When we have flat infinite parallel plates, making a capacitor, as in the image below:
then the electric potential between these plates will vary as:
I understand how this can be derived using charge densities on the plates then calculate the electric field using Gauss law and then we can get the electric potential from this but my question is now, how do we derive this spatial dependence of the electric potential V(x) using curved plates.
The only thing we know is the potential on the plates, and the shape of the plates. This is in relation to lab experiments using curved electrodes to get a high electric field at the tip of the electrodes. We don't know the charge densities on the plates.
The easiest approach I would take is to change distance between the plates d to be d(y) if y is the vertical spatial component in the above picture. But I don't know how to formally derive this correctly.