tailieunhanh - Behaviour of Electromagnetic Waves in Different Media and Structures Part 14

Tham khảo tài liệu 'behaviour of electromagnetic waves in different media and structures part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 378 Behaviour of Electromagnetic Waves in Different Media and Structures We prefer to express physical quantities by laboratory time Xo rather than proper time . To do so we first calculate dt _ 4 0 d Jio O Zo exp 2 o B2 4 . . . . . 61 4 0 exp o B 4 Xo O exp 2 o B2 4 z0 fromEq. 54 . Using integral formula J x2 1 1 2dx cosh 2 x we obtain f _ J 4 0 exp o B2 4 d 0 Jxo2 O exp 2 o B2 4 Zo 2 11 62 4 0 . irXo O 2 11-4 0 Jr- cosh lUi exp o B 4 cosh o B2 4 z0 z0 z0 in getting this result we have assumed that initial laboratory time t 0 corresponds to initial proper time 0 . The distance of electron away from the origin varies with the laboratory time f can be written out according to Eqs. 54 and 58 and the result is B2 4 . 63 dt dt d 4 0 K If we can express the right hand side of this equation by the expression of f then we get the equation describing r changing with laboratory timef. From Eq. 62 after some calculation we can obtain a quadratic equation of exp 0 B2 4 which is exp 2 o B2 4 2exp o B2 4 cosh o B2 4 f 1 . z 2 Ur 2 I J 1X1 _ n -T7 -Lrircosh2 o B2 4 f 0 4 0 4 0 It is easy to solve this equation and the result is exp o B2 4 cosh o B2 4 f . 1 sinh o B2 4 f . 64 4 0 We know that when the electron is stationary 4 0 1 and f .So we should take the in Eq. 64 and Eq. 63 becomes dr _ Zo di Xo 0 cosh o B2 4 f sinh o B2 4 f The Influence of Vacuum Electromagnetic Fluctuations on the motion of Charged Particles 379 This equation can be easily solved and the result is r Zoldf r 0 r t f ------- - 0 Xo 0 cosh o B2 4 f sinh o B2 4 f 2 Zo dexpty o B2 4 J xo O l exp 2y xo O 1 2 ll- I Xq 0 1 r 7 . . ll- I Xq 0 1 . Itan 1 J . exp o B2 4 f J tan . o B2 4 l Xo O 1 L yxo O 1 So the final expression of the electron s distance away from the origin expressed by the laboratory time is r f r 0 2 itan 2 JXo exw o B2 4 t tan 65 o B2 4 l L Xo O 1 Vx 1 which explicitly shows the inward spiral characteristic of the electron s planar motion with a constant magnetic field along its normal .