# How can we derive the spatial dependence of the electric potential when having a curved plate capacitor?

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.

The electrostatic problem of the parallel plates is relatively simply and easily solved by Gauss' law because the form of the field, due to the symmetries of the shape of the domain. Beware that the above formula is accurate only for infinite plates. At the edges the potential will have a much more complicated form (see a very beautiful picture of the potential lines from Maxwell's treatise on electricity and magnetism).

For arbitrarily curved plates there is no easy way out (in most cases): you have to solve Poisson's partial differential equation $\nabla^2V=-\frac{q}{\varepsilon}$ with appropriate boundary conditions. This equation can be solved only numerically in most of the cases. However, for some simple geometries there are workarounds. For a spherical capacitor (two concentric spherical electrodes) there is an analytical solution that can be derived with Gauss' law. For other structures (e.g. some combinations of flat surfaces) one can use some tricks from complex analysis (like conformal and Schwarz-Christoffel mappings) to get an analytical equation for the field. See here for links to some software and here for theoretical explanations and calculations.