Answer to Question #128308 in Electricity and Magnetism for christopher seebaran

Question

A hollow rectangular waveguide has dimensions a = 2.28 cm., b = 1.01 cm. What TE modes will propagate in this waveguide if the frequency of the electromagnetic waves is 1.7 x 1010 Hz? What range of frequencies can be used to excite only one TE mode?

Answer 1

As per the given question,

Dimensions of the wave guide $a = 2.28 cm, b = 1.01 cm$

Frequency of the electromagnetic wave $(f)=1.7\times 10^{10}Hz$

Range of frequencies =?

Cut off wave guide for the dominant $TE_{10}$ mode $=\frac{2}{\sqrt{(\frac{m}{a})^2+(\frac{n}{b})^2}}$

$=\frac{2}{\sqrt{(\frac{1}{2.28})^2+(\frac{0}{1.01})^2}}$

$=0.046 cm$

For the higher wavelength $=\frac{2}{\sqrt{(\frac{m}{a})^2+(\frac{n}{b})^2}}$

$=\frac{2}{\sqrt{(\frac{1}{2.28})^2+(\frac{1}{1.01})^2}}$

$=4.02 cm$

Hence the required frequencies

$f_c =\frac{c}{2a} =\frac{3\times 10^8}{2\times 2.28 \times 10^{-2}}Hz$

$=0.657 \times 10^{10}Hz$

and

$f_c=\frac{c}{2b}$

$=\frac{3\times 10^8}{2\times 1.10 \times 10^{-2}}Hz$

$=1.36\times 10^{10}Hz$