Physics 515 Problem Set #6 May 13, 2004
Due Thursday, May 20, 2004

1. (25 pts) An electron in a linear accelerator has 4-momentum $p^\mu(\tau) = (p^0(\tau),0,0,p^3(\tau))$, with $p^2 + m_e^2 = 0$. Show that $(dp/d\tau)^2 = (dE/dz)^2$, where $E = cp^0$ is the electron's energy. Evaluate the power radiated and compare the rate at which energy is lost to radiation to the rate at which the electron gains energy due to its acceleration. If $E_{\rm Rad}$ denotes the energy lost to radiation, show that $(dE_{\rm Rad}/dz) \Big/ (dE/dz) = {2\over3} (c/v) (r_0 / m_ec^2) (dE/dz)$, where $r_0 = (e^2 / 4\pi)/ (m_e c^2) \approx 2.8 \times 10^{-13}$ cm is the ``classical electron radius''. Use this to estimate the fractional loss of energy to radiation for the SLAC accelerator, which is 2 miles long and accelerates electrons to an energy of 40 GeV.
2. (25 pts) An electron with momentum $\p = p \, \hat \z$ flies by a positron with momentum $\p' = -p \, \hat \z$ at an impact parameter $b$. Both particles are ultra-relativistic, $p \gg mc$, and the impact parameter is sufficiently large so that the relative change in momentum of either particle is small. Estimate (in the lab frame):
   1. The maximum value of the electric field (due to the positron) acting on the electron.
   2. The maximum value of the magnetic field (due to the positron) acting on the electron.
   3. The duration of the electromagnetic field pulse which acts on the electron (i.e., the time interval during which the field is within a factor of two of its peak value).
   4. The total energy radiated during the ``collision''.
   5. The frequency at which the power spectrum $dE_{\rm Rad}/d\omega$ is peaked.
   (``Estimate'' means determine the dependence on all relevant parameters, without worrying about overall pure numerical factors.)
3. (30 pts) The mechanical stress-energy tensor for a swarm of particles with arbitrary spacetime trajectories may be written as $T^{\mu\nu}_{\rm M}(x) = \sum_a m_a c \int d\tau_a \> u_a^\mu(\tau_a) \, u_a^\nu(\tau_a) \, \delta^4(x{-}z_a(\tau_a))$, where $m_a$ is the mass of the $a$'th particle, $z_a^\mu(\tau_a)$ is its spacetime location (as a function of its proper time), and $u^\mu_a(\tau_a)$ is its 4-velocity.
   1. Perform the proper time integral and express the various components of $T^{\mu\nu}_{\rm M}(x)$ in terms of the spatial positions, 3-velocities, spatial momentum, and/or energies of the particles.
   2. What is $Q^\mu(x^0) \equiv \int d^3x \> T^{\mu 0}_{\rm M}(x)$?
   3. Let $q_a$ be the charge of the $a$'th particle, and that all particles move under the influence of an electromagnetic field. Prove that $\partial_\nu T^{\mu\nu}_{\rm M}(x) = {1 \over c} \, F^{\mu\nu}(x) \, j_\nu(x)$.
   4. Explain why the above results imply that $F^{0\nu}(x) j_\nu(x)$ gives the rate at which the electromagnetic field increases the kinetic energy density of a system of moving charges (the rate of ``Joule heating''), and explain why $(1/c) F^{k\nu}(x) j_\nu(x)$ gives the rate at which the electromagnetic field increases the ($k$'th component of the) momentum density of the moving charges.
   5. Using Maxwell's equations in covariant form, prove that $\partial_\nu \, T^{\mu\nu}_{\rm EM}(x) = -{1 \over c} \, F^{\mu\nu}(x) \, j_\nu(x)$, where $T^{\mu\nu}_{\rm EM}(x) = \epsilon_0 \left[ F^{\mu\lambda}(x) F^\nu_{\;\;\... ... {1 \over 4} \, g^{\mu\nu} \, F^{\alpha\beta}(x) F_{\alpha\beta}(x) \right]$ is the electromagnetic stress-energy tensor. Combined with part (a), this proves that the total stress-energy tensor, $T^{\mu\nu}(x) \equiv T^{\mu\nu}_{\rm M}(x)+T^{\mu\nu}_{\rm EM}(x)$, is exactly conserved, $\partial_\nu \, T^{\mu\nu}(x) = 0$.