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 $$E=-\nabla V \Leftrightarrow V=-\phi.$$

But how to set the distribution of voltages in the problem? Intuitively, I would assume that the accelerator element is pushed full of electrons as consequence of the applied voltage, therefore the solution disregarding a beam should be equivalent to assuming constant charge density for the accelerator element. Is this a correct intuition?

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.