Hi Tony,
You may like to build first the resonant tank (cylinder and a capacitor in parallel).
I guess you will need two disks for the edges and the resonant capacitor could be connected between their centres.
About the formula, maybe one can start, as a rough approximation, with the one of a wire with skin effect:
Lw=0.2H[Ln(4*H/D)-1]
Lw in uH
H in mm
D in mm
Of course, this formula assumes that H>>D and there is no return current inside the cylinder and both are not valid here. The latter could be made true if the capacitor is connected outside the cylinder but this may not be desirable.
The other related formula could be of two concentric cylinders.
In your case, the inner cylinder diameter (d) is of the wire connecting the two terminals of the resonant capacitor with the edge disk centres.
Lc = 4π*10-7 * [Ln(D/d) + 1/4]
Lc in H/m
D and d could be in mm
Also this formula assumes that the cylinder length >> D which is not the case here.
Lw gives 130 uH (I think the value is rather too high)
Lc gives 33 uH assuming the inner wire diameter is 1mm
It will be interesting to find out empirically the right value of L by measuring the resonant frequency of the tank with a known C and inner wire diameter.
Kerim