Two equal magnitude, opposite sign charges are located at either end of a molecule of mass M and length l. The molecule rotates end over end (a nonrelativistic tumbling motion) with an initial rotational period T (cT >> l). How long will it take the molecule to lose 1/10 of its rotational energy by electromagnetic radiation?
Let E0 = E0 k. The Abraham-Lorentz
force equation for a damped, charged, oscillator driven by an electric field
E0exp(-iωt) in the dipole approximation is
d2r'/dt2 + Γ dr'/dt - τ d3r'/dt3 + ω02 r' = (q/m)E0exp(-iωt),
where Γ, τ, and ω0 are constants, q is the charge and m is the mass of the oscillator.
Using this and the expression for the radiation electric field, Erad(r,t) = -(4πε0)-1[(q/(c2r'')]a⊥(t - r''/c), where r'' = r - r'(t - |r - r'|/c), show that the differential cross section for scattering of radiation of frequency ω and polarization n = (θ/θ) is
dσ/dΩ = (e2/(mc2))2 (k∙n)2[ω4/((ω02 - ω2)2 + ω2Γt2)],
where e2 = q2/(4πε0) and Γt = Γ + ω2τ.
For a dipole antenna at the origin with complex electric
dipole moment p(t) = p0exp(-iωt) k the electric and
magnetic fields in the far (radiation) zone are
E(r,t) = Eθ(θ/θ), Eθ = -[1/(4πε0)](p0/k2)(sinθ/r)exp(i(kr - ωt)), k = ω/c,
B(r,t) = Bφ(φ/φ), Bφ = Eθ/c.
(a) If the real dipole moment is p(t) = p0cos(ωt) k with p0 real, what is the instantaneous power radiated per unit solid angle, dP(r,θ,φ,t)/dΩ? What is the time average?
(b) If there is a second dipole at -d0 with p(t) = p0cos(ωt + α) k, with kd0 << 1, what is the instantaneous power radiated by the pair of dipoles in the x-z plane, dP(r,θ,φ,t)/dΩ?
What is the time average? What are the minimum and maximum values of the time average with respect to variations of the phase α?
(c) A large number, N, of dipoles, as above, is located near the origin with Nkd0 << 1 and with phases α0 randomly distributed. What is the time averaged dP(r,θ,φ,t)/dΩ for the system?
A thin linear antenna of length d is excited in
such a way that the sinusoidal current makes a full wavelength of oscillation as shown in
(a) Calculate the power radiated per unit solid angle and plot the angular distribution of the radiation.
(b) Determine the total power radiated and find a numerical value for the radiation resistance R by setting <P> = <I2>R. You can leave your answer in integral form.
A non-relativistic positron of charge qe and velocity v1 (v1 << c) impinges head-on on a fixed nucleus of charge Zqe. The positron which is coming from far away (∞), is decelerated until it comes to rest and then accelerated again in the opposite direction until it reaches a terminal velocity v2. Taking radiation loss into account (but assuming it is small), find v2 as a function of v1 and Z.