### Conductors

Let the linear density on the inner plate be $\lambda$. Gauss theorem gives
$E_r\cdot 2\pi rl=4\pi k (\lambda + \pi \rho(r^2-R_2^2))l ^$
so that
$E_r=\frac{2k(\lambda-\pi \rho R_2^2)}{r}+ 2\pi \rho r ^$
The potential will be
$\phi=-\int_{R_1}^r E_r(r') dr'= 2k(\lambda-\pi\rho R_2^)\ln\frac{R_1}{r} -2\pi \rho (r^2-R_1^2)\\ \phi_0=-\int_{R_1}^{R_2} E_r(r') dr'= 2k(\lambda-\pi\rho R_2^)\ln\frac{R_1}{R_2} -2\pi \rho (R_2^2-R_1^2)\\ ^$
Find $\lambda$ from the second equation and substitute into the first.