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  {\bf University of Kentucky, Physics 404G \\
    Homework \#5, Rev. D, due Monday, 2018-10-08 }
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\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}{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 return the magnetic field. Warning: Don't try this at home without also including the space-charge effect and its associated ``pinwheel''!}

{\bf 2. Phase space of a Harmonic Oscillator}---All of our computer simulations have converted second order ODEs involving $\ddot x$ into first order equations in $u=(x,v)$.  This is formalized in a more symmetric way by Hamilton's equations of the canonical conjugate variables $\bs a=(x,p)$, as discussed in class.

{\bf \quad a)} The Lagrangian for the harmonic oscillator is $\mathcal L=T-V=\frac12 mv^{2}-\frac12 kx^{2}$, where $v=\dot x$.  Calculate the Canonical momentum $p=\partial\mathcal{L}/\partial v$, and solve Lagrange's equation of motion~on~$x$.

{\bf \quad b)} Calculate $\mathcal H = p\dot q-\mathcal L$ and solve Hamilton's equations by determining the time derivative of the complex coordinate $a=\frac{1}{\sqrt{2}}(x\sqrt{k}+ip/\sqrt{m})$, the same pattern used for cyclotron motion.

{\bf \quad c)} Plot the trajectory $a$ in the complex plane.  Show that $\mathcal H=a^{*}a$.  Note that the polar representation $|a|e^{i\phi}$ is called {\it action-angle} coordinates, an alternative to $(x,p)$ with trivial equations of motion.  In quantum mechanics $a$ is made dimensionless by factoring out $\sqrt{\hbar\omega}$ so \mbox{that $\mathcal H=\hbar\omega(a\ad a+\frac12)$}.

\newpage

{\bf 3.} There is a direct mechanical-electrical \href{https://en.wikipedia.org/wiki/Impedance_analogy}{impedance analogy} between electrical circuits and mechanical oscillators, matching each quantity in the drag force $F=vb$ with a corresponding quantity in Ohm's law $V=IR$, which allows direct computation of one system from the solution of the other.

{\bf \quad a)} What are the analogs of capacitance $Q=CV$ and inductance $V=L\dot I$?  Write the Lagrangian for an $LC$ circuit, and calculate the canonical momentum of $Q$.  Find the resonant frequency by analogy with the solution for a mass $m$ on a spring $k$.

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