AP Physics C: Electricity & Magnetism 10.4 A Describe How a Dielectric Inserted Between the Plates of a Capacitor Changes the Properties of the Capacitor Questions

An isolated, air-filled, charged capacitor consists of two conducting, coaxial, cylindrical shells that each have length L. The inner shell has radius $R_{1}$ and the outer shell has radius $R_{2}$, as shown in Figure 1, where $R_{1}<R_{2} \ll L$. The surface charge densities (amounts of charge per unit area) of the inner and outer shells are $+\sigma_{1}$ and $-\sigma_{2}$, respectively. The absolute values of the total charges on the shells are equal.

Note: Figures not drawn to scale.

A material of dielectric constant $\kappa$ is inserted into the isolated, charged capacitor such that the material fills the region $R_{1}<r<R_{2}$, as shown in Figure 3.

Derive an expression for the capacitance C of the capacitor with the material inserted in terms of $L, R_{1}, R_{2}, \kappa$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.