# Tutorial: Resonant cavities

1. (0 pts) Carefully study sections 8.7 and 8.8 in Jackson.
2. (4 pts) Provide all the details leading to Eq. (8.74)-(8.79).
3. (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 $$R/d$$.
4. (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.
5. (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. (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 $$\mathbf{B}$$).
7. (3 pts) Identify the lowest resonant frequency of the cubic cavity and specify its degeneracy.
8. (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 $$a$$?
9. (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.
10. (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.