Problem set 10

Problem 1

In cylindrical electromagnetic cavities the transverse electric modes are noted by TE${}_{mnp}$, where $p$ is the index for the component in the $z$-direction (the axis of the cylinder), $m$ is associated with the angle $\varphi$, and $n$ with the radial coordinate $r$. We consider a cylindrical cavity with height $d$ and radius $a$.

Use the general expressions for the TE${}_{mnp}$ modes derived in Problem set 10 2014. When the cylindrical waveguide is closed at $z=0$ and $d$ we have to add $H_r\sim J^{\prime }({\chi }_{01}^{\prime }r/a)\cos (\pi z/d)$ to the TE${}_{011}$ mode.

(a) Show how the TE${}_{011}$ mode fulfills the boundary conditions and sketch the field lines for its components.

(b) Find the asymptotic forms of the field components as $r<<a$ in the center of the cylinder.

(c) Find a vector potential $\mathbf{A}$ that can deliver the asymptotic fields in the center of the cylindrical cavity.

The problem is due Monday March 24 2025 at 20:00