Topic 7: Electrodynamics of spinning particles Revised Draft

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1 Lectue Seies: The Spin of the Matte, Physics 4250, Fall Topic 7: Electodynamics of spinning paticles Revised Daft D. Bill Pezzaglia CSUEB Physics Updated Nov 28, 2010 Index: Rough Daft 2 A. Classical Electodynamics B. Toques C. Gadient Foces D. Refeences 1

2 A. Classical Electodynamics 3 1. Loentz Foce 2. Relativistic Fom 3. Types of Cuents A. Classical Electodynamics 4 1. Loentz Foce (point paticle) Wok-enegy Loentz foce & ξ = ee v P = e ( E + v B) Did Maxwell talk about magnetic foce in 1861? Thomson in 1881 is fist to wite (1/2)qvxB Heaviside 1889 shows the (1/2) was wong Loentz (1892) wites the total foce law. Who did the wok-enegy theoem equation? Joule? 2

3 2. Special Relativity fom 5 Unify Phenomena with 4 th Dimension Combine scala law with vecto law with 4 vectos Scala & ξ = ee v Vecto P = e( E + v B) & p = e m p F 4 vecto: p = p 1 ê 1 + p 2 ê 2 +p 2 ê 3 +(E/c)ê 4 Field F= E 1 ê 4 ê 1 +E 2 ê 4 ê 2 +E 3 ê 4 ê 3 + Bivecto B 1 ê 2 ê 3 +B 2 ê 3 ê 1 +B 3 ê 1 ê 2 3. The natue of cuent 6 Moe coect : & p = j ν F ν 4 cuent: j = J 1 ê 1 + J 2 ê 2 +J 2 ê 3 +(ρ/c)ê 4 Chage density: ρ Rowland cuent: (convection cuent) j = ex& = e m p 3

4 4. Othe Cuents 7 Given: & p = j ν Should cuent include: Maxwell s Displacement cuent? Roentgen Cuent (1888)? [change in dipole] ν j F = x& σ U& σ J D D = t B. Magnetic Moment 8 1. Toque on Dipole 2. Relativistic Dipole Tenso 3. Gyomagnetic Ratio 4

5 1. Toque on Dipole 9 (a) Toques on electic and magnetic moment τ = B + p E (b) What about moving magnetic dipole? τ B c ( v 1 E) = 2 (c) Toque on moving magnetic and electic dipole? 1 τ B c v E + p E + v B ( ) ( ) = 2 2. Dipole Tenso (a) Moving magnetic dipole ceates an electic dipole p = v = v p 10 (b) Relativistic Dipole Tenso Dipole U= p 1 ê 4 ê 1 +p 2 ê 4 ê 2 +p 3 ê 4 ê 3 + Bivecto 1 ê 2 ê ê 3 ê ê 1 ê 2 (c) Relativistic Toque equation: In tenso fom: S& = U F υ S& = U σ F συ 5

6 3. Spin and Dipole Moment 11 (a) Analogy of Rowland cuent: we assume magnetic dipole is due to otational convection of chage (assume chage density popotional to mass density) (b) Lande s Anomalous g facto: fo electon, g 2 hence we get: (c) Relativistic Dipole tenso U = ge = 2 m συ = e 2m ge 2m S S = S e m συ S Moe on Gyomagnetic Ratio 12 Rotating Ball of Chage (e.g. electon) should have magnetic moment popotional to spin: Gyomagnetic Ratio: = γ γ = ge 2m Lande s Anomalous g facto g=1 fo classical objects g=2 howeve fo pointlike electon (why?) S 6

7 (Joseph) Lamo Pecession 13 (a) Toque on Magnet (b) Pecession (c) Lamo Fequency τ = B ds dt = γ S B S ω = & = γ B S 4. Othe Toques? 14 (a) Its possible that thee ae othe souces of dipole moment othe than convection of otating chage, such that dipole moment U will not be paallel to Spin. ge 2m S? (b) Thee appea to be othe toques due to momentum not paallel to velocity S& = x& p (c) Inseting canonical momentum into above gives unattactive gauge dependent toque, which appeas often in liteatue. S& = j A & S = J A 7

8 C. Gadient Foces Toque on Dipole 2. Mass is dynamic 3. New foce and toque equations 1. Sten-Gelach Foce 16 Gadient foces due to dipole moment (a) Enegy of Static Dipole ξ = B p E (b) Enegy of Moving Magnetic Dipole (c) Sten-Gelach Foce (d) Relativistic Fom: ξ & B c = 2 ( v 1 E) F = ξ 1 αβ ( F ) p = U 2 αβ 8

9 Famous People 17 Otto Sten (1943 Nobel Pize) Walte Gelach Expeiment in Dynamic Mass 18 Relativistic Gadient Foce equation p = e m p F ( U F ) Is inconsistent with constant est mass. Must include mass change: p & = mv& + mv & Contacting foce equation with the fou-velocity yields an equation fo dynamic est mass dm dτ ' c d dτ 1 [ ( B v E) ] = c 9

10 2b. Mass changed by Dipole Inteaction 19 Baut & othes popose est mass is hence: m' 0 1 = m c 1 ( B v E) c Hence a dipole in a constant B field will have moe inetia? (o less depending upon spin oientation). Has this eve been tested? In a big enough field, the mass would be ZERO? Question: How big of magnetic field would be needed fo the est mass of an electon to be zeo? 3a Modified Foce Equation 20 Baut s Lagangian L 1 αβ ( m Uαβ F ) x& x& ex& 0 + A = 2 This dynamical mass modifies the foce equation to the messy fom (has it eve been expeimentally veified?): 1 αβ σ σ 1 σ σ αβ ( m U F ) x& = ex& F + U ( g x& x& ) F 0 2 αβ & 2 αβ 10

11 gavitational consequences 21 It appeas that dynamical mass will eplace the both the inetial and gavitational mass. Does this peseve the equivalence pinciple? 3b Modified Spin Equation 22 In ode to peseve Dixon s equation, I get (unpublished) that thee should be an additional gadient toque tem: S& = 2 σ j A + U F + λ j Would this be obsevable in any way? σ F 11

12 Refeences 23 Fenkel J. (1925): Zu Elektodynamik punktf omige Elektonen, Z. Physik. 32, Fenkel J. (1926): Die Elektodynamik des otieenden Elektons,... W.G. Dixon, Dynamics of extended bodies in geneal elativity, Poc. Roy. Soc. Lond. A314, (1970) Baut, Electodynamics and Classical Theoy of Fields and Paticles Tomonaga, The Stoy of Spin, Univ Chicago Pess (1997) Ideas 24 If a moving magnetic dipole ceates an electic dipole moment, then what does an ACCELERATING dipole ceate? (i.e. what is affect of Thomas pecession hee?) 12