Consider a cylinder-like solid whose base is described by a half-annulus of inner radius \(a\) and outer radius \(b\). The solid is composed of two leaky dielectrics, with parameters \(\varepsilon_{r 1}\) and \(\sigma_{1}\) for the region \(0 \leq \varphi<\theta_{0}\) and parameters \(\varepsilon_{r 2}\) and \(\sigma_{2}\) for the region \(\theta_{0}<\varphi \leq \pi\), as shown in figure. Now we coat each flat rectangular face of the solid with a metallic film, apply a steady potential \(V_{0}\) between the two conductors (see figure for polarity), and wait until the system reaches a steady state. We are given the vaccum permittivity \(\varepsilon_{0}\), that \(\varepsilon_{r 1}\) and \(\varepsilon_{r 2}\) are large, and that \(\sigma_{1}\) and \(\sigma_{2}\) are small. Neglect any fringe effects. Take \(V=0\) on the left metallic film.

1  ^12.00 Find the magnitude of the electric field \(E\) and the electric potential \(V\) everywhere within the dielectric.

2  ^4.00 Find the total accumulated charge \(Q\) at \(\varphi=\theta_{0}\).

3  ^12.00 Obtain the resistance and capacitance across the regions \(0 \leq \varphi<\theta_{0}\left(R_{1}, C_{1}\right)\) and \(\theta_{0}<\varphi \leq \pi\) \(\left(R_{2}, C_{2}\right)\).

4  ^8.00 Suppose we disconnect the potential source at time \(t=0\). In order to model the subsequent behaviour, we can design a circuit based on the given parameters of the solid. Draw this circuit. Hence, find the time dependence \(V(t)\) of the potential difference between the metal films on each end.