I am attempting to compute electric field in a vacuum (preferably by finite volume method) near a high voltage accelerator element, as used in electron guns, analog televisions and so forth. By Maxwell's equations

$$\nabla \cdot E=\nabla \cdot \nabla \phi=\rho,$$ where $\phi$ is a scalar field, which is easy to solve from a known charge (density) $\rho$. However, in the problem description the potential $V$, not charge $C$ of the accelerator element is known. The voltage $V$ is defined as the negative of the scalar field $$E=-\nabla V \Leftrightarrow V=-\phi.$$ Solving the equation is easy, but something is still missing. If a constant voltage of say $10 \text{kV}$ is assumed for accelerator element and the surrounding vacuum is assumed to be at $0$ the electric field exists only very near the accelerator element, which is incorrect. The classic isopotential lines are not obtained.

I would assume that the accelerator element is pushed full of electrons as consequence of the applied voltage, thus the electric field should extend further. How to correctly attack the problem? Computing $V$ properly so that the correct solution for $E$ is obtained seems to be the important and missing part. The naive assumption that the accelerator element only is at the specified high voltage $V$ seems to fail.

Edit: From the request, a picture of the problem, although in the problem the anode ($= 0 V$) does not have a gap to pass through, but rather is the direct target. The cap is a cylindrical object whose field is the primary interest.

Edit 2: An interesting matter is whether the modelling can be carried out assuming constant charge density $\rho$. I am assuming no, so then why is the intuition of the accelerator being pumped full of charge in an uniform way incorrect?