1. Consider TE011 mode in a cubical cavity of side  a. Write the expressions for the electric and magnetic fields and show that they are orthogonal.

By definition of TE mode, E_z=0. Further, for 011 modes, . This gives . We have,

2. For the above problem determine the total energy stored in the electric and magnetic fields.

Total energy in the electric field is obtained by

One can similarly find the magnetic field energy from  . This gives,



Using the relation , this expression can be written as . So that the total energy in the field is .

3. A cubical cavity of side 3 cm has copper walls (conductivity  S/m). Calculate the Q factor of the cavity for TE101 mode.

The resonant frequency of the cavity for TE₁₀₁ mode  is given by  rad/s. Using the expression given in the text, the Q factor can be written as follows for the case of

Equivalent expressions for Q factor can be written using  and the preceding expression for  as


Where . Substituting these values we get .

 

4. Consider a cylindrical cavity resonator. Show that the dominant frequency is TE mode of the resonator.

For the TE mode of the cavity, must vanish at the surface of the end faces. The z –component of the magnetic field is, therefore, given by

The resonant frequencies are given by



where  are the zeros of the derivatives of Bessel functions. Note that unlike in the case of TM modes, in this case  because that would make the fields vanish.  For TM modes, the dominant frequency is TM010, for which the frequency is . The dominant mode for TE is TE₁₁₁ for which the frequency is given by