Consider a cylindrical capacitor consisting of an inner conducting cylinder of outer radius $R_1$ and an outer conducting cylinder of inner radius $R_2.$ The length of the capacitor is $L.$ The capacitor is oriented ertically and is fully immersed into a dielectric liquid of susceptibility ${?}_0$ and mass density $n_{}$ that fills the region between the two conductors. The conductors are charged to $+-Q$ with positive charge on the nner cylinder. Neglect any effects of fringe fields at the cylinder ends. 

 

$($a$)$ Find the energy of this configuration.

$($b$)$ The capacitor is discharged, and is then raised so that the cylinders are only partially immersed in the liquid. The capacitor is charged to voltage $V$ with a battery, and the liquid rises a height $h$ in the space between the tube walls. Determine $h$ in terms of the parameters given above.