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.
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
