Electric Field Locus

At first we'll denote the Cillynder-Sphere system as the superposition of the whole cillynder ( $c$ ) with charge density $\rho$ and a sphere ( $s$ ) with charge density $-\rho$ . So, for using Gauss's Law we have: $\oint \vec{E}_{c}\cdot d\vec{A}=\frac{Q_{enclosed}}{\epsilon_{0}}\Rightarrow \vec{E}_{c}(r)=\frac{R^2\rho}{2\epsilon_{0}r}\hat{r}$ $\oint \vec{E}_{s}\cdot d\vec{A}=\frac{Q_{enclosed}}{\epsilon_{0}}\Rightarrow \vec{E}_{s}(r)=-\frac{R^3\rho}{3\epsilon_{0}r^2}\hat{r}$ $\Rightarrow E(r)=\frac{R^2\rho}{\epsilon_{0}}(\frac{1}{2r}-\frac{R}{3r^2})\Rightarrow \max{E(r)}=E(\frac{4}{3}R)=\frac{3R\rho}{16\epsilon_{0}}$ All of this is for $r\geq R$ , but it's easy to find out that, by the same way, the electric field for $r<R$ gets: $E(r)=\frac{r\rho}{6\epsilon_{0}}\Rightarrow \max(E(r))=E(R)=\frac{R\rho}{6\epsilon_{0}}<\frac{3R\rho}{16\epsilon_{0}}$ So the locus on the whole plane in which the electric field is maximum is a circle with center at the origin and radius $\frac{4}{3}R$