Section 7.1 · Further exercises · 2 of 5
Exercise 7.1.2: An unknown guide with an unknown filling
About the further exercises of Section 7.1
Five further exercises on the general analysis of uniform guides, graded from easy to harder. None of them needs the shape of the cross-section: (or a measurement) is all that is known.
An unknown guide with an unknown filling
#A metal guide of unknown cross-section is filled with an unknown lossless nonmagnetic dielectric. With a slotted line in its lowest mode one measures at and at . (a) Find the speed in the filling, its relative permittivity and the cutoff frequency of the mode. (b) Find , and the wave impedance at if the mode is TE. (c) The filling is poured out. What are the cutoff frequency and the guide wavelength at now?
(from with ), so . One measurement gives one equation in the two unknowns and , two measurements give both. The geometry fixes , hence when the filling changes.
Subtracting the two equations eliminates :
Then and , so .
, :
is unchanged, and grows with : . At , and .
At : , the second measurement.
The filled guide carries at , the empty one only at , close to cutoff, which explains the longer guide wavelength. A dielectric filling lowers every cutoff by : the same signal fits through a smaller pipe, at the price of dielectric loss (Exercise 7.1.5).
cf. guide wavelength