Classical Electrodynamics · Problem 14.20

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-magnetic-moment-flipping

Problem

Apply part a of Problem 14.19 to the radiation emitted by a magnetic moment at the origin flipping from pointing down to pointing up, with components, μz=μ0tanh⁡(νt),μx=μ0sech⁡(νt),μy=0\mu_z = \mu_0\tanh(\nu t), \qquad \mu_x = \mu_0\operatorname{sech}(\nu t), \qquad \mu_y = 0 where ν−1\nu^{-1} is characteristic of the time taken to flip. (a) Find the angular distribution of radiation and show that the intensity per unit frequency interval is dImagdx=43(νc)3μ02{16(x/π)4[cosech⁡2x+sech⁡2x]}\frac{dI_{\text{mag}}}{dx} = \frac{4}{3}\left(\frac{\nu}{c}\right)^3 \mu_0^2\left\{16(x/\pi)^4[\operatorname{cosech}^2 x + \operatorname{sech}^2 x]\right\} where x=πω/2νx = \pi\omega/2\nu is a dimensionless frequency variable and the quantity in curly brackets is the normalized frequency distribution in xx. Make a plot of this distribution and find the mean value of ω\omega in units of ν\nu. (b) Apply the method of Problem 9.7 to calculate the instantaneous power and total energy radiated by the flipping dipole. Compare with the answer in part a.

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