# Electric field in coaxial cavity

I have a geometry as follows:

This is a hollow metallic cylinder with a small metallic rod inside. The rod is connected to the bottom of the cylinder but is a bit shorter than the outer cylinder. It is some type of coaxial cavity. The larger radius is rc, the length is l, the rod has radius rr and length l-d. What I want to do now is to solve the wave equation in this domain. I don't know whether it is possible to analytically do that, but I read in an IEEE paper that it should work.

To solve my problem, I first took the scalar Helmholtz equation $$\frac{1}{\varrho} \, \frac{\partial}{\partial \varrho} \left( \varrho \, \frac{\partial \psi}{\partial \varrho} \right) + \frac{1}{\varrho^2}\,\frac{\partial^2 \psi}{\partial \varphi^2} + \frac{\partial^2\psi}{\partial z^2} + k^2\,\psi = 0$$ and tried it to separate as follows: $$\psi = R(\varrho) \, \Phi(\varphi) \, Z(z)$$ I am now interested on the z component of the electric field. The boundary conditions are: the z component must vanish at the outer wall of the cylinder from z = 0 to z = l, and it must also vanish at the inner wall from z = 0 to z = l-d. Using above separation, one finds the three ODEs $$Z'' + k_z^2\,Z = 0$$ and $$\Phi'' + n^2\,\Phi = 0$$ and $$\varrho^2\,R'' + \varrho\,R' + R \left( \varrho^2\,a^2 - n^2 \right) = 0$$ and I do know that I need the Bessel function of the first kind to solve the last ODE. However, I don't know what to do with the boundary conditions because I have two different ranges for z, where psi must vanish. How can I proceed?

• I've attempted to give a sketch of a solution. If more detail is wanted, you're likely to get better answers on the Physics SE or Computational Science SE. If you want search terms to research further, you're probably looking for Green's functions. – The Photon Jul 26 '16 at 16:46

Then you can express the field in the lower section as a sum of modes with the appropriate boundary conditions (longitudinal field going to 0 at $r=r_r$), and the field in the upper section as a sum of different modes (with no interior boundary). You'd then have a boundary condition at the interface between the two "cavities" where the total field must be equal for the two solutions at $z=l_d$.