Classical Electrodynamics · Problem 9.15

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · two-rotating-dipoles-radiation

Problem

Two fixed electric dipoles of dipole moment pp are located in the xx-yy plane a distance 2a2a apart, their axes parallel and perpendicular to the plane, but their moments directed oppositely. The dipoles rotate with constant angular speed ω\omega about a zz axis located halfway between them. The motion is nonrelativistic (ωa/c≪1\omega a/c \ll 1). (a) Find the lowest nonvanishing multipole moments. (b) Show that the magnetic field in the radiation zone is, apart from an overall phase factor, H=cpa2π k3[(x^+iy^)cos⁡θ−z^sin⁡θ eiϕ]cos⁡θ eikrr\mathbf{H} = \frac{cpa}{2\pi}\, k^3[(\hat{\mathbf{x}} + i\hat{\mathbf{y}})\cos\theta - \hat{\mathbf{z}}\sin\theta\, e^{i\phi}]\cos\theta\, \frac{e^{ikr}}{r} (c) Show that the angular distribution of the radiation is proportional to (cos⁡2θ+cos⁡4θ)(\cos^2\theta + \cos^4\theta) and the total time-averaged power radiated is P=415πϵ0 ck6p2a2P = \frac{4}{15\pi\epsilon_0}\, ck^6 p^2 a^2 Hint: Problem 6.21 is relevant.

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