week5 - 1 Monday October 17 Multi-particle Systems For...

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1 Monday, October 17: Multi-particle Sys- tems For non-relativistic charged particles, we have derived a useful formula for the power radiated per unit solid angle in the form of electromagnetic radiation: dP d Ω = q 2 4 πc 3 [ a 2 q sin 2 Θ] τ , (1) where q is the electric charge of the particle, ~a q is its acceleration, and Θ is the angle between the acceleration vector ~a q and the direction in which the radiation is emitted. The subscript τ is a quiet reminder that for any observer, we must use the values of a q and Θ at the appropriate retarded time τ rather than the time of observation t . By integrating over all solid angles, we found the net power radiated by a non-relativistic charged particle: P = 2 q 2 3 c 3 [ a 2 q ] τ . (2) Because the charge of the electron (or proton) is small in cgs units, and the speed of light is large in cgs units, we expect the power radiated by a single electron (or proton) to be small, even at the large accelerations that can be experienced by elementary particles. 1 The energy that the charged particle is radiating away has to come from somewhere. If the only energy source is the particle’s kinetic energy, E = mv 2 q / 2, the characteristic time scale for energy loss is t E = E P = 3 c 3 m 4 q 2 " v 2 q a 2 q # τ . (3) As a concrete example, consider an electron moving in a circle of radius r q with a speed v q = βc . The acceleration of the electron will be a q = β 2 c 2 /r q , the power radiated will be P = 2 q 2 e β 4 c 3 r 2 q = 4 . 6 × 10 - 17 erg s - 1 ˆ β 0 . 01 ! 4 r q 1 cm - 2 . (4) 1 In problem set 2, the electron being bombarded by red light had a maximum accel- eration of a q 10 19 cm s - 2 ; the proton in Lawrence’s cyclotron had an acceleration of a q 4 × 10 16 cm s - 2 . 1
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The time scale for energy loss will be t E = 3 cm e r 2 q 4 q 2 e β 2 = 8 . 9 × 10 5 s ˆ β 0 . 01 ! - 2 r q 1 cm 2 . (5) It is informative to compare the time scale for energy loss with th orbital period of the electron: t P = 2 πr q βc = 2 . 1 × 10 - 8 s ˆ β 0 . 01 ! - 1 r q 1 cm . (6) For the electron’s orbit to be stable, rather than a steep inward death spiral, we require t E t P , which implies an orbital radius r q 8 πq 2 e 3 c 2 m e β = 8 πr 0 3 β , (7) where r 0 q 2 e / ( m e c 2 ) = 2 . 8 × 10 - 13 cm is the classical electron radius . According to classical electromagnetic theory, then, an electron on an orbit smaller than βr 0 in radius will radiate away its energy and spiral in to the origin on a time scale comparable to its orbital time. The radiation by electrons on circular orbits was what spelled the doom of classical atomic theory. Around 1911, Ernest Rutherford proposed a theory in which electrons went on circular orbits around a positively charged nucleus. The radius r of the orbits was determined by the requirement that the centripetal force be provided by the electrostatic force between the electron (charge q e ) and the nucleus (charge - Zq e ): Zq 2 e r 2 = m e β 2 c 2 r , (8) or r = Zq 2 e β 2 m e c 2 = Z β 2 r 0 . (9) However, it was known that the radius of the hydrogen atom ( Z = 1) was r 0 . 5 ˚ A 0 . 5 × 10 - 8 cm 2 × 10 4 r 0 . This implied β = ( r/r 0 ) - 1 /
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