Question #80153

A vertical cylindrical capacitor consists of two concentric conducting tubes of inner radius $a \text{ (m)}$ and outer radius $b \text{ (m)}$. It is partially immersed in a large bath of a non-conducting polar liquid of density $\rho \text{ (kg/m}^3\text{)}$, surface tension $\gamma \text{ (N/m)}$, and dielectric constant $K \text{ (dimensionless)}$. The contact angle of the liquid with both the inner and outer tubes is $\theta_0 = 120^\circ$. Initially, the liquid level inside the annular region is depressed due to capillary action. A DC voltage $V \text{ (V)}$ is applied between the tubes. What is the minimum voltage $V$ required to raise the liquid level inside the capacitor so that it is exactly flush with the free surface of the liquid bath outside? (Assume the liquid bath is large enough that its level remains constant, and edge effects at the bottom of the capacitor are negligible. Permittivity of free space is $\varepsilon_0 \text{ (F/m)}$.)