Physics 515 Problem Set #7 May 20, 2004
Due Tuesday, June 1, 2004

1. (20 pts) Show that the result derived in lecture for the proper-time rate at which 4-momentum is radiated per unit solid angle, ${d \over d\tau} {d p_{\rm Rad}^\mu \over d\Omega} = -{1\over c} \left( e \... ...a^2 \over (u\cdot n)^3} - c^2 {(a\cdot n)^2 \over (u\cdot n)^5} \right\}$ [with $n^\mu \equiv (1,\hat \r)$], is consistent with our previous result for the total 4-momentum radiated per unit proper-time, $d p_{\rm Rad}^\mu/d\tau$. A recommended approach is to show that $\int d\Omega \> {d \over d\tau} {d p_{\rm Rad}^\mu \over d\Omega} = {e^2 \o... ...l u^\beta} \right\} {1 \over 2} {\partial \over \partial u_\mu} I(u)$, where $I(u) \equiv \int {d\Omega \over 4\pi} \> (u.n)^{-2}$ and $u$ is temporarily regarded as an arbitrary 4-vector (not constrained by $u^2 = -c^2$). Evalute the remaining scalar integral explicitly and show that $I(u) = -1/u^2$. Perform the indicated derivatives and derive the expected result for $d p^\mu_{\rm Rad} / d\tau$.
2. (30 pts) A particle of charge $q$ and mass $m$ executes periodic oscillations along the $z$-axis with dynamics controlled by a non-electromagnetic force $\F(\r) = -m \alpha {\rm sgn}(z) \hat \z$. Let $\tau$ denote the period of the resulting oscillations, which are non-relativistic.
   1. Ignoring back-reaction due to radiation, the particle's acceleration $\a(t) = \mp \alpha \hat \z$ where the upper sign applies for $0 \lt [t \bmod \tau] \lt \tau / 2$, and the lower sign for $\tau / 2 \lt [t \bmod \tau] \lt \tau$. What is the Fourier series representation for the acceleration $\a(t)$? What is the Fourier series representation for the dipole moment $\p(t)$?
   2. At what frequencies will this system emit electromagnetic radiation? How much power is emitted at each frequency?
   3. What is the (time averaged) doubly differential power spectrum ${d^2 P(\omega,\hat \r) \over d\omega d\Omega}$, defined such that ${dP(\hat \r)\over d\Omega} = \int_0^\infty d\omega \> {d^2 P\over d\omega d\Omega}$ is the power radiated per unit solid angle in direction $\hat \r$, and $\Delta P = \int_\omega^{\omega+\Delta\omega}d\omega \int d\Omega \> {d^2 P \over d\omega d\Omega}$ is the power radiated in any direction with frequency between $\omega$ and $\omega{+}\Delta\omega$.
3. (30 pts) If the motion of a charged particle is periodic in time, with period $T$, then the emitted radiation has a discrete spectrum consisting only of integer multiples of the fundamental frequency $\omega_0 = 2\pi/T$. The angular distribution of the (time-averaged) power emitted in the $n$'th harmonic is ${dP_n \over d\Omega} = {e^2 \over 4\pi \epsilon_0} {\omega_0^4 n^2 \... ...\exp \left[ i n \omega_0 (t - \hat \r \cdot \z(t)/c \right] \right\vert^2$, where $\z(t)$ is the trajectory of the particle. Derive this.
4. (70 pts) An ultrarelativistic electron spiraling around a magnetic field line has 3-momentum (as a function of proper time) $\p(\tau) = p_\perp (\hat x \cos \omega_p \tau + \hat y \sin \omega_p \tau) + p_z \hat z$, with $p_z$ and $p_\perp$ positive. Let the $z$-axis coincide with the magnetic field line around which the electron spirals, and choose the origin so that the electron passes through the $x$-$y$ plane at coordinate time (and proper time) zero.
   1. How are the radius $R$ of the trajectory and the magnetic field $B$ related to the frequency $\omega_p$ and the electron momentum?
   2. What is the total power radiated by the electron?
   3. An observer is at rest at location $\r = r \hat \x$ (with $r \gg R$). Describe the radiation seen by this observer. In particular:
      1. Sketch the energy flux (i.e., power per unit area, or intensity) as a function of (coordinate) time. What is the manimum energy flux? At what time does the maximum flux occur? Estimate the duration of the largest pulse of radiation.
      2. Near the time of peak intensity, what is the polarization of the radiation received (i.e., in what direction does the electric field point)?
      3. Sketch the frequency distribution of the radiation. Is it continuous or discrete? Where is the peak in the frequency distribution?