Classical Electrodynamics · Problem 14.18

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · synchrotron-radiation-helical-spectrum

Problem

(a) By comparison of (14.91) with (14.79), show that the frequency spectrum of the received power for the situation in Problem 14.17 is dPdω=(3e2γωB2πccos⁡α)G(ωωc)\frac{dP}{d\omega} = \left(\frac{\sqrt{3}e^2\gamma\omega_B}{2\pi c\cos\alpha}\right)G\left(\frac{\omega}{\omega_c}\right) where G(x)=x∫x∞K5/3(t) dtG(x) = x\int_x^\infty K_{5/3}(t)\, dt and the other symbols are as in Problem 14.17. This expression shows that the shape of the power spectrum in units of ω/ωc\omega/\omega_c is unchanged by the spiraling. (b) Show that the integral over frequencies yields P=2e2ωB2γ43cP = \frac{2e^2\omega_B^2\gamma^4}{3c} Comparison with (14.31) shows that the total received power is independent of the pitch angle of motion. [In doing the integration over solid angles in part a, note that ψ=0\psi = 0 corresponds to θ=π/2−α\theta = \pi/2 - \alpha.]

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