elettromagnetica Giovanni Romano 1 febbraio 2013 Accademia di Scienze Fisiche e Matematiche in Napoli Una teoria consistente dell induzione

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1 Accademia di Scienze Fisiche e Matematiche in Napoli 1 febbraio 2013

2 ED=Electrodynamics and DG=Differential Geometry

3 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED

4 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Johann Carl Friederich Gauss ( )

5 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Hermann Günther Grassmann ( )

6 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Bernhard Riemann ( )

7 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Marius Sophus Lie ( )

8 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Felix Klein ( )

9 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Gregorio Ricci-Curbastro ( )

10 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Tullio Levi-Civita ( )

11 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Elie Cartan ( )

12 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Henri Cartan ( )

13 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Charles Ehresmann ( )

14 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Hassler Whitney ( )

15 ED=Electrodynamics and DG=Differential Geometry ED is an important source of inspiration for DG and DG is the natural tool to develop a mathematical modeling of ED Founders of DG Jean-Louis Koszul ( )

16 Electromagnetic Story Starring

17 Electromagnetic Story Starring Luigi Galvani ( )

18 Electromagnetic Story Starring Alessandro Volta ( )

19 Electromagnetic Story Starring Gian Domenico Romagnosi ( )

20 Electromagnetic Story Starring Hans Christian Ørsted ( )

21 Electromagnetic Story Starring André-Marie Ampère ( )

22 Electromagnetic Story Starring Johann Carl Friedrich Gauss ( )

23 Electromagnetic Story Starring Francesco Zantedeschi ( )

24 Electromagnetic Story Starring Joseph Henry ( )

25 Electromagnetic Story Starring Michael Faraday ( )

26 Electromagnetic Story Starring James Clerk-Maxwell ( )

27 Electromagnetic Story Starring Joseph John Thomson ( )

28 Electromagnetic Story Starring

29 Vector fields in

30 Vector fields in Josiah Willard Gibbs ( )

31 Vector fields in Oliver Heaviside ( )

32 Vector fields in Heinrich Rudolf Hertz ( )

33 Zero spatial velocity g E = µ ( θ=0 B) Henry Faraday(1831) Σ in Σ in µ B = 0 Σ out Gauss(1831) g H = Σ out µ ( θ=0 D + J) Σ out Ampère(1826) µ D = ρ µ Σ out Σ out Maxwell(1861) Gauss(1835) with Σ out a bounded connected surface and Σ out bounded connected domain in S. Applying Ampère law to closed surfaces Σ out = Ω, we get the equation of continuity. θ=0 ρ + div J = 0

34 Relativity Story Length contraction and time dilation effects Woldemar Voigt ( )

35 Relativity Story Length contraction and time dilation effects George Francis FitzGerald ( )

36 Relativity Story Length contraction and time dilation effects Hendrik Antoon Lorentz ( )

37 Relativity Story Length contraction and time dilation effects Albert Einstein ( )

38 Relativity Story Space-time metric Henri Poincaré ( )

39 Relativity Story Space-time metric Hermann Minkowski ( )

40 Relativity Story General relativity Christian Felix Klein ( )

41 Relativity Story General relativity David Hilbert ( )

42 Albert Einstein (1905)

43 ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 1905 It is known that Maxwell s electrodynamics as usually understood at the present time when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise assuming equality of relative motion in the two cases discussed to electric currents of the same path and intensity as those produced by the electric forces in the former case. Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the light medium, suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. 1 We will raise this conjecture (the purport of which will hereafter be called the Principle of Relativity ) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent of the electrodynamics of moving bodies based on Maxwell s for stationary bodies. The introduction of a luminiferous ether will 1 The preceding memoir by Lorentz was not at this time known to the author. 1

44 Maxwell-Hertz equations - (Einstein 1905) 1 X c t = N y M z 1 Y c t = L z N x 1 Z c t = M x L y 1 L c t = Y z Z y 1 M c t = Z x X z 1 N c t = X y Y x

45 Differential forms vs. vector fields ω 1 E = g E ω 2 B = ω 3 µ B ω 1 H = g H ω 2 D = ω 3 µ D electric field magnetic vortex magnetic field electric flux ω 1 B = g A ω 2 J = ω 3 µ J ω 3 ρ = ρ ω 3 µ ω 0 E = V E magnetic potential electric current electric charge electric potential

46 Space-time split Z R = dt Z P = I R field of time-arrows projector on time-arrows projector on space slices

47 Space-time of electromagnetics ω 2 B = i Ω 2 B Λ 2 (VS ; R) ω 1 E = i Ω 1 E Λ 1 (VS ; R) with i spatial immersion. magnetic vortex electric field Ω 2 B := P Ω 2 F Ω 1 E := P (Ω 2 F V) P projector on the spatial slices V = Z + PV space-time velocity v = i (PV) spatial velocity. Representation formula Ω 2 F = Ω 2 B dt (Ω 1 E + Ω 2 B V).

48 Faraday law in space-time Closedness of Faraday two-form in the trajectory manifold is equivalent to the spatial Gauss law for the magnetic vortex and to the spatial Henry-Faraday induction law, i.e. d Ω 2 F = 0 d S ω 2 B = 0 L S V ω 2 B + d S ω 1 E = 0 with the integral formulation θ=0 ω ϕ 2 Sθ (Σin) B = Σ in ω 1 E for any inner-oriented surface Σ in in a spatial slice.

49 Faraday law ω 1 E = L S V ω 1 B + d S ω 0 E = L S Z ω 1 B + L v ω 1 B + d S ω 0 E = L S Z ω 1 B + (d S ω 1 B) v + d S (ω 1 B v) + d S ω 0 E = L S Z ω 1 B + ω 2 B v + d S (ω 1 B v) + d S ω 0 E Usual formulation with d S (ω 1 B v) dropped ω 1 E = L S Z ω 1 B + ω 2 B v + d S ω 0 E

50 Synoptic table ( v = 0 ) new (E, E ) (γ E, E ) (E, γ (E + w B)) (B, B ) (B, γ (B (w/c 2 ) E)) idem (H, H ) (γ H, H ) (H, γ (H w D)) (D, D ) (D, γ (D + (w/c 2 ) H)) idem (J, J ) (J, γ J ) (γ (J ρ w), J ) ρ γ (ρ g(w/c 2, J)) idem old V E V E γ (V E g(w, A)) (A, A ) (γ (A (w/c 2 )V E ), A ) idem

51 Denoting by the Euclid connection in spatial slices and assuming that the observer measures 1) a magnetic potential independent of time, L S Z ω 1 B = 0 2) a spatially constant scalar electric potential, d S ω 0 E = 0 3) a spatially constant magnetic vortex field, ω 2 B = 0 the formula for the electric field may be evaluated to get ω 1 E = (d S ω 1 B) v + d S (ω 1 B v) = ω 2 B v 1 2 ω2 B v = 1 2 ω2 B v, or in terms of vector fields E = 1 2 v B which is just one-half of what is improperly called the Lorentz force.