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\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Homework \#5, Rev. A, due Friday, 2019-10-18 }
\end{center}

\vspace{.3 in}

{\bf 1.} It has been claimed that World War II was won by the \href{https://web.archive.org/web/20180219122811/http://www.radartutorial.eu/08.transmitters/Magnetron.en.html}{\bf Magnetron}, a vacuum device which converts electrical energy into microwave radiation.  It enabled short-wavelength tactical radar, which used a frequency too high for electrical circuits of the time.  Radiation is emitted from several cylindrical microwave cavities tangent to a central cylindrical bore of a solid copper waveguide under vacuum.  A thermionic cathode rod of radius $a$ in the center emits electrons, which accelerate from the negative high voltage cathode to the grounded outside walls of the cavity at radius $b$.  A longitudinal magnetic field $\bs B$ sweeps the electrons in a spiral path around the cylinder, exciting resonant currents on the walls of the microwave cavities they pass by.  The electric and magnetic fields are perpendicular in this {\it crossed-field} device.

{\bf \quad a)} We already calculated the equation of motion for an electron in a magnetic field, $\dot\eta=-i\omega\eta$, using complex coordinates $\xi=x+iy$ and $\eta=\dot\xi=v_{x}+iv_{y}$, where $\omega=qB/m$ is the cyclotron frequency.  Add the force $\bs F=q\bs E$ due to the electric field $\bs E={\bh \rho V}/{\rho\ln(b/a)}$, where $a=0.5~\q{cm}$ and $b=2.5~\q{cm}$ are the inner and outer radii of the cylindrical cavity, respectively, with negative voltage $V$ applied to the {\it cathode} (central electron-emitting elecrode), and the {\it anode} at ground (0 V). {\it Hint: the final equations should be of the form $\dot\xi=\eta$ and $\dot\eta=-i\omega\eta + \lambda \xi/|\xi|^{2}$, for some constant $\lambda$ to be determined.}

{\bf \quad b)} Integrate these equations in Matlab, for an electron emitted from the cathode essentially at rest.  Before you start, what will the motion look like qualitatively?  Determine the {\it Hull cut-off} magnetic field $B_{c}$, such that an electron barely grazes the surface of the anode and returns to the cathode, using $V=-4~\q{kV}$. [bonus: Determine the function $B_{c}(V)$.]

{\bf \quad c)} The electrons from part a) excite eight cylindrical resonant cavities of radius $r=1~\q{cm}$ distributed around the central cavity, each one separated from the anode (at $\rho=b$) by a $w=4~\q{mm}$ long, $t=5~\q{mm}$ wide channel cut in the solid cylindrical copper block of thickness $d$.  Calculate the resonant frequency using the inductance $L=N^{2}\mu_{0}\pi r^{2}/d$ and capacitance $C=\epsilon_{0}w d/t$.  Optimize both the voltage $V_{0}$ and magnetic field $B$ such that the electrons just graze past the resonators at the resonant frequency.  {\it Hint: use conservation of energy to determine the required voltage and then determine the magnetic field. Warning: Don't try this at home without also including the space-charge effect and its associated ``pinwheel''!}

{\bf 2. Damped Driven Pendulum}  A mass $m$ swings at the end of a massless rod $\ell$ rotating about a fixed point at the top.

{\bf \quad a)} Calculate the Lagrangian $\mathcal L(\dot\theta,\theta,t)$ and resulting equation of motion for the undamped, undriven case.  Express it in terms of the natural frequency $\omega_{0}=\sqrt{g/\ell}$ at small angles.  [bonus: Calculate the exact trajectory, starting from rest at angle $\theta_{0}$, in terms of the Jacobi elliptic function $sn(u;m)$ and calculate the frequency $\omega(\theta_{0})$.]

{\bf \quad b)} Calculate the canonical momentum $p_{\theta}$ and Hamilton's equations for the undamped, undriven case.  Calculate the maximum velocity $v_{1}$ of the pendulum, starting from rest at the top, using $m=1~\q{kg}$ and $\ell=1~\q{m}$.  Plot trajectories in phase space starting from equilibrium $\theta=0$ with initial velocity $v_{k}=k v_{1}$ for $k=0, \pm\frac15, \pm\frac25,\frac 35, \ldots 2$.  You can obtain a more complete flow of phase space by starting from more equilibrium points: $\theta_{0}=0,\pm 2\pi, \pm 4\pi$.

\newpage

{\bf \quad c)} Recalculate the trajectories, adding a frictional force $F=-bv$, starting with $b=1~\q{kg/s}$, normalized to $2\beta=b/m$.  Determine the critical damping coefficient $b_{c}$, above which oscillations entirely disappear.  Repeat the phase space plots for $b=b_{c}/2, b_{c}$, and $2b_{c}$.

{\bf \quad d)} Recalculate the trajectories, adding a driving force $F=F_{0}\cos(\omega t)$ on resonance, $\omega=\omega_{0}$, with $F_{0}=\gamma mg$, and $\gamma=1$ for this problem.  Repeat the trajectories $\theta(t)$ and phase space diagrams $[p_{\theta},\theta](t)$ for the same values of $b$ as part c).

% [bonus] {\bf \quad e)} Symplectic integrator: conserve the Hamiltonian, which in this case has the interpretation of total energy of the system.

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