Scanning with electrostatic fields

Posed by: Jan Teuber for Amfitech Aps.

This problem aims to investigate scanning of dielectric objects using stationary (electrostatic) fields.

Consider a standard plate capacitor with a voltage V=Q/C across the plates, where C=\epsilon_0 A/d is the capacitance of the system.

If now a dielectric object E is inserted between the plates,

the capacitance (or, equvivalently charge and/or potential) will change in a way characteristic of the dielectric and geometric properties of the object E. For instance, for a homogeneous sphere of radius R made from a material with dielectric constant K, inserted between plates so large that boundary fields can be ignored, a surface element containing a charge Q on one of the plates will experience a decrease in potential equal to

\Delta \varphi = R^3\frac{K -1}{K + 2} Q/r^4

If we correspondingly replace the pair of simple capacitor plates with a pair of plates each of which is actually subdivided into a number of smaller segments (e.g., in a checkerboard pattern),

such that separate voltages can be applied to each individual segment, a number of voltage constellations between the upper and lower plates is now possible. Each one of these will provide information about the object E (held stationary through all the voltage combinations).

The problem posed to the Study Group is to provide an estimate of the amount of geometric and dielectric information about E that may be obtained by this procedure.

The setup is to be considered as quite general, and various other geometries (of the capacitor part of the circuit) is also of interest.