A proposed experiment for measuring the speed of propagation of the Coulomb force.

Transcription

1 A proposed experiment for measuring the speed of propagation of the Coulomb fore. January 29, Introdution The eletri field at a time t due to an eletrial harge moving with veloity v is given, using the Lienard-Wiehert retarded potentials, by 1 v2 R(τ) e E(t) =e 2 (R(τ) v )+ (R(τ) R(τ) v ) 3 2 (R(τ) R(τ) v ) R(τ) [(R(τ) v ) v ] 3 (1) where the vetor radius R(τ)is evaluated at the time τ = t R(τ), being t the time when we want to have the field E(t). It is known [1, 2, 3] that, very surprisingly, the first term, the field obtained for onstant veloity, is idential to and E(t) = er(t) R(t) 3 1 v 2 2 (1 v2 2 sin 2 (θ) 3 2 (2) H = 1 v E (3) where θ is the angle between v and R(τ) (Eqs and 38.9 of [1]) and R(t) is the vetor radius between the point where the harge is loated at the time t and the point P where we want the field, at the same time t 1. This an be easily seen by substituting 2 in Eq.1 the expression: R(t) =R(τ) v R(τ) As well known, if the partile veloity is not onstant, the seond term in Eq. 1 in non null and represents an eletromagneti field whih propagates, in vauum, with the speed of light (em wave). 1 In Landau s words:...the distane R(t) at preisely the moment of observation (see pag.162 in [1] ). 2 For the ase of onstant veloity v. (4) 1

2 Figure 1: The two omponents of the eletrial field, indiated with the blue olor: the first one direted from the retarded position of the harge and the seond one direted as -v. In other words, we have two ontributions to the eletrial field: the first one of Eq.1 produing a field that one ould alulated by onsidering the propagation of the field with infinite veloity and a seond term produing a field that propagates with the speed of light. The idea that the first term ould be attributed to a field propagating with infinite veloity is not, by most of the sientifi ommunity, onsidered valid. Aording to Feynman [4] the: influene of the harge omes, in a ertain sense, from the retarded position; but beause the motion is exatly speified, the retarded position is uniquely given in terms of then present position. Aording to Carlip [5]: This effet does not mean that the eletri field propagates instantaneously, rather, the field of a moving harge has a veloity-dependent omponent that anels the effet of propagation delay to first order. That the field of a moving harge have a veloity-dependent omponent that anels the effet of propagation delay is orret as an be seen in the Eq.1 and in fig.1 In our opinion the ontribution of the veloity-dependent omponent does not have a physial meaning, and appears just a mathematial term whih ompensate the wrong physial assumption of using the retarded potential method for the Coulomb field. 2

3 It is also interesting to report the onsiderations in the text book by Beker [2]: Il ampo di un elettrone in moto qualunque risulta dalla sovrapposizione di due ampi, dei quali il primo può interpretarsi ome ampo elettrostatio he aompagna il moto della partiella, mentre il seondo ha il arattere di un onda elettromagnetia. No explanation is given for the ampo elettrostatio he aompagna il moto della partiella. Does this means that the entire oulomb field travels together with the eletron, thus without retarded effets inside this field? 2 The fore of gravity There is a strong similarity between the eletrial and gravitational fores. From ref. [6] In the simple Newtonian model, gravity propagates instantaneously: the fore exerted by a massive objet points diretly toward that objet s present position. For example, even though the Sun is 500 light seonds from the Earth, Newtonian gravity desribes a fore on Earth direted towards the Sun s position now, not its position 500 seonds ago. Putting a light travel delay into Newtonian gravity would make orbits unstable, leading to preditions that learly ontradit Solar System observations. The problem of the instability of the solar system planetary orbits, if gravity does not propagates instantaneously, was well known to Newton himself and to Laplae, who alulated an upper limit for the veloity of propagation of gravity of the order of 10 8 times the veloity of light [7](1825, pp of translation). The problem is also onsidered by Eddington [3] who, however, assumes a veloity of propagation of gravity equal to that of light beause the same problem ours in eletromagnetism where, as well known, the maximum veloity is the veloity of light. In other words, the problem of propagation of the Coulomb and Newton field exists, but it is often either ignored without a lear explanation. Hene, the neessity to add experimental data to our knowledge 3. 3 Calulating the Coulomb eletrial field In order to investigate this problem we onsider the ase treated by Feynman of an eletri harge moving with onstant veloity. The alulation of the eletri field properly done by means of the retarded potential gives, eventually, the two 3 Per una aria elettria in moto uniforme il ampo puó essere alolato in due modi: a)mettendosi nel sistema di riferimento in ui la aria è ferma e muovendo il punto in ui si misura il ampo. In questo aso hiaramente il risultato è quello espresso dalla Eq. 2. b) Oppure stando fermi on lo strumento di misura e muovendo la aria elettria. Seondo la relativitá si deve avere lo stesso risultato e osì infatti troviamo, ma non possiamo non domandari: il ampo elettrio di Coulomb si muove assieme alla aria?. Se fosse osí si avrebbero grosse onsequenze quando si dovesse appliare questo al ampo gravitazionale delle stelle lontane. 3

4 Table 1: The times of the signals in the four sensors for the various models. model t 1 t 2 t 3 t 4 ns ns ns ns e.m.wave oulomb with light veloity oulomb with infinite veloity omponents q E y = 4πɛ o 1 v 2 q E x = 4πɛ o 1 v 2 y [ (x vt)2 1 v 2 + y 2 ] 3 2 x vt [ (x vt)2 1 v 2 + y 2 ] 3 2 Fig.2 shows that the alulated field at (x,y) and at the time t is just that obtained onsidering the position of the harge at the same time t. It is lear that the eletri field is entral, referred to the present position of the harge. If the harge veloity is near the veloity of light the relativisti effets an be relevant, as illustrated in the fig.3. The relativisti effet enhane the intensity of the field when the harge is nearest and diminish the intensity in the other plaes. This is learly shown in the fig.4 referred to the proposed experiment. (5) (6) 4 The proposed experiment We propose to use the eletrons produed by the linear aelerator of DAΦNE. The DAΦNE Beam Test Faility (BTF) is a beam transfer line optimized for the prodution of a defined number of eletrons or positrons, in a wide range of multipliities and down to single-eletron mode, in the energy range between 50 and 800 MeV. The typial pulse duration is 1ns or 10 ns and the maximum repetition rate is 50 Hz. We propose to measure the eletrial field as funtion of time in the predetermined loations shown in the fig.5. These loations have been hosen onsidering that we have three possibilities: The signals observed by the sensors are due to eletromagneti waves emitted when the eletrial harges exit from the linear aelerator. The signals are due to the oulomb field whih travels with the veloity of light The signals are due to the oulomb field whih propagates with infinite veloity. For eletrons exiting the aelerators at time zero the four sensors shown in the fig.5 observe signals at the times indiated in the Table 1. From preliminary tests we have verified that with the available instrumentation signals due the eletron beam is observed with SNR greater than one order of magnitude and that it is possible to measure a time differene between two sensors of the order of 0.2 ns. 4

5 Figure 2: The field is alulated at the point with oordinates (x,y) at the time t. The harge begins its motion at x=y=0 in the x diretion. 5

6 6 Figure 3: In the upper graph the eletri field lines of fore in the referene system where the harge is at rest; in the lower graph the lines of fore where the harge is moving with veloity v =0.9.

7 Figure 4: The expeted eletri field, at x=y=1 m, versus the position of the harge= eletroni harges, for various total energies. 7

8 Figure 5: The position of the four sensors. The red arrow indiates the eletron beam. 8

9 Referenes [1] L.D.Landau and E.M.Lifshitz The lassial theory of fields, pag.162, Pergamon Press, Oxford (1971) [2] R.Beker Teoria della elettriità, pag Sansoni Ed. Sientifihe (1950) [3] A.Eddington Spae, Time and Gravitation, Harper Torhbooks, pag 94 (1959) [4] R. Feynman, R.B. Leighton, M.L. Sands, The Feynman Letures on PhysisAddison-Wesley, Redwood City, vol. II, Chapter 21 (1989) [5] S.Carlip Physis Letters A267, (2000) [6] speed.html [7] Laplae, P., Mehanique Celeste, volumes published from , English translation reprinted by Chelsea Publ., New York (1966). 9