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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.

Electron source

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  • $\begingroup$ A diagram of the situation you actually want to model would be helpful. Does your accelerator element have something like a "screen" electrode to terminate the electric field while allowing electrons to pass ballistically? $\endgroup$
    – The Photon
    Commented Oct 10, 2023 at 17:32
  • $\begingroup$ The surrounding vacuum is not at zero - where are your V=0 boundaries defined? $\endgroup$
    – Jon Custer
    Commented Oct 10, 2023 at 18:34
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    $\begingroup$ "Solving the equation is easy, but something is still missing." So, it sounds like solving the equation is actually not so easy after all. $\endgroup$
    – hft
    Commented Oct 11, 2023 at 1:19
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    $\begingroup$ "the surrounding vacuum is assumed to be at 0" -- why do you assume this? Zero potential should occur infinitely far away from the system or at surfaces that are grounded. $\endgroup$
    – Andrew
    Commented Oct 11, 2023 at 1:41
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    $\begingroup$ @Magemathician Well, the assumption that $\phi=0$ in the vacuum is very much at odds your statement that "the equation is trivial." The solution to Laplace's equation $\nabla^2\phi=0$ depends on the boundary conditions. You cannot conclude that $\phi=0$ directly from the differential equation. In fact there are some quite non-trivial solutions of Laplace's equation eg en.wikipedia.org/wiki/Laplace%27s_equation#Analytic_functions. $\endgroup$
    – Andrew
    Commented Oct 11, 2023 at 13:08

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What you are describing seems to be fairly simple: You have two conducting objects connected by a conducting wire, in which there is inserted a high voltage supply. The high voltage supply imposes a voltage (i.e. a potential difference!) between the two objects, cap and anode. Since the absolute value of potential doesn't matter (only potential differences do) you can set the potential to zero on one of the objects, as I'm guessing from your question you do that with the anode. Once you do this the potential in the cap should be (plus or minus) the potential difference given by the voltage supply.

Now you know the potential on anode and cap and want to calculate the field (and potential) in the surrounding vacuum. In vacuum you have $\rho = 0$. Insert this in Gauß' law and you find $$ \Delta \phi = 0 $$ in vacuum and given values for $\phi$ on the boundary of the vacuum (the two objects). This is called a boundary value problem for the Poisson equation (with Dirichlet boundary conditions). A pretty standard problem both in electrostatics and numerics...

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  • $\begingroup$ So the condition is $\rho = 0$ in the vacuum and $\phi = -V$ on the boundary surface? What is $\rho$ of the cap at the potential (and or similarly the anode)? Obtaining $\rho$ from the voltage was actually the first formulation of the question. $\endgroup$
    – Magemathician
    Commented Oct 11, 2023 at 12:27
  • $\begingroup$ If you want to solve for the field in vacuum you can do that with the above boundary value problem. The value of $\rho$ inside anode and cap, or what causes the boundary values of $\phi$ is of no importance for the solution in the vacuum once you fix the boundary values according to the give voltage (from the voltage supply). $\endgroup$
    – kricheli
    Commented Oct 11, 2023 at 13:56
  • $\begingroup$ @krichelil Eh, I take it I can assume $\phi = -V$ in all of the cap (and possibly $\phi=0$ for the anode) and solve the entire domain? This may be outside the question, but is there a reference on what is the preferred way to implement the constraints for conjugate gradient method? $\endgroup$
    – Magemathician
    Commented Oct 11, 2023 at 14:21
  • $\begingroup$ Yes, $\phi=$const. in either the anode or the cap (because they are conductors). As for your second question, yes this is outside the question, and sadly I can offer no expertise on this. $\endgroup$
    – kricheli
    Commented Oct 11, 2023 at 15:07
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    $\begingroup$ @Magemathician This set of notes gives a matlab implementation of the conjugate gradient method for the Poisson equation: sites.cs.ucsb.edu/~gilbert/cs240a/old/cs240aSpr2011/hw2/hw2.pdf. The vector $b$ in the last equation of Section 1 encodes the boundary values. $\endgroup$
    – Andrew
    Commented Oct 11, 2023 at 16:23

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