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Consider a parallel-plate capacitor that is partially filled with a dielectric of dielectric constant k. the dielectric has the same same height as the separation of the plates of the capacitor but fills a fraction f of the area of the capacitor. the capacitance of the capacitor when the dielectric is completely removed is c0.

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The formula for the capacitance of the capacitor made of two parallel plates with a dielectric in between them is
C = k A f /d

When the dielectric is removed, air is between the plates with a permissivity of 1
Co = A / d

The height of the dielectric is the same as the separation of the plates. So,
k h lf / C = h l / Co
The ratio of the lengths of the dielectric and the plates is
lf / l = C / k Co

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