Lorentz Transformations in Electrodynamics

phy 712 electrodynamics 9 9 50 am mwf olin 105 n.w
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Dive into Lecture 24 of PHY 712 on Electrodynamics, covering equations in cgs units, special theory of relativity, Lorentz transformation relations, and basics of special relativity. Explore the invariance of physical laws in different frames of reference and the constant speed of light. Discover the v notation for Lorentz transformations and examples of applicable 4-vectors, shedding light on the complexities of relativistic physics.

  • Electrodynamics
  • Lorentz Transformations
  • Special Relativity
  • Physics Education
  • Relativistic Physics
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  1. PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105 Plan for Lecture 24: Start reading Chap. 11 A. Equations in cgs (Gaussian) units B. Special theory of relativity C. Lorentz transformation relations 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 1

  2. 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 2

  3. 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 3

  4. 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 4

  5. 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 5

  6. More relationships MKS (SI) = D CGS (Gaussian = + D E ) E + = = E P E P 4 0 1 1 1 = = = = H B M B H B M B 4 0 A 1 c A = E = E t t = B A = B A m u / / 0 0 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 6

  7. Notions of special relativity The basic laws of physics are the same in all frames of reference (at rest or moving at constant velocity). The speed of light in vacuum c is the same in all frames of reference. y y v x x y x y x 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 7

  8. Convenient v notation : Lorentz transformations c 1 2 1 Stationary frame Moving ct + frame y y ( x ) = + ' ' ct x ( ' ) = x ' ' ct v = y y x = z ' z x y x y x 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 8

  9. 1 v c Lorentz transformations -- continued 2 1 = v : v x For the moving frame with 0 0 0 0 0 0 0 0 = = 1 - L L 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 ' ' ct ct ct ct ' ' x x x x = = 1 - L L ' ' y y y y ' ' z z z z Notice : = 2 2 2 2 2 2 2 2 2 2 ' ' ' ' c t x y z c t x y z 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 9

  10. Examples of other 4-vectors applicable to the Lorentz transformation: = v : v x 1 For the moving frame with v c 2 1 0 0 0 0 0 0 0 0 = = 1 - L L 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 / / ' k / ' k / c c c c ' ' k k x x x x = = = 1 - L L k r k r Note In free space: : ' ' t ' ' t ' ' k k k k y y y y ' ' k k k k 2 2 z z z z ' = = 2 2 ' 0 k k ' ' E E E E c c ' ' p c p c p c p c x x x x = = = 1 2 2 2 2 2 2 - L L Note : ' ' E p c E p c ' ' p c p c p c p c y y y y ' ' p c p c p c p c z z z z 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 10

  11. The Doppler Effect = v : x v For the moving frame with 0 0 0 0 0 0 0 0 = = 1 - L L 0 0 1 0 0 0 1 0 c 0 0 0 1 0 0 0 1 / / ' k / ' k / c c c ' ' k k x x x x = = = r 1 - L L k r k Note : ' ' t ' ' t ' ' k k k k y y y y ' ' k k k k z z z z ( ) ( ) = = / ' k / k = ' / c = c k k k c x x x ' ' k k y y z z 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 11

  12. The Doppler Effect -- continued ( ) ( ) = = / ' k / k = ' / c = c k k k c x x x ' ' k k y y z z More generally = : ( ) ( ) y k / ' / / cos c c c k y ( ) ( ) ' = = k k / ' cos ' cos / c k k c k = k k ' v = = For 0 : ( ) k /c 1 ( ) k = = ' 1 ' + 1 x = For 0 : ( ) k /c sin x = tan ' ( ) cos 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 12

  13. Electromagnetic Doppler Effect (=0) 1 ' 1 + source v detector v = c source v detector v v c = More precisely: (Thanks to E. Carlson) v 1 c source detector 2 Sound Doppler Effect ( =0) 1 / detector v c = ' s 1 / source v c s 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 13

  14. Lorentz transformation of the velocity Stationary frame Moving ct + frame ( x ) = + ' ' ct x ( ' ) = x ' ' ct = y y = z ' z For an infinitesimal increment: frame Stationary Moving frame ( ) = + ' ' cdt cdt dx ( ) = + dx dx' cdt' = dy dy' = dz dz' 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 14

  15. Lorentz transformation of the velocity -- continued frame Stationary Moving frame ( ) = + ' ' cdt cdt dx ( ) = + dx dx' cdt' = dy dy' = dz dx dz' dy dz Define : u u u x y z dt dt dt ' ' ' dx dy dz ( ( dt ' ' ' u u u x y z ' ' ' dt dt v / dt ) ) + ' + ' u dx dx' cdt' / ' = = = x u x + ' + 2 1 dt dx c vu c x ' u ' dy dy + ' y = = = u ( 1 ) ( ) y + 2 / ' ' / dt dt dx c vu c x 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 15

  16. Summary of velocity relationships + ' u v = x u x + 2 1 ' / x u vu vu c ' ' u vu + y y = u ( ) ( ) y + 2 2 1 ' / x 1 ' / x c c v ' ' u vu u vu = z z u ( ) ( ) z + + 2 2 1 ' / x 1 ' / x c c v 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 16

  17. Example of velocity variation with (u x/c=u y/c=0.5) ux/c uy/c 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 17

  18. Extention to tranformation of acceleration 3/2 2 v c v u 1 2 = a a ' 3 ' + 1 2 c 2 v c 1 2 v ( ) = + a a a u ' ' ' 3 2 c v u ' + 1 2 c 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 18

  19. Velocity transformations continued: ' Consider: 1 ' / x vu + ' u vu + + ' u v u vu y = = = . x z u u u ( ) ( ) x y z 2 + 2 2 c 1 ' / x 1 ' / x c c v v + 2 1 ' / x 1 vu c ( ) = = + 2 Note that 1 ' / x vu c ' u v u ( ) ( ) ( ) 2 2 2 1 / 1 '/ 1 / u c u c v c ( ) = = = = + ' c c v u u v + ' ' u v u u ' y x ( u ) ( ) = + v u c ' ' u u ' c u u ' ' ' ' u x v u x u v u u x u u ' ' u y u u z u z c ' u u u u u ' ' ' ' u x u x = L Velocity 4-vector: v ' u y u y PHY 712 Spring 2019 -- Lecture 24 u z u ' u z 03/25/2019 19

  20. Some details: 2 2 2 2 ' u c v v c u c u c ( ) = + = + 2 1 ' / x 1 1 1 1 x vu c ' u v u 2 2 2 2 ' u vu + + ' ' u v u vu y = = = where . x z u u u ( ) ( ) x y z + 2 + 1 ' / x 2 2 vu c 1 ' / x 1 ' / x c c v v 2 2 2 y 2 y ' u c u c 2 x 2 z 2 z 2 ' ' u c u c v u u c v c u c v c + + + = + + + 1 1 x x 2 2 2 2 2 2 2 c 2 2 2 2 2 2 ' u c v u c v u c u c v c v c + = + + 1 1 1 1 x x 2 2 2 2 2 2 2 2 2 2 ' u c v u c u c v c + = 1 1 1 1 x 2 2 2 2 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 20

  21. c u u Significance of 4-velocity vector: u x u u y u u z Introduce the rest mass m of particle characterized by velocity u: 2 c E mc u u u p c mu c u x x = = u x mc u p c mu c u y y u y u p c mu c u z z u z Properties of energy-moment 4-vector: ' ' E E E E ' ' p c p c p c p c x x x x = = = 1 2 2 2 2 2 2 - L L Note : ' ' E p c E p c ' ' p c p c p c p c y y y y ' ' p c p c p c p c z z z z 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 21

  22. Properties of Energy-momentum 4-vector -- continued = c mu y u 2 E mc u p c mu c x u x p c y p c mu c z u z ( 1 ) 2 2 2 2 u ( ) 2 u mc u 2 0, y = = = 2 2 2 2 2 2 2 Note : 1 ' ' x z E p c mc E p c 2 c c c E u = 2 p Notion of "rest energy": For mc = + 2 2 2 2 4 2 Define kinetic energy: E E mc p c m c mc K 2 p p = + 2 Non-relativistic limit: If 1, 1 1 E mc K mc mc 2 p 2 m 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 22

  23. Summary of relativistic energy relationships = c mu c p z u z 2 E mc u p c mu c x u x p c mu c y u y = + = 2 2 2 4 2 E p c m c mc u 2 2 + = + = 2 2 2 4 2 2 Check : 1 p c m c mc mc u u u = 2 Example electron an for : c E 5 . 0 MeV m = for 200 GeV E = = 5 4 10 u 2 mc 1 1 12 = 1 1 1 3 10 u 2 2 2 u u 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 23

  24. Special theory of relativity and Maxwells equations + = J Continuity equation: 0 t + = 2 t 1 c = A Lorentz gauge condition: 0 t 2 1 c 2 Potential equations: 4 2 2 t A 1 c 4 = 2 A J 2 2 c A 1 c = E Field relations: t = B A 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 24

  25. More 4-vectors: ct x position and Time : x y z J x c Charge current and : J J y J z A x Vector and scalar potentials : A A y A z 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 25

  26. 0 0 Lorentz transformations v v v 0 0 v v v = L v 0 0 1 0 0 0 0 1 v L = L Time and space : ' ' x x x v v L = L Charge current and : ' ' J J J v v L = L Vector and scalar potential : ' ' A A A v 03/25/2019 PHY 712 Spring 2019 -- Lecture 24 26

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