On the charge density measured by the capacitive probe

In investigation on behaviour of excess charge on an electric insulator, a capacitive probe is usually used to detect this charge. The relation between response of the probe and the excess charge density varies with the shape of the insulator. For very thin film, the record of the probe is 1/ε_r times of the excess charge density; and 2ε₀/(ε₀+ε₁ ) times of the excess charge density for thick slab. In the case of right cylindrical spacer, the response of the probe is equal to the excess charge density when its distribution on the circumference is uniform