the energy exhange between moleules and radiation as a result of the statistial laws (A), (B) and (B ) shall not disturb the distribution of moleular states present as a onsequene of equation (5). For this, it is neessary and suffiient that on the average, per unit time, as many elementary proesses of type (B) take plae as proesses (A) and (B ) together. This ondition gives as a result of (5), (A), (B), (B ), for the elementary proesses orresponding to the index ombination (m, n) the equation p n e ɛ n kt B m n ρ = p m e ɛ m kt (B n m ρ + A n m) If, further, ρ is to beome infinite as T does, the onstants Bn m and Bm n must satisfy the relation p n Bn m = p m Bm n (6) We then obtain as the ondition for dynamial equilibrium the equation ρ = e An m/b n m ɛ m ɛ n kt 1 This is the dependene of the radiation density on the temperature that is given by the Plank law. From the Wien displaement law (1) it then following immediately that A n m Bm n = αν 3 (8) 6 (7)
7 and ɛ m ɛ n = hν (9) where α and h are universal onstants. To obtain the numerial values of α and h we must have an exat theory of eletrodynami and mehanial proesses; we ontent ourselves for the moment with the Rayleigh law in the limit of high temperatures, where the lassial theory is valid in the limit. Equation (9) is, as we know, the seond prinipal rule in Bohr s theory of spetra, about whih we may assert, following upon Sommerfeld s and Epstein s ompletion of the theory, that it belong to the most fully verified domain of our siene. It also ontain impliitly the photohemial equivalent law, as I have already shown. Method for Calulating the Motion of Moleules in Radiation Fields We now turn our attention to the investigation of the motion imparted to our moleules by the radiation field. we make use in this of a method that is known to us from the theory of Brownian motion and whih I have often used in investigating motions in a region ontaining radiation. To simplify the alulation, we shall arry it through for the ase in whih the motion ours only along the Xdiretion of the oordinate system. We further ontent ourselves with alulating the mean value of the kineti energy of the translation motion, and thus dispense with proof that these veloities v are distributed aording to the Maxwell law. Let the mass M of the moleule be large enough so that higher powers of v an be negleted relative to lower ones; we an then apply the usual mehanis to the moleule. Moreover, without any loss in generality, we may arry out the alulation as through the states with indies m and n were the ones the moleule an be in. The momentum Mv of a moleule undergoes two kinds of hanges in the short time τ. Even though the radiation is the same in all diretions, the moleule, beause of its motion, will experiene a resistane to its motion that stems from the radiation. Let this opposing fore be Rv, where R is a onstant to be determined later. This fore would ultimately bring the moleule to rest if the randomness of the ation of the radiation field were not suh as to transfer to the moleule a momentum of alternating sign and varying magnitude; this random effet will, in opposition to the previous one, sustain a ertain amount of motion of the moleule. At the end of the 7
8 given short time τ the momentum of the moleule will equal Mv Rvτ + Sine the veloity distribution is to remain onstant in time, the mean of the absolute value of the above quantity must equal that of the quantity Mv; thus, the mean values of the squares of both quantities averaged over a long time or a large number of moleules must be equal: (Mv Rvτ + ) 2 = (Mv) 2 Sine we have taken into aount the influene of v on the momentum of the moleule separately, we must disard the mean value v. On developing the left hand side of the equation we thus obtain 2 = 2RMv 2 τ (10) The mean value v 2 whih the radiation of temperature T by its interation imparts to the moleule must just equal the mean value v 2 whih the gas moleule aquires at temperature T aording to the gas law and the kineti theory of gases. For otherwise the presene of our moleules would disturb the thermal equilibrium between thermal radiation and an arbitrary gas of the same temperature. We must therefore have Equation (10) thus goes over into Mv 2 2 = kt 2 (11) 2 τ = 2RkT (12) The investigation is now to be arried through as follows. For a given radiation density (ρ(ν)) we shall be able to ompute 2 and R by means of our hypotheses about the interation between radiation and moleules. If we put this result into (12), this equation will have to be identially satisfied when ρ is expressed as a funtion of ν and T by means of Plank s equation (4). Computing R Let a moleule of given kind be in uniform motion with speed v along the Xaxis of the oordinate system K. We inquire about the momentum transferred on the average from the radiation to the moleule per unit time. To 8
9 alulate this we must onsider the radiation from a oordinate system K that is at rest with respet to the given moleule. For we have formulated our hypotheses about emission and absorption only for moleules at rest. The transformation to the system K has often been performed in the literature. Nevertheless, I shall repeat the simple onsiderations here for the sake of larity. Relative to K the radiation is isotropi, that is, the quantity of radiation in a solid angle dκ in the diretion of the radiation in a frequeny range dν is ρdν dκ (13) 4π where ρ depends only on the frequeny ν but not on the diretion of the radiation. This speial beam orresponds to a speial beam in the system K whih is also haraterized by a frequeny range dν and a solid angle dκ. The volume density of this speial beam is ρ (ν, φ )dν dκ 4π (13 ) This defines ρ. It depends on the diretion of the radiation whih, in the familiar manner, is defined by the angle φ it makes with the X axis and whih its projetion on the Y, Z plane makes with the Y axis. These angles orrespond to the angles φ and ψ whih in an analogous manner determine the diretion of dκ in K. To begin with, it is lear that the same transformation law between (13) and (13 ) must hold as between the amplitudes A 2 and A 2 of a plane wave moving in the orresponding diretion. Hene, to our desired approximation we have ρ (ν, φ )dν dκ = 1 2 v os φ (14) ρ(ν)dνdκ or ρ (ν, φ ) = ρ(ν) dν dν dκ (1 dκ 2 v ) os φ (14 ) The relativity theory further gives the formulae, valid to the desired approximation, ν = ν (1 v ) os φ (15) os φ = os φ v + v os2 φ (16) ψ = ψ (17) 9
10 from (15) it follows, to the same approximation that ν = ν ( 1 + v os φ ) or Hene, again to the desired approximation ρ(ν) = ρ (ν + v ) ν os φ ρ(ν) = ρ(ν ) + ρ(ν ) ν ( v ν os φ ) Further, aording to (15), (16), and (17) dν dν = (1 + v ) os φ dκ dκ = sin φ dφ dψ sin φ dφ dψ = d(os φ) d(os φ ) = 1 2 v os φ (18) As a result of these two equations and equation (18), equation (14 ) goes over into [ ρ (ν, φ ) = (ρ) ν + v ( ) ] ρ ν os φ (1 3 v ) ν os φ (19) With the aid of (19) and our hypotheses about the radiation from and radiation onto moleules, we an easily alulate the average momentum transferred to the moleule per unit time. Before we an do this, however, we must say something to justify our proedure. It may be objeted that equations (14), (15), (16) are based on maxwell s theory of the eletromagneti field that is not onsistent with the quantum theory. This objetion deals, however, more with the form than with the substane of the problem. For, no matter how the theory of eletromagneti proesses may be formulated, in any ase the Doppler priniple and the law of aberration still remain, and hene also the equations (15) and (16). Moreover, the validity of the energy relationship (14) ertainly extends beyond that of the wave theory; this transformation law is also valid, for example, aording to relativity theory, for the energy density of a mass of infinitesimally small rest density that is moving with the [quasi] speed of light. We may therefore assert the validity of equation (19) for any theory of radiation. The radiation belonging to the solid angle dκ would, aording to (B), give rise to B m n ρ (ν, φ ) dκ 4π 10 ν
11 elementary proesses per seond of radiation events of the type Z n Z m if the moleule after eah suh proess immediately returned to state Z n. Atually, however, the time of lingering in state Z n, aording to (5), is where we have used the abbreviation 1 S p ne ɛ n kt S = p n e ɛ n kt + p m e ɛ m kt (20) The number of these proesses per seond is therefore atually 1 S p ne ɛ n kt B m n ρ (ν, φ ) = dκ 4π. In eah of these elementary proesses the momentum ɛ m ɛ n os φ is transferred to the moleule in the diretion of the X axis. In an analogous manner we find, based on (B ) that the orresponding number of elementary proesses of radiation events of type Z m Z n per seond is 1 S p me ɛ m kt Bmρ n (ν, φ ) dκ 4π and in eah suh elementary proess the momentum ɛ m ɛ n os φ is transferred to the moleule. The total momentum transferred to the moleule per unit time by inident radiation is, keeping in mind (6) and (9), hν S p nb m n ( e ɛ n kt e ɛ m kt ) ρ (ν, φ ) os φ dκ where the integration is to be taken over all solid angles. Carry this out, and we obtain with the aid of (19) value hν ( 2 S ρ (1/3) ν ρ ν ) p n B m n ( 4π e ɛ n kt e ɛ m kt ) v. 11
12 Here we have represented the effetive frequeny again with ν and not with ν. This expression gives, however, the total momentum transferred on the average to a moleule moving with speed v. For it is lear that those elementary proesses of emission of radiation not indued by the ation of the radiation field have no preferred diretion as seen from system K and hene, on the average, annot transfer any momentum to the moleule. We thus obtain as the final of our onsiderations R = hν ( 2 ρ 1/3 ν ρ ) p n B mn e ɛ ( n kt 1 e hν ) kt (21) S ν Calulating 2 It is muh easier to alulate the random effet of the elementary proesses on the mehanial behavior of the moleule. For we alulate this for a moleule at rest for whih the approximation whih we have been using applies. Let some event ause the momentum λ to be transferred to a moleule in the X diretion. This momentum is to be of varying magnitude and diretion from moment to moment. However, let λ obey a statistial law suh that its average value vanishes. Then let λ 1, λ 2... be the momenta whih are transferred to the moleule in the Xdiretion by various operating auses that are independent of eah other so that the total momentum that is transferred is = Σλ ν We then have (if for the individual λ ν vanish) 2 = Σλ ν 2 (22) If the mean values λ ν 2 of the individual momenta are all equal to eah other (= λ 2 ) and if l is the total number of proesses giving rise to momenta, we have the relation 2 = lλ 2 (22a) Aording to our hypothesis, in eah proess of inident radiation and outflowing radiation, the momentum λ = hν os φ 12
13 is transferred to the moleule. Here φ is the angle between the Xaxis and some randomly hosen diretion. Hene, we obtain λ 2 = 1 3 ( ) hν 2. Sine we assume that all the elementary proesses that are present are to be onsidered sas events that are independent of eah other, we may apply (22a), l is then the number of all elementary proesses that our in the time τ. This is twie as large as the number of radiationinident proesses Z n Z m in the time τ. We thus have From (23), (24) and (22) we thus obtain l = 2 S p nb m n e ɛ n kt ρτ (24) 2 τ = 2 3S ( ) hν 2 p n Bn m e ɛ n kt ρ (25) Results In order now to show that the momenta transferred from the radiation to the moleule aording to our basi hypotheses never disturb the thermodynami equilibrium, we need only introdue the values for 2 τ and R alulated in (25) and (21) respetively after the quantity ( ρ ( 1 / 3 ) ν ρ ) ( 1 e hν ) kt ν in (21) is replaed by ρhν 3RT from (4). We then see that our fundamental equation (12) is satisfied identially. The above onsideration lends very strong support to the hypotheses introdued earlier for the interation between matter and radiation by means of absorption and emission, and through inident and outgoing radiation. I was led these hypotheses in trying to postulate in the simplest possible way a quantum behavior of moleules that is analogous to the Plank resonators of lassial theory. We obtained, without effort, from the general 13
14 quantum assumption for matter, the seond Bohr rule (equation (9)) as well as Plank s radiation formula. Most important, however, appears to me the result about the momentum transferred to the moleule by inoming and outgoing radiation. If one of our hypotheses were altered, the result would be a violation of equation (12); it appears hardly possible, exept by way of our hypotheses, to be in agreement with this relationship whih is demanded by thermodynamis. We may therefore onsider the following as pretty well proven. If beam of radiation has the effet that a moleule on whih it is inident absorbs or emits an amount of energy h ν in the from of radiation by means of an elementary proess, then the momentum h ν / is always transferred to the moleule, and, to be sure, in the ase of absorption, in the diretion of the moving beam and in the ase of emission in the opposite diretion. If the moleule is subjet to the simultaneous ation of beams moving in various diretions, then only one of these taken part in any single elementary proess of inident radiation; this beam alone then determined the diretion of the momentum transferred to the moleule. If, through an emission proess, the moleule suffers a radiant loss of energy of magnitude h ν without the ation of an outside ageny, then this proess, too, is a direted one. emission in spherial waves does not our. Aording to the present state of the theory, the moleule suffers a reoil of magnitude h ν / in a partiular diretion only beause of the hane emission in that diretion. This property of elementary proesses as expressed by equation (12) makes a quantum theory of radiation almost unavoidable. The weakness of the theory lies, on the one hand, in its not bringing us loser to a union with the wave theory, and, on the other hand, that it leaves the time and diretion of the elementary proesses to hane; in spite of this, I have full onfidene in the trustworthiness of this approah. Only one more general remark. Almost all theories of thermal radiation rest on the onsiderations of the interation between radiation and moleules. But, in general, one is satisfied with dealing only with the energy exhange, without taking into aount the momentum exhange. One feels justified in this beause the momentum transferred by radiation is so small that it always drops out as ompared to that from other dynamial proesses. But for the theoretial onsiderations, this small effet is on an equal footing with the energy transferred by radiation beause energy and momentum are very intimately related to eah other; a theory may therefore be onsidered orret only if it an shown that the momentum transferred aordingly from the radiation to the matter leads to the kind of motion 14
15 that is demanded by thermodynamis. 15
From (2) follows, if z0 = 0, then z = vt, thus a2 =?va (2.3) Then 2:3 beomes z0 = z (z? vt) (2.4) t0 = bt + b2z Consider the onsequenes of (3). A ligh
Chapter 2 Lorentz Transformations 2. Elementary Considerations We assume we have two oordinate systems S and S0 with oordinates x; y; z; t and x0; y0; z0; t0, respetively. Physial events an be measured
More informationLectureXXI. Doppler effect & Relativistic dynamics
LetureXXI Speial Theory of Relativity Doppler effet & Relativisti dynamis The Doppler Effet: The Doppler effet of sound (in introdutory physis) is represented by an inreased frequeny of sound as a soure
More informationOn the recoil and Doppler shifts
Journal of Modern Optis Vol. 00, No. 00, DD Month 200x, 1 5 On the reoil and Doppler shifts Stephen M. Barnett Department of Physis, SUPA, University of Strathlyde, Glasgow G4 0NG, UK and Department of
More informationChapter 2 Particle properties of waves
Chapter Partile properties of waves Eletronis: partiles Eletromagneti wave: wave harge, mass Wave? diffration, interferene, Polarization partile?.1emwaves Wavepartile Duality Changing magneti field urrent
More informationTheory of linear elasticity. I assume you have learned the elements of linear
Physial Metallurgy Frature mehanis leture 1 In the next two letures (Ot.16, Ot.18), we will disuss some basis of frature mehanis using ontinuum theories. The method of ontinuum mehanis is to view a solid
More informationMirror plane (of molecule) 2. the Coulomb integrals for all the carbon atoms are assumed to be identical. . = 0 : if atoms i and j are nonbonded.
6 Hükel Theory This theory was originally introdued to permit qualitative study of the πeletron systems in planar, onjugated hydroarbon moleules (i.e. in "flat" hydroarbon moleules whih possess a mirror
More informationChapter 5 Single Phase Systems
Chapter 5 Single Phase Systems Chemial engineering alulations rely heavily on the availability of physial properties of materials. There are three ommon methods used to find these properties. These inlude
More informationClassical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk
Classial Eletromagneti Doppler Effet Redefined Copyright 04 Joseph A. Rybzyk Abstrat The lassial Doppler Effet formula for eletromagneti waves is redefined to agree with the fundamental sientifi priniples
More informationPY1002: Special Relativity
PY100: Speial Relativity Notes by Chris Blair These notes over the Junior Freshman ourse given by Dr. Barklie in Mihaelmas Term 006. Contents 1 Galilean Transformations 1.1 Referene Frames....................................
More informationChapter 2 Michelson Morley Experiment and Velocity of Light
Chapter 2 Mihelson Morley Experiment and Veloity of Light 2. Attempts to Loate Speial Privileged Frame Sientists tried to determine relative veloity of light with respet to earth. For this purpose, the
More informationRole of the Reference Frame in Angular Photon Distribution at ElectronPositron Annihilation
Journal of Modern Physis,, 5, 353358 Published Online April in SiRes. http://www.sirp.org/journal/jmp http://dx.doi.org/.36/jmp..565 Role of the Referene Frame in Angular Photon Distribution at EletronPositron
More informationExperimental demonstration of Doppler spectral broadening using the PC sound card
APPARATUS AND DEMONSTRATION NOTES Jeffrey S. Dunham, Editor Department of Physis, Middlebury College, Middlebury, Vermont 05753 This department welomes brief ommuniations reporting new demonstrations,
More informationRelativity Worked Solutions
Physis Option Problems  Relativisti Addition of Veloities (from Gianoli 6 th edition 43. (I A person on a roket traveling at.5 (with respet to the Earth observes a meteor ome from behind and pass her
More informationLecture 5. Gaussian and GaussJordan elimination
International College of Eonomis and Finane (State University Higher Shool of Eonomis) Letures on Linear Algebra by Vladimir Chernyak, Leture. Gaussian and GaussJordan elimination To be read to the musi
More informationTaylor s Formula G. B. Folland
Taylor s Formula G. B. Folland There s a lot more to be said about Taylor s formula than the brief disussion on pp.113 4 of Apostol. Let me begin with a few definitions. Definitions. A funtion f defined
More informationExponential Maps and Symmetric Transformations in ClusterSpin. YouGang Feng
Exponential Maps and Symmetri Transformations in ClusterSpin System for Lattie Ising Models YouGang Feng Department of Basi Sienes, College of Siene, Guizhou University, Caijia Guan, Guiyang, 550003
More informationSpecial Relativity and Linear Algebra
peial Relativity and Linear Algebra Corey Adams May 7, Introdution Before Einstein s publiation in 95 of his theory of speial relativity, the mathematial manipulations that were a produt of his theory
More informationSet Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos)
A.1 Set Theory and Logi: Fundamental Conepts (Notes by Dr. J. Santos) A.1. Primitive Conepts. In mathematis, the notion of a set is a primitive notion. That is, we admit, as a starting point, the existene
More information) ( )( ) ( ) ( )( ) ( ) ( ) (1)
OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples
More informationChapter 38A  Relativity. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University
Chapter 38A  Relativity A PowerPoint Presentation by Paul E. Tippens, Professor of Physis Southern Polytehni State University 27 Objetives: After ompleting this module, you should be able to: State and
More informationNotes on Coaxial Transmission Lines
ELEC344 Mirowave Engineering Handout#4 Kevin Chen Notes on Coaxial Transmission Lines TEM Fields in Coaxial Transmission Lines Beause it is a twowire transmission line, oaxial line an support TEM waves.
More informationRelativistic Analysis of Doppler Effect and Aberration based on Vectorial Lorentz Transformations
Uniersidad Central de Venezuela From the SeletedWorks of Jorge A Frano June, Relatiisti Analysis of Doppler Effet and Aberration based on Vetorial Lorentz Transformations Jorge A Frano, Uniersidad Central
More informationAn analysis of the classical Doppler effect
Home Searh Colletions Journals About Contat us My IOPsiene An analysis of the lassial Doppler effet This artile has been downloaded from IOPsiene. Please sroll down to see the full text artile. 2003 Eur.
More informationThe Doppler effect from a uniformly moving mirror
The Doppler effet from a uniformly moing mirror Aleksar Gjurhinoski Department of Physis, Faulty of Natural Sienes Mathematis, Sts. Cyril Methodius Uniersity, P. O. Box 16, 1 Skopje, Maedonia Eletroni
More informationChemical Equilibrium. Chemical Equilibrium. Chemical Equilibrium. Chemical Equilibriu m. Chapter 14
Chapter 14 Chemial Equilibrium Chemial Equilibriu m Muh like water in a Ushaped tube, there is onstant mixing bak and forth through the lower portion of the tube. reatants produts It s as if the forward
More informationMansuripur s Paradox
1 Problem Mansuripur s Paradox Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (May 2, 2012) An eletrially neutral urrentloop, with magneti dipole m 0,thatisatrestinastati,
More informationImpact Simulation of Extreme Wind Generated Missiles on Radioactive Waste Storage Facilities
Impat Simulation of Extreme Wind Generated issiles on Radioative Waste Storage Failities G. Barbella Sogin S.p.A. Via Torino 6 00184 Rome (Italy), barbella@sogin.it Abstrat: The strutural design of temporary
More informationRestricted Least Squares, Hypothesis Testing, and Prediction in the Classical Linear Regression Model
Restrited Least Squares, Hypothesis Testing, and Predition in the Classial Linear Regression Model A. Introdution and assumptions The lassial linear regression model an be written as (1) or (2) where xt
More informationSequences CheatSheet
Sequenes CheatSheet Intro This heatsheet ontains everything you should know about real sequenes. It s not meant to be exhaustive, but it ontains more material than the textbook. Definitions and properties
More informationCapacity at Unsignalized TwoStage Priority Intersections
Capaity at Unsignalized TwoStage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minorstreet traffi movements aross major divided fourlane roadways
More information10.1 The Lorentz force law
Sott Hughes 10 Marh 2005 Massahusetts Institute of Tehnology Department of Physis 8.022 Spring 2004 Leture 10: Magneti fore; Magneti fields; Ampere s law 10.1 The Lorentz fore law Until now, we have been
More informationCHAPTER 14 Chemical Equilibrium: Equal but Opposite Reaction Rates
CHATER 14 Chemial Equilibrium: Equal but Opposite Reation Rates 14.1. Collet and Organize For two reversible reations, we are given the reation profiles (Figure 14.1). The profile for the onversion of
More informationFundamentals of Chemical Reactor Theory
UNIVERSITY OF CALIFORNIA, LOS ANGELES Civil & Environmental Engineering Department Fundamentals of Chemial Reator Theory Mihael K. Stenstrom Professor Diego Rosso Teahing Assistant Los Angeles, 3 Introdution
More informationSPECIAL RELATIVITY. MATH2410 KOMISSAROV S.S
SPECIAL RELATIVITY. MATH2410 KOMISSAROV S.S 2012 2 Contents Contents 2 1 Spae and Time in Newtonian Physis 9 1.1 Spae............................................ 9 1.1.1 Einstein summation rule..............................
More informationPHY 140Y FOUNDATIONS OF PHYSICS Tutorial Questions #12 Solutions December 3/4
PHY 40Y FOUNDTIONS OF PHYSICS 0000 Tutorial Questions # Solutions Deember 3/4 orentz Veloity ddition, Time Dilation, and orentz Contration. Two spaeships are traelling with eloities of 0.6 and 0.9 relatie
More informationSolution Derivations for Capa #12
Solution Derivations for Capa #1 1) An ideal stepdown transformer has a primary oil of 300 turns and a seondary oil of 5 turns. It is plugged into an outlet with 10 V (AC) from whih it draws a urrent
More informationIsaac Newton. Translated into English by
THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY (BOOK 1, SECTION 1) By Isaa Newton Translated into English by Andrew Motte Edited by David R. Wilkins 2002 NOTE ON THE TEXT Setion I in Book I of Isaa
More informationSebastián Bravo López
Transfinite Turing mahines Sebastián Bravo López 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that
More informationarxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 USA
LBNL52402 Marh 2003 On the Speed of Gravity and the v/ Corretions to the Shapiro Time Delay Stuart Samuel 1 arxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrene Berkeley National Laboratory
More informationVibration Theory Fundamentals
Vibration Theory Fundamentals 1 1.1 INTRODUCTION Mehanial vibrating systems onsist of elements suh as a spring for storing potential energy, mass and inertia for kineti energy, and damper for dissipating
More informationChapter 5 lecture Notes
Lebanese Amerian University From the SeletedWorks of Ghassan Dibeh 04 Chapter 5 leture Notes Ghassan Dibeh, Lebanese Amerian University Available at: http://works.bepress.om/ghassan_dibeh/00/ Chapter
More informationCONDITIONAL PROBABILITY (continued) ODDS RATIO ( ) P( H ) Provides nformation on how much more likely that an event A occurs than that it does not.
CONDITIONAL PROBABILITY (ontinued) ODDS RATIO Def: The odds ratio of an event A is defined by 1 Provides nformation on how muh more likely that an event A ours than that it does not. Its value an hange
More informationPhase Velocity and Group Velocity for Beginners
Phase Veloity and Group Veloity for Beginners Rodolfo A. Frino January 015 (v1)  February 015 (v3) Eletronis Engineer Degree from the National University of Mar del Plata  Argentina Copyright 014015
More informationChapter 1 Microeconomics of Consumer Theory
Chapter 1 Miroeonomis of Consumer Theory The two broad ategories of deisionmakers in an eonomy are onsumers and firms. Eah individual in eah of these groups makes its deisions in order to ahieve some
More informationINTRODUCTION.
"Explaining the Illusion of the Constant Veloity of Light." Paul Marmet 401 Ogilvie Road, Glouester, Ontario, Canada K1J 7N4 EMail: Paul.Marmet@Ottawa.om ABSTRACT. Considering that photons travel at the
More informationExperimental Determination of the Speed of Light by the Foucault Method
Experimental Determination of the Speed of Light y the Fouault Method R. Prie and J. Zizka University of Arizona 1 Introdution The speed of light was measured using the Fouault method of refleting a eam
More informationDerivation of Einstein s Equation, E = mc 2, from the Classical Force Laws
Apeiron, Vol. 14, No. 4, Otober 7 435 Derivation of Einstein s Equation, E = m, from the Classial Fore Laws N. Hamdan, A.K. Hariri Department of Physis, University of Aleppo, Syria nhamdan59@hotmail.om,
More informationVapor Pressure of a Solid by Knudsen Effusion
Vapor Pressure of a Solid by Knudsen Effusion Readings: Atkins (8 th ed) pages 1718 and 746757; Garland, Nibler, and Shoemaker (GNS) (8 th ed) pages 11917, 587594, and 597. Introdution Methods of measuring
More information10 UNSTEADY FLOW IN OPEN CHANNELS
0 UNTEY FLOW IN OEN CHNNEL 0. Introdution Unsteady flow in open hannels differs from that in losed onduits in that the eistene of a free surfae allows the flow rosssetion to freely hange, a fator whih
More informationChannel Assignment Strategies for Cellular Phone Systems
Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious
More informationThe Doppler effect from a uniformly moving mirror
The Doppler effet from a uniformly moing mirror Aleksar Gjurhinoski Department of Physis, Faulty of Natural Sienes Mathematis, Sts. Cyril Methodius Uniersity, P. O. Box 16, 1 Skopje, Maedonia Eletroni
More informationComputer Networks Framing
Computer Networks Framing Saad Mneimneh Computer Siene Hunter College of CUNY New York Introdution Who framed Roger rabbit? A detetive, a woman, and a rabbit in a network of trouble We will skip the physial
More informationA novel active mass damper for vibration control of bridges
IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 37 July 008, Seoul, Korea A novel ative mass damper for vibration ontrol of bridges U. Starossek & J. Sheller Strutural
More informationIntuitive Guide to Principles of Communications By Charan Langton www.complextoreal.com
Intuitive Guide to Priniples of Communiations By Charan Langton www.omplextoreal.om Understanding Frequeny Modulation (FM), Frequeny Shift Keying (FSK), Sunde s FSK and MSK and some more The proess of
More informationb) Estimate the relevant electrooptical coefficient in the material (0.7p)
Exam. Advaned Optis and Lasers 85 (8) Allowed tools: Textbook, your own notes, a alulator, your favourite physis and mathematis tables & formulas. Some plots and tables are shown on a separate sheet
More informationMagnetic Materials and Magnetic Circuit Analysis
Chapter 7. Magneti Materials and Magneti Ciruit Analysis Topis to over: 1) Core Losses 2) Ciruit Model of Magneti Cores 3) A Simple Magneti Ciruit 4) Magneti Ciruital Laws 5) Ciruit Model of Permanent
More informationA OneDimensional Solution of the Photoacoustic Wave Equation and its Relationship with Optical Absorption
Int J Thermophys (2013) 34:1473 1480 DOI 10.1007/s1076501315201 A OneDimensional Solution of the Photoaousti Wave Equation and its Relationship with Optial Absorption D. Cywiak M. D. BarreiroArgüelles
More informationAmplitude Modulation
Amplitude Modulation Table of ontents Modulation theory... 3 Amplitude modulation... 4 Modulation Index... 6 Power in an AM waveform... 9 Double Side Band Suppressed Carrier... 10 Single Sideband Modulation
More informationRelativistic Kinematics a project in Analytical mechanics Karlstad University
Relativisti Kinematis a projet in Analytial mehanis Karlstad University Carl Stigner 1th January 6 Abstrat The following text is a desription of some of the ontent in hapter 7 in the textbook Classial
More informationHEAT CONDUCTION. q A q T
HEAT CONDUCTION When a temperature gradient eist in a material, heat flows from the high temperature region to the low temperature region. The heat transfer mehanism is referred to as ondution and the
More informationGravity and the quantum vacuum inertia hypothesis
Ann. Phys. (Leipzig) 14, No. 8, 479 498 (2005) / DOI 10.1002/andp.200510147 Gravity and the quantum vauum inertia hypothesis Alfonso Rueda 1, and Bernard Haish 2, 1 Department of Eletrial Engineering,
More informationIntroducing Faraday's Law
Introduing Faraday's Law L. L. Tankersley and Eugene P. Mosa Physis Department United tates Naval Aademy Annapolis, MD 21402 tank@usna.edu Faraday's Law Tankersley and Mosa 5/27/2013 page 1 This note is
More informationLecture 5: Voltage Drops
1 Leture 5: Voltage Drops 1. Soures of voltage drops  Ohmi drop  Conentration overpotential  Surfae overpotential 2. Porous eletrodes 1. Soures of voltage drops in an eletrohemial ell The equilibrium
More informationISOMETRIES OF THE HYPERBOLIC PLANE
ISOMETRIES OF THE HYPERBOLIC PLANE KATHY SNYDER Abstrat. In this paper I will define the hyperboli plane and desribe and lassify its isometries. I will onlude by showing how these isometries an be represented
More informationOn the Notion of the Measure of Inertia in the Special Relativity Theory
www.senet.org/apr Applied Physis Researh Vol. 4, No. ; 1 On the Notion of the Measure of Inertia in the Speial Relativity Theory Sergey A. Vasiliev 1 1 Sientifi Researh Institute of Exploration Geophysis
More informationON THE ELECTRODYNAMICS OF MOVING BODIES
ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 905 It is known that Maxwell s eletrodynamis as usually understood at the present time when applied to moing bodies, leads to asymmetries
More informationON THE ELECTRODYNAMICS OF MOVING BODIES
ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 905 It is known that Maxwell s eletrodynamis as usually understood at the present time when applied to moing bodies, leads to asymmetries
More informationAUDITING COST OVERRUN CLAIMS *
AUDITING COST OVERRUN CLAIMS * David PérezCastrillo # University of Copenhagen & Universitat Autònoma de Barelona Niolas Riedinger ENSAE, Paris Abstrat: We onsider a ostreimbursement or a ostsharing
More informationOptical Doppler Effect in a Medium. C. H. Chang, J. H. Hsiao, and T. M. Hong
CHINESE JOURNAL OF PHYSICS VOL. 47, NO. 4 AUGUST 2009 Optial Doppler Effet in a Medium C. H. Chang, J. H. Hsiao, and T. M. Hong Department of Physis, National Tsing Hua University, Hsinhu 30043, Taiwan,
More informationHEAT EXCHANGERS2. Associate Professor. IIT Delhi Email: prabal@mech.iitd.ac.in. P.Talukdar/ MechIITD
HEA EXHANGERS2 Prabal alukdar Assoiate Professor Department of Mehanial Engineering II Delhi Email: prabal@meh.iitd.a.in Multipass and rossflow he subsripts 1 and 2 represent the inlet and outlet, respetively..
More informationSpecial Theory of Relativity An Introduction
Speial Theory of Relatiity An Introdution DJM 8/ Bakground Most basi onept of relatiity is as old as Galilean and Newtonian Mehanis. Crudely speaking, it is: The Laws of Physis look the same in different
More informationINCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS
Virginia Department of Taxation INCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS www.tax.virginia.gov 2614086 Rev. 07/14 * Table of Contents Introdution... 1 Important... 1 Where to Get Assistane... 1 Online
More informationPolynomials INTRODUCTION CHAPTER 3 OUTLINE. Chapter 3 :: Prerequisite Test 182. Exponents and Polynomials 183
bar92103_h03_a_181219.qxd 9/19/09 12:03 PM Page 181 C H A P T E R hapter The MGrawHill Companies. All Rights Reserved. The Streeter/Huthison Series in Mathematis Beginning Algebra 3 > Make the Connetion
More informationPlasma Waves Reflection from a Boundary with Specular Accommodative Boundary Conditions 1
ISSN 965545, Computational Mathematis and Mathematial Physis,, Vol. 5, No. 8, pp. 4 446. Pleiades Publishing, Ltd.,. Published in Russian in Zhurnal Vyhislitel noi Matematiki i Matematiheskoi Fiziki,,
More informationCurious George and the. Choose 3 out of 4 problems Fall 2006 Test 1 09/27/06
Contemporary Physis SOLUTION KEY Curious George and the. Choose 3 out of 4 problems Fall 006 Test 1 09/7/06 1) Curious George is on a roket ship that is heading out to a planet that is 6 lightyears away
More informationConversion of short optical pulses to terahertz radiation in a nonlinear medium: Experiment and theory
PHYSICAL REVIEW B 76, 35114 007 Conversion of short optial pulses to terahertz radiation in a nonlinear medium: Experiment and theory N. N. Zinov ev* Department of Physis, University of Durham, Durham
More informationNonideal gases in diesel engine processes
First Biennial Meeting and General Setion Meeting of The SandinavianNordi Setion of the Combustion Institute, Chalmers Univ. of Tehnol., Gothenburg, (Sweden) April 18, 1 Nonideal gases in diesel engine
More informationDoppler Phenomenon for Electromagnetic Waves and New Equations for Quantum Relativity
Doppler Phenomenon or Eletromagneti Waes and New Equations or Quantum Relatiity Mohammadreza Tabatabaei ontat@ki00.net What is the Doppler eet? Beause o the Doppler phenomenons importane in osmology, it
More informationLinear description of ultrasound imaging systems
Linear desription of ultrasound imaging systems Notes for the International Summer Shool on Advaned Ultrasound Imaging Tehnial University of Denmark July 5 to July 9, 1999 Release 1.01, June 29, 2001 Jørgen
More information2 Absorption and Emission of Radiation
Chapter, page Absorption and Emission of Radiation. Eletromagneti Radiation In the year 886, Heinrih Hertz experimentally demonstrated the existene of eletromagneti waves and their equivalene to light
More informationEffects of InterCoaching Spacing on Aerodynamic Noise Generation Inside Highspeed Trains
Effets of InterCoahing Spaing on Aerodynami Noise Generation Inside Highspeed Trains 1 J. Ryu, 1 J. Park*, 2 C. Choi, 1 S. Song Hanyang University, Seoul, South Korea 1 ; Korea Railroad Researh Institute,
More informationRelativity in the Global Positioning System
Relativity in the Global Positioning System Neil Ashby Department of Physis,UCB 390 University of Colorado, Boulder, CO 8030900390 NIST Affiliate Email: ashby@boulder.nist.gov July 0, 006 AAPT workshop
More informationElectric/magnetic dipole in an electromagnetic field: force, torque and energy
Eur. Phys. J. Plus (2014) 129: 215 DOI 10.1140/epjp/i201414215y Regular Artile THE EUROPEAN PHYSICAL JOURNAL PLUS Eletri/magneti dipole in an eletromagneti field: fore, torque and energy Alexander Kholmetskii
More informationChapter 3 Cosmology 3.1 The Doppler effect
Chapter 3 Cosmology 3.1 The Doppler effet Learning objeties: Why does the waelength of waes from a moing soure depend on the speed of the soure? What is a Doppler shift? How an we measure the eloity of
More informationEE 201 ELECTRIC CIRCUITS LECTURE 25. Natural, Forced and Complete Responses of First Order Circuits
EE 201 EECTRIC CIUITS ECTURE 25 The material overed in this leture will be as follows: Natural, Fored and Complete Responses of First Order Ciruits Step Response of First Order Ciruits Step Response of
More information5. Hypothesis Tests About the Mean of a Normal Population When σ 2 is Not Known
Statistial Inferene II: he Priniples of Interval Estimation and Hypothesis esting 5. Hypothesis ests About the Mean of a Normal Population When σ is Not Known In Setion 3 we used the tdistribution as
More informationMath 115 HW #9 Solutions
1. Solve the differential equation Math 115 HW #9 Solutions sin x dy dx + (os x)y = x sin(x2 ). Answer: Dividing everything by sin x yields the linear first order equation Here P (x) = os x sin x and Q(x)
More informationTHICKNESS MEASURING OF NICKEL COATINGS ON STEELS WITH VARIOUS MAGNETIC PROPERTIES
THICKNESS MEASURING OF NICKEL COATINGS ON STEELS WITH VARIOUS MAGNETIC PROPERTIES A. A. Lukhvih, O. V. Bulatov, A. L. Lukyanov, and A. A. Polonevih Institute of Applied Physis of the National Aademy of
More informationPhysics 53. Wave Motion 1
Physics 53 Wave Motion 1 It's just a job. Grass grows, waves pound the sand, I beat people up. Muhammad Ali Overview To transport energy, momentum or angular momentum from one place to another, one can
More information2D waves and other topics
Chapter 7 2D waves and other topis David Morin, morin@physis.harvard.edu This hapter is fairly short. In Setion 7.1 we derive the wave equation for twodimensional waves, and we disuss the patterns that
More informationUser s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities
File:UserVisfit_2.do User s Guide VISFIT: a omputer tool for the measurement of intrinsi visosities Version 2.a, September 2003 From: Multiple Linear LeastSquares Fits with a Common Interept: Determination
More informationA Keyword Filters Method for Spam via Maximum Independent Sets
Vol. 7, No. 3, May, 213 A Keyword Filters Method for Spam via Maximum Independent Sets HaiLong Wang 1, FanJun Meng 1, HaiPeng Jia 2, JinHong Cheng 3 and Jiong Xie 3 1 Inner Mongolia Normal University 2
More informationTHERMAL TO MECHANICAL ENERGY CONVERSION: ENGINES AND REQUIREMENTS Vol. I  Thermodynamic Cycles of Reciprocating and Rotary Engines  R.S.
THERMAL TO MECHANICAL ENERGY CONVERSION: ENGINES AND REQUIREMENTS Vol. I  Thermodynami Cyles of Reiproating and Rotary Engines  R.S.Kavtaradze THERMODYNAMIC CYCLES OF RECIPROCATING AND ROTARY ENGINES
More informationIdentification Method in the Dynamics of Rotor Bearing Systems Using Measurement Data of Unbalance Responses
International Journal of Emerging Tehnology and Advaned Engineering Website: www.ijetae.om ISSN 50459, ISO 900:008 Certified Journal, Volume, Issue 8, August 0 Identifiation Method in the Dynamis of otor
More informationWeighting Methods in Survey Sampling
Setion on Survey Researh Methods JSM 01 Weighting Methods in Survey Sampling Chiaohih Chang Ferry Butar Butar Abstrat It is said that a welldesigned survey an best prevent nonresponse. However, no matter
More informationINCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS
Virginia Department of Taxation INCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS www.tax.virginia.gov 2614086 Rev. 01/16 Table of Contents Introdution... 1 Important... 1 Where to Get Assistane... 1 Online File
More informationTrigonometry & Pythagoras Theorem
Trigonometry & Pythagoras Theorem Mathematis Skills Guide This is one of a series of guides designed to help you inrease your onfidene in handling Mathematis. This guide ontains oth theory and exerises
More informationENERGY METEOROLOGY. UNIT 1: Basics of Radiation. Energy transfer through radiation. Concept of blackbody radiation, Kirchhoff s law
UNIT 1: Basics of Radiation Energy transfer through radiation Concept of blackbody radiation, Kirchhoff s law Radiation laws of Planck, StefanBoltzmann and Wien Radiation quantities Examples Radiative
More informationINVESTIGATION OF STABILITY ON NONLINEAR OSCILATIONS WITH ONE DEGREE OF FREEDOM
91 Kragujeva J. Math. 25 (2003) 91 96. INVESTIGATION OF STABILITY ON NONLINEAR OSCILATIONS WITH ONE DEGREE OF FREEDOM Julka KneževićMiljanović 1 and Ljubia Lalović 2 1 Mathematis faulty of Belgrade, Serbia
More informationComay s Paradox: Do Magnetic Charges Conserve Energy?
Comay s Paradox: Do Magneti Charges Conserve Energy? 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (June 1, 2015; updated July 16, 2015) The interation energy
More information