Today in Physics 218: radiation reaction

Transcription

1 Today in Physics 18: radiaion reacion Radiaion reacion The Abraham-Lorenz formula; radiaion reacion force The pah of he elecron in oday s firs example (radial decay grealy exaggeraed) 6 March 004 Physics 18, Spring 004 1

2 Synchroron radiaion If v is perpendicular o a (he oher simple geomery), as in he case of uniform circular moion, P and dp dω can be calculaed wih jus a lile more effor han he previous problem (This in fac is problem!1116 in he book, which will no be assigned) The answers are ( ) ( ) dp q a 1 β cosθ 1 β sin θ cos φ =, 3 5 dω emied 4π c ( 1 β cosθ) y, B q a 4 P = γ 3 3 c q The coordinae sysem is described x, a a righ z, v 6 March 004 Physics 18, Spring 004

3 Synchroron radiaion (coninued) The mos common way o see charges in uniform circular moion in naure is of course o pu some in moion in a uniform magneic field This resul shows ha he radiaion sill ends o be beamed in he forward direcion Charges used o be acceleraed o high energies like his, in variable-b machines called synchrorons, and he radiaion resuling from he cenripeal acceleraion, for which he oal power is given by he expression above, has been called synchroron radiaion ever since Mos of he radio radiaion by normal galaxies is produced in his way, by elecrons spiraling around in inersellar magneic fields 6 March 004 Physics 18, Spring 004 3

4 Radiaion reacion This is a very deep opic ha, unlike everyhing else you ve seen in elecrodynamics, sill conains many imporan unresolved problems, and many aspecs ha seem o indicae ha elecrodynamics migh no be inernally consisen We will only ouch on some of he imporan feaures Take a charge and a region of uniform magneic field Give he charge an impulse of kineic energy, wih is iniial velociy lying in he plane perpendicular o B Wha happens? The charge moves in circles, a speed v and acceleraion a= v r And i radiaes, according o he formula jus discussed 6 March 004 Physics 18, Spring 004 4

5 Radiaion reacion (coninued) Radiaion akes energy away from he charge Where does his energy come from? Only one source exiss: he charge s kineic energy The uniform magneic field is he only oher energy densiy around, and magneic fields can do work on charges So he radiaion decreases he kineic energy and speed of he paricle, an effec known as radiaion reacion This decrease in kineic energy can be characerized as work, and hus expressed in erms of a force ha does he work And he force is wha causes all he confusion 6 March 004 Physics 18, Spring 004 5

6 Example: radiaion reacion and cycloron moion Consider he charge we sared wih, given an impulse of kineic energy and allowed o fly in circles in a uniform magneic field Describe he subsequen moion, and work ou how long i akes he charge s speed o decay away Soluion Mos radiaion reacion problems are mos easily and less confusingly solved wih energy echniques We ll assume he moion is insananeously circular, and sar wih q q qvb P = a = c c mc de d 1 dv = = mv = mv d d d This can be rearranged for inegraion: 6 March 004 Physics 18, Spring 004 6

7 Example: radiaion reacion and cycloron moion (coninued) v v 0 dv v v q B ln = 3 5 v0 3 mc 4 q B v = v0exp = v 3 5 0e 3 mc τ = = 3 3mc q B q B mc 0 d τ, where v 0 6 March 004 Physics 18, Spring 004 7

8 Example: radiaion reacion and cycloron moion (coninued) Some numbers may help: suppose ha he magneic field srengh were B = 1 esla = 10 4 gauss (a ypical value in he laboraory) and he charge is an elecron: 3 5 3mc τ = = 516 sec 4 q B Shor, bu he period of he elecron s orbi in he magneic field is much shorer: qvb v F = ma m cgs unis, remember c r πr πmc 11 T = = = sec v qb 6 March 004 Physics 18, Spring 004 8

9 Example: radiaion reacion and cycloron moion (coninued) So he elecron makes T τ = orbis before he velociy has decayed o 1/e of is original value This means he iniial assumpion of circular orbis is a very good approximaion The orbial radius, given as above by r = mvc qb decreases as he speed decreases: he elecron spirals inward, as seen (wih he radial decay exaggeraed so as o render i visible) on he nex page 11, 6 March 004 Physics 18, Spring 004 9

10 Example: radiaion reacion and cycloron moion (coninued) 6 March 004 Physics 18, Spring

11 Example: radiaion reacion and cycloron moion (coninued) A closely relaed problem is of grea hisorical imporance in physics As you will show in Problem 1114 on he nex homework se, he elecron in he Bohr model of he hydrogen aom, a poin charge in orbi abou a poin nucleus, also exhibis his sor of radial decay The ime i akes he elecron o spiral in o he nucleus, due o radiaion losses in orbi, urns ou o be very shor Aoms are hus unsable in he (oherwise quie successful) Bohr model This led o he successful supposiion of wave properies of he elecron, which subsequenly led o all of quanum mechanics as we know i 6 March 004 Physics 18, Spring

12 Reacion force, and he Abraham-Lorenz formula Here we will derive a formula for he force corresponding o radiaion reacion in non-relaivisic siuaions Consider an elecric charge q undergoing acceleraion a() A any insan i radiaes power according o he Larmor formula: q a () P = 3 3 c So if nohing else happens o he charge, i will no gain kineic energy a he rae one migh expec from he acceleraion, on accoun of he flow of some of his energy ino radiaion I s as if an exra force is a work: 6 March 004 Physics 18, Spring 004 1

13 Reacion force, and he Abraham-Lorenz formula (coninued) Supposing ha he power in radiaion is a resul of work done agains he exra force, we can find ou how big i is: Work done agains ( Power radiaed) F vd = 1 1 rad 3 3 q c F rad a d = d F rad Uncharged m a= F m Charged q,m ( ) a= F F rad m F F 6 March 004 Physics 18, Spring

14 Reacion force, and he Abraham-Lorenz formula (coninued) d d Since a = v v, he righ-hand side of his expression d d can be inegraed by pars: F vd = 1 1 rad 3 q dv dv x = dv d, d 3 c d d dy = ( dv d) d; y = v q dv d v = v v d 3 3 c d d 1 1 Suppose ha we are pushing he charge in such a way ha i s in he same sae v and a a he iniial and final imes, bu allow he sae o change in beween 6 March 004 Physics 18, Spring

15 Reacion force, and he Abraham-Lorenz formula (coninued) Then he inegral on he righ-hand side jus represens he ime average of a beween hose imes I also means ha dv v = 0, d 1 because v and a are he same a he wo imes Thus q Frad vd = v a d a = 3 3 c 1 1 Equae he inegrands: F = rad 3 3 q c a 6 March 004 Physics 18, Spring d d Abraham-Lorenz formula v

16 How o use he radiaion reacion force (coninued) You will ge pracice in he use of he radiaion reacion force in Problem 1117 on his week s homework To help you ge sared Example (114 in he book): Calculae he radiaion damping of a charged paricle aached o a spring wih naural frequency ω0, driven a frequency ω Soluion: The radiaion reacion force involves one more ime derivaive han is usual in kinemaic problems, which ofen complicaes oherwise simple siuaions In his case periodic moion keeps his one from geing ou of hand F = ma Fspring + Frad + Fdriving = mx 6 March 004 Physics 18, Spring

17 How o use he radiaion reacion force (coninued) q mω0x+ x + F0 cos ω = mx 3 3 c Have no fear If he sysem is driven, i will oscillae a frequency ω The wors he exra erm can do is produce a (poenially complicaed) phase delay So x( ) = x0 cos( ω+ δ) x = ωx sin ω+ δ ( ) cos( ) ( ) ( ) x = ω x ω+ δ x = ω sin ω+ δ = ω x!, and we re lef wih q ω mω0x x + F 3 0 cos ω = mx 3 c 6 March 004 Physics 18, Spring

18 How o use he radiaion reacion force (coninued) Thus q ω F0 x + x + ω0 x = cos ω ; 3 3 mc m F0 x + γx + ω0 x = cos ω m We have solved his damped harmonic oscillaor problem many imes (for insance, in lecure on 18 February 004): iω F0 m x = x0e, x0 = ω ω iωγ ( ) 0 6 March 004 Physics 18, Spring