Time-varying electromagnetic fields E(x,t) and B(x,t) of finite duration act on a charged particle of charge e and mass m bound harmonically to the origin with natural frequency ω0 and small damping constant Γ. The fields may be caused by a passing charged particle or some other external source. The charge's motion in response to the fields is nonrelativistic and small in amplitude compared to the scale of spatial variation of the fields (dipole approximation). Show that the energy transferred to the oscillator in the limit of very small damping is
ΔE=mπe2∣E(ω0)∣2
where E(ω) is the symmetric Fourier transform of E(0,t):
E(0,t)=2π1∫−∞∞E(ω)e−iωtdω;E(ω)=2π1∫−∞∞E(0,t)eiωtdt