Not all dielectrics that separate the plates of a capacitor are rigid. For example, the membrane of a nerve axon is a bilipid layer that has a finite compressibility. Consider a parallel-plate capacitor whose plate separation is maintained by a dielectric of dielectric constant k = 3.0 and thickness d = 0.2 mm when the potential across the capacitor is zero. The dielectric, which has a dielectric strength of 40 kV/mm, is highly compressible, with a Young's modulus for compressive stress of 5 × 106 N/m2. The capacitance of the capacitor in the limit V → 0 is C0.

(a) Derive an expression for the capacitance as a function of voltage across the capacitor.

(b) What is the maximum voltage that can be applied to the capacitor? (Assume that k does not change under compression.)

(c) What fraction of the total energy of the capacitor is electrostatic field energy and what fraction is mechanical stress energy stored in the compressed dielectric when the voltage across the capacitor is just below the breakdown voltage?