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I know the field difference across a charged sheet-but what's the potential on a charged sheet? Is it just asymptotic? If we had a thin cylindrical sheet would it be the same? Zero field inside the cylinder, and jump to infinity on the sheet, and q/r outside?

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If you take $U_{\infty}=0$ then $U_{sheet}=\infty$ (for infinite sheet)

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Remember that potential is not an absolute so the actual value of the potential at a point has no meaning. For the cylinderical case $\vec{E} = \frac{\lambda}{2\pi \epsilon_0 r} \hat{r}$ outside the cylinder and zero inside. and $V = \frac{-\lambda}{2\pi \epsilon_0}\ln r$ outside and $V=constant$ inside.

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