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Consider a torus made of electrically conductive material that is given some charge
$Q$. What would be the charge distribution on it? I tried using Gauss's law.

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I suppose this problem is solvable by using toroidal coordinates. These are an orthogonal coordinate system. The surfaces of constant $\tau$-coordinate are tori. An electrostatic field is orthogonal to the surface of the conductor. The absence of a tangential field component means that the surfaces of constant electrostatic potential are also tori in this problem. I think this should be true, at least, near the surface of the conductor. Knowledge of the constant potential surfaces allows for the finding of the electric field on the surface of the conductor. And only now one can use Gauss's law to find the charge's surface density.

Update. In the general case, there is no simple solution to this problem. I was able to find the following discussion: link to the outer forum. It contains a reference to the article about torus's electrostatic field.

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  • $\begingroup$ Can you give a simpler solution ? All I understood is you are telling me to use a toroidal gaussian surface $\endgroup$
    – Protein
    Apr 16, 2020 at 23:59
  • $\begingroup$ @Protein, I am afraid there is no simple solution to this problem. See my updated answer. $\endgroup$
    – Gec
    Apr 17, 2020 at 7:45

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