Classical Electrodynamics · Problem 16.3

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-damping-classical-models-charged-particles

Problem

An electron moving in an attractive Coulomb field (−Ze2/r)(-Ze^2/r) with binding energy ϵ\epsilon and angular momentum LL has an elliptic orbit, 1r=Ze2mL2[1+1−2ϵL2Z2e4mcos⁡(θ−θ0)]\frac{1}{r} = \frac{Ze^2 m}{L^2}\left[1 + \sqrt{1 - \frac{2\epsilon L^2}{Z^2 e^4 m}}\cos(\theta - \theta_0)\right] The eccentricity of the ellipse is given by the square root multiplying the cosine. (a) By performing the appropriate time averages over the orbit, show that the secular changes in energy and angular momentum are dϵdt=23/23Z3e8m1/2c3ϵ3/2L5(3−2ϵL2Z2e4m)\frac{d\epsilon}{dt} = \frac{2^{3/2}}{3}\frac{Z^3 e^8 m^{1/2}}{c^3}\frac{\epsilon^{3/2}}{L^5}\left(3 - \frac{2\epsilon L^2}{Z^2 e^4 m}\right) dLdt=−25/23Ze4m1/2c3ϵ3/2L2\frac{dL}{dt} = -\frac{2^{5/2}}{3}\frac{Ze^4}{m^{1/2}c^3}\frac{\epsilon^{3/2}}{L^2} (b) If the initial values of ϵ\epsilon and LL are ϵ0\epsilon_0 and L0L_0, show that ϵ(L)=Z2e4m2L2[1−(LL0)3]+ϵ0L0L\epsilon(L) = \frac{Z^2 e^4 m}{2L^2}\left[1 - \left(\frac{L}{L_0}\right)^3\right] + \frac{\epsilon_0}{L_0}L Calculate the eccentricity of the ellipse, and show that it decreases from its initial value as (L/L0)3/2(L/L_0)^{3/2}, showing that the orbit tends to become circular as time goes on. (c) Compare your results here to the special case of a circular orbit of Problem 16.2. Hint: In performing the time averages make use of Kepler's law of equal areas (dt=mr2 dθ/L)(dt = mr^2\,d\theta/L) to convert time integrals to angular integrals.

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