Physics 515 Problem Set #1 April 1, 2004
Due Thursday, April 8, 2004

1. (50 pts) Complete the derivation of eigenmodes in a straight, infinite waveguide with perfectly conducting walls and vacuum in the interior, Let $\hat z$ be the longitudinal direction of the waveguide. The cross-section of the waveguide is some unspecified shape which occupies a region $\cal S$ in the $x$-$y$ plane. Specifically:
   1. Explain why appropriate boundary conditions at the walls of the waveguide are $\hat\n \cdot \B = 0$ and $\hat\n \times \E = 0$, where $\hat\n$ is the normal to the surface.
   2. Find TE modes, for which $\E(\r,t) = \Grad \times [\hat z \psi(x,y) e^{i k z -i \omega t}]$. In order to satisfy all Maxwell equations, and the appropriate boundary conditions, show that $\psi(x,y)$ must be an eigenfunction of the transverse Laplacian $\nabla^2_t \equiv \nabla_x^2 + \nabla_y^2$ (up to a physically irrelevant additive constant), with Neumann boundary conditions. How is the frequency $\omega$ related to the longitudinal wavenumber $k$ and the eigenvalue of $-\nabla^2_t$? What is the cut-off frequency $\omega_c$ for this mode? (The cut-off frequency for a particular mode is the minimum frequency for which solutions propagating down the waveguide exist.)
   3. Repeat the above for TM modes, for which $c \B(\r,t) = \Grad \times [\hat z \chi(x,y) e^{i k z -i \omega t}]$. What eigenvalue equation and what boundary conditions must $\chi$ satisfy?
   4. When can a TEM mode (for which both $\E$ and $\B$ are transverse to $\hat z$) exist? What conditions must hold on the eigenvalue spectrum of $\nabla_t^2$, and ultimately on the shape of the region $\cal S$?
   5. Suppose a single mode, either TE or TM, is excited. Compute the time-averaged energy per unit length (for physical fields which are the real parts of the complex solutions above), and show that $<\hbox{energy}/\hbox{length}> = \half \epsilon_0 (\omega_c/c)^2 \int_{\cal S} dx \> dy \> \vert\psi\vert^2$ or for TM modes, the same expression with $\phi \to \chi$.
   6. Again with a single mode excited, compute the energy flow per unit time along the waveguide. Show that $<\hbox{energy flow}/\hbox{time}> = v_g \hat z <\hbox{energy}/\hbox{length}>$, where $v_g = \partial \omega/\partial k$ is the group velocity of the mode.
   7. Suppose the waveguide has a circular cross-section. For what range of frequencies does a single mode propagate down the waveguide?
2. (30 pts) Consider harmonically varying electromagnetic fields near the surface on an imperfect conductor. Inside the conductor, the frequency-dependent conductivity $\sigma(\omega)$ is non-zero and finite. Outside the conductor is vacuum. Assume that $\omega \ll \sigma(\omega)/\epsilon_0$. Let $\hat\n$ be an outward-pointing normal at some point on the surface of the conductor. Assume that the radius of curvature of the surface of the conductor is much larger than the skin depth (at the given frequency). Evaluate the time-averaged energy flux into the conductor, and show that at the surface of the conductor $\langle \hat n \cdot \S \rangle = - \coeff 14 \omega \delta \vert\B\vert^2/\mu_0$.
3. (50 pts) An empty cavity with perfectly conducting walls has length $L$ in the $z$ direction, and a perpendicular cross-section of some arbitrary (simply connected) shape which is independent of $z$.
   1. Construct TE eigenmodes within the cavity for which $\E(\r,t) = \grad \times \hat z f(\r) e^{-i \omega t}$, by suitably superposing the TE waveguide modes found earlier. What boundary conditions must $f(\r)$ satisfy at $z = 0$ and $L$? What is $\B(\r,t)$? What are the resonant frequencies (in terms of appropriate eigenvalues of the two-dimensional Laplacian)?
   2. Repeat the above analysis for TM modes for which $c\B(\r,t) = \grad \times \hat z g(\r) e^{-i \omega t}$.
   3. Let the cavity be a cube of size $L$. What are the resonant frequencies? Which mode(s) have the lowest frequency? Show that the number of modes $dN$ with resonant frequencies between $\omega$ and $\omega+d\omega$ is given by $dN = 2 (4\pi \omega^2 d\omega/c^3) (L/2\pi)^3$, provided $L \gg c/\omega$.
   4. Now suppose the conductors forming the walls of this cubical cavity have a non-zero skin depth $\delta$ (with $\delta \ll L$). Evaluate the $Q$ of the cavity in its lowest mode.
   5. Extra credit: Return to the case of ideal conducting walls, and let $\alpha$ label a individual eigenmode (so $\alpha$ really denotes three integers plus an indication of TE vs. TM). ``Quantize'' the electromagnetic field: suppose that the energy of mode $\alpha$ with eigenfrequency $\omega_\alpha$ can only take on values which are some non-negative integer $N_\alpha$ times $\hbar \omega_\alpha$. Suppose that the probability that a specific value of $N_\alpha$ occurs is given by $Z_\alpha^{-1} \exp(-\beta N_\alpha \hbar \omega_\alpha)$, where $Z_\alpha = \sum_{N=0}^\infty \exp(-\beta N_\alpha \hbar \omega_\alpha)$. Show, in the limit where the cavity volume becomes very large, that the average total electromagnetic field energy is given by $<U>_\beta = (V / \pi^2 c^3) \int_0^\infty {\hbar \omega^3 d\omega} \> [e^{\beta \hbar\omega} -1]^{-1}$. This correctly describes blackbody radiation at temperature $T = 1/\beta$ confined to a cavity.