Today in Physics 218: radiation from moving charges

Transcription

1 Today in Physics 218: adiation fom moving chages Poblems with moving chages Motion, snapshots and lengths The Liénad-Wiechet potentials Fields fom moving chages Radio galaxy Cygnus A, obseved by Rick Peley et al with the VLA The jets ae steams of matte, ejected at elativistic speeds, in which the electons adiate by vitue of thei acceleation in magnetic fields 19 Mach 2004 Physics 218, Sping

2 Moving chages Last time we consideed, among othe things, the adiation by acceleating chages, bound to othe chages by spings, and obtained the Lamo fomula fo the powe such acceleating chages adiate in all diections: P = q a The use of the spings was delibeate: it keeps the chages fom moving vey fa Hee s what else you have to account fo, if the chages can move a long ways o acceleate to vey high speeds c 3 19 Mach 2004 Physics 218, Sping

3 Moving chages (continued) Conside a point chage that moves along the path as shown What ae the potentials at? Because the chage moves, the situation is diffeent fom last time Not only ae the potentials at (, t) dependent upon the state of the chage, w( t ) at the ealie time t to account fo the popagation delay; but we also have to conside that the chage was at a diffeent point along its path at t Thus, t = t c, but x z Instead, = w t q v y 19 Mach 2004 Physics 218, Sping

4 Moving chages (continued) This might seem like a simple change, but thee s moe to it Suppose, fo instance, we naively compute (, t ) ρ V(, t) = dτ and ρ, t = qδ t q V(, t) = 3 [ ] We wouldn t be done, because it tuns out in this case that q q The eason? An object in motion looks like its size is diffeent fom its est size, because light fom one end of the object took longe to get thee than the othe end, and the object was simply in a diffeent place: 19 Mach 2004 Physics 218, Sping

5 Snapshots of moving objects c t v t θ A L L, M ae equidistant fom the obseving point B M ˆ v Suppose the obsevation point is vey fa away compaed to the object s size ( ) To each the obsevation point at the same time as light fom point B, light fom point A had to leave t ealie, whee ( v t+ )cos θ = c t 19 Mach 2004 Physics 218, Sping

6 Snapshots of moving objects (continued) Thus, v tcosθ + c t = cosθ cos t = θ c vcosθ The moving chage, which has length along the diection of its motion in eal life, looks in a snapshot as it its length is v cosθ c vcosθ + vcosθ = + v t = + = c vcosθ c vcosθ = = v 1 1 cosθ 1 ˆ v c c 19 Mach 2004 Physics 218, Sping

7 Snapshots of moving objects (continued) It looks longe if the angle between the diections of motion and obsevation point is acute, and shote if it s obtuse It looks its natual size if the angle is 90 Note that this has nothing whatsoeve to do with elativity; it s just geomety and the finite speed of light at wok So we tied to do that integal too quickly Let s conside the snapshot effect on the infinitesimal volume elements into which the chage is divided One of its sides lines up with the diection of motion (if we like) The othe two ae pependicula, and thei appeaance unaffected by motion Thus it looks fom at an instant to have a diffeent volume than it does at est: 19 Mach 2004 Physics 218, Sping

8 The Liénad-Wiechet potentials dτ dτ = 1 1 ˆ v c Now we can do that supposedly simple integal: 3 (, t ) δ [ ] ρ t dτ V(, t) = dτ = q 1 1 ˆ v c q = 1 1 ˆ v c 19 Mach 2004 Physics 218, Sping

9 The Liénad-Wiechet potentials (continued) Similaly, 1 J(, t) 1 ρ (, t) v A(, t) = dτ = dτ c c 3 [ t ] δ v dτ v q v = q = = V(, t) 1 ˆ 1 1 v c 1 ˆ c c v c These esults ae called the Liénad-Wiechet potentials 19 Mach 2004 Physics 218, Sping

10 Fields fom moving chages As usual, we e afte the fields and the powe emitted by the moving chage Evaluation of the fields fom the potentials is hade than it sounds, though, because hee is a etaded position, evaluated at the etaded time: = w t, and v = w is the velocity that counts ( ) ( t ) Nevetheless we must poceed: 1 A E = V, B = A c t It will be handiest to compute deivatives of t fist, since they appea often in the expession 19 Mach 2004 Physics 218, Sping

11 Fields fom moving chages (continued) Fist, t t Fom the definition of etaded time, = c( t t ), so c t t = = 2 t 2c ( t t ) 1 = 2 t t t c 1 = t t, Now, = w and is a fixed point in space, so t w w t t = = = v t t t t t, 19 Mach 2004 Physics 218, Sping

12 Fields fom moving chages (continued) and so t t c 1 = v t t t t c = ( c v) u t t t c =, t u whee we have defined u = c v Next, t : t = ( t) = = ( ) c c 2c 1 = ( 2 [ ] + 2 [ ] ) using poduct 2c ule #4 19 Mach 2004 Physics 218, Sping

13 Fields fom moving chages (continued) We ll have to use the chain ule caefully hee: = = + + x y z ( ) ( ) [ t ] ( w ) ( w[ t ]) x y z t d t d t d = + + x y z x dt y dt z dt t t t dw = + + = = + w w ( t ) v x y z x y z dt w w z y wy 0 ˆ wx wz w ˆ x = + ˆ + + y z x y z x x y z 19 Mach 2004 Physics 218, Sping

14 Fields fom moving chages (continued) wz t w y t w ˆ x t wz t = ˆ t y t z x + y t z t x wy t wx t + ˆ t x t y z = v t ; ( ) = ( v t ) = v( t ) + t ( v) Combine these last two with the fomula at the stat: 1 t ( [ ] [ ] ) = c 1 = v( t) + t ( v) + ( t) v c 19 Mach 2004 Physics 218, Sping

15 Fields fom moving chages (continued) o 1 t = - t v c Solving now fo t, we get t c v = t ; = c v Next time, we ll use these esults to simplify the deivatives of the fields 19 Mach 2004 Physics 218, Sping