Scattering 1 Classical scattering of a charged particle (Rutherford ...

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Thus the radiated power into solid angle dΩ is ;dPdΩ=q216π 2 ǫc [ˆn × ˆn × ⃗˙β] 2If a charged particle is accelerated by the ⃗ E field of an EM wave we have;⃗E I = E 0I e i(⃗ k·⃗r − ωt) ⃗η 0⃗F(force) = m⃗a = mc ⃗˙β = q ⃗ EI⃗˙β = (q/mc)E 0I e i(⃗ k·⃗r−ωt) ˆη 0Ignore phase differences (take the instantaneous time) and let (ˆη be the polarization direction,Figure 4.〈 dPdΩ 〉 = c ǫ ( q 24πmc 2 ) 2 |ˆη · ˆη 0 | 2The incident flux is the time-averaged Poynting vector of the incident wave so the crosssection is;dσdΩ = 〈(dP/dΩ)〉〈S in 〉dσdΩ = ( q 24πǫmc 2 ) 2 |ˆη · ˆη 0 | 2The evaluation of the factor |ˆη · ˆη 0 | 2 is obtained from Figure 4.ˆη 1 = cos(θ)[ˆxcos(φ) + ŷ sin(φ)] − ẑ sin(θ)ˆη 2 = −ˆx sin(φ) + ŷ cos(φ)Choose ˆη 0 = ˆx and sum over the final polarizations to obtain;|ˆη · ˆη 0 | 2 = cos 2 (θ) cos 2 (φ) + sin 2 (φ)This result can be substiuted above to obtain the scattered differential power. Note we haveneglected the effect of the magnetic field on the motion of the accelerated charge, but thiseffect was worked out to first order in an earlier example.10
IncidentDirectionzθn^ScatteredDirection^η 0xφθη^η^φ 21yFigure 4: The geometry used to evaluate the angles in Thompson scattering8 BremsstrahlungThe general form of the radiated energy,dWdΩ= ∫ dt dPdΩ ;Thus use ⃗ S = ǫc|E| 2ˆn and ⃗ E =dWdΩ= A ∫ dt |E| 2 R 24πǫ q × ˆn ×⃗˙β[ˆnR] Ret and write;In the above A is a constant depending on the charge. Take the Fourier transform of thefield using the time dependence of the field, e iωt .E(ω) = 1 √2π∫dt E(t) eiωtE(t) = 1 √2π∫dω E(ω) e−iωtThen substitute to obtain;dWdΩ= (AR 2 /2π) ∫ dt ∫ dω ∫ dω ′ E ∗ (ω ′ ) E(ω) e i(ω−ω′ )tdWdΩ= (AR 2 /2π) ∫ dω |E(ω)| 2Therefore we identify the frequency spectrum of the radiation where I is the intensity(power/area) as;d 2 IdωdΩ = 2A|E(ω)|2 R 2Now substitute for the radiation field of an accelerated particle (use the relativistically correctexpression);11
Page 1 and 2: Scattering1 Classical scattering of
Page 3 and 4: sClosestApproachθxθθFigure 2: Th
Page 5 and 6: Now a properly normalized outgoing
Page 7 and 8: dσ ⊥dΩ = k4 η 62 |ǫ − 1ǫ +
Page 9: a spherical system, written in term
Page 13 and 14: q/2vvq/2φ oFigure 6: Radiation fro
Page 15 and 16: xEρyrPzFigure 7: A schematic of a
Page 17 and 18: RetardedPointn^R =c(t − t’)θβ
Page 19 and 20: cos(θ) = 1/(1.5β)and gives a cut-
Page 21 and 22: oth energy and momentum, write;⃗p
Page 23 and 24: ⃗B(k, ω) = i/c 2 ( ⃗ k × ⃗v

Scattering 1 Classical scattering of a charged particle (Rutherford ...

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