Problem $1$ Consider a concentric cylindrical capacitor consisting of two thin cylindrical metallic shells of radius $R_1=R$ and $R_2=1\mathrm{.}01R,$ and length $L$ separated by a homogeneous dielectric with dielectric constant $=5\mathrm{.}0$ as shown in the figure, $LR.$ The inner conductive shell of radius $R_1$ is charged with charges $Q_1=Q,$ the outer one $(R_2)$ with charges $Q_2=-2Q.$ Obtain the magnitude of the displacement vector $D(r)$ as a function of $r,$ the distance from the axis of the capacitor. Then find the values of the quantities $)=(\frac{R_1}{2})=(\frac{R_1+R_2}{2}$ and $)=(2R_2.$ Obtain the magnitude of the electric field $E(r),$ then calculate the values of the quantities $)=(\frac{R_1}{4})=(\frac{R_1+R_2}{2}$ and $)=(3R_2$ in terms of $\frac{Q}{2{}_{0}RL}$ units, respectively.