(0 pts) Carefully study sections 8.7 and 8.8 in Jackson.
(4 pts) Provide all the details leading to Eq. (8.74)-(8.79).
(3 pts) Consider a circular-cylindrical cavity discussed in section 8.7. Introduce a reduced (dimensionless) frequency \( \tilde\omega=\omega R/c \), where \( R \) is the radius of the cavity’s cross-section and \( c \) the speed of light in the material that fills it. Find the reduced resonant frequencies \( \tilde\omega \) corresponding to the TE and TM resonant modes, expressing them as functions of the geometrical parameter \( R/d \).
(3 pts) Find the lowest 5 resonant frequencies of the cavity from step #2 in the limit \( R/d\to 0 \). Note the degeneracies, if any.
(4 pts) For the modes found in step #3, use a computer to plot the dependence of the reduced resonant frequencies on the parameter \( R/d \) in the range from 0 to 3.
(6 pts) Consider a perfectly conducting cubic cavity with an edge \( a \). Align the “waveguide axis” \( z \) with one of the cube edge and classify the eigenmodes of the cavity. Find the corresponding resonant frequencies and field configurations (all components of \( \mathbf{E} \) and \( \mathbf{B} \)).
(3 pts) Identify the lowest resonant frequency of the cubic cavity and specify its degeneracy.
(2 pts) Suppose we can now change the height \( h \) of the cavity while keeping the other two dimensions fixed. If we want to make the lowest resonant frequency non-degenerate, should we make h greater or smaller than \( a \)?
(1 pts) For a cavity of an arbitrary shape (not a capped cylinder), we need to return to Eq. (8.17) and solve it along with (8.16) and the boundary conditions. Argue that the eigenvalue problem (wave equation + divergence equation + boundary condition) for \( \mathbf{E} \) is real, and therefore the solutions can be chosen to be real. Note that this means that \( \mathbf{E} \) oscillates in phase throughout the whole cavity.
(2 pts) Using a curl equation from (8.16), show that the \( \mathbf{E} \) and \( \mathbf{B} \) fields are exactly out of phase. This means that the field configuration oscillates between purely electric and purely magnetic fields, like it does in an \( LC \) circuit.