Narrow Gaps in Microwave Problems

An explicit expression, $E_x=f\left(x\right)\mathrm{}$, is given for the $x$ component of the electric field on the $y=0\mathrm{}$ plane in the gap between two infinite, oppositely charged conducting masses bounded by the $y=0\mathrm{}$, $x=b$ planes and the $y=0\mathrm{}$, $x=-b$ planes, respectively. The formula can be handled analytically and its maximum deviation from the rigorous value is one part in 5000. The coefficients of the Fourier expansion of $E_x$ and $V$ are given and used to write down a rapidly convergent series for the potential in the narrow gap between two right circular, coaxial conducting cylindrical electrodes. The resonant frequency of a re-entrant cylindrical cavity with a narrow gap is computed to about 0.03 percent. The radiation pattern from the open end of a rectangular wave guide terminating in an infinite conducting plane and transmitting only the $T{E}_{10}$ mode is calculated with four-place accuracy.