= Partially filled coaxial capacitor
{title2=$C=2\pi\epsilon_0[L+(\epsilon_r-1)x]/\ln(b/a)$}
A long <coaxial cylindrical capacitor> of length $L$ and radii $a,b$ has a dielectric-filled overlap of length $x$ and <relative permittivity> $\epsilon_r$. Neglecting end and interface fringing, its filled and vacuum portions are in parallel because both share the same two conductors. The <electric potential difference> $V$ produces $E_r=V/[r\ln(b/a)]$ in either portion; the <electric displacement field> and electrode charge differ by the local permittivity. Adding their capacitances gives the title formula. The dielectric boundary is axial while the interior <electric field> is radial and tangential to that interface, consistent with the <electrostatic interface boundary conditions>.
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