An infinitely long cylindrical surface of circular cross-section is uniformly charged lengthwise, with the surface density $\sigma ={\sigma }_0\mathrm{cos\phi }$, where $\mathrm{\phi }$ is the polar angle of the cylindrical coordinate system, whose $z$-axis coincides with the axis of the given surface. Find the magnitude and direction of the electric field strength vector on the $z$-axis.

A ball of radius $R$ carries a positive charge whose volume density depends only on a separation $r$ from the ball's centre, as $\rho ={\rho }_0(1-\frac{r}{R}),$ where ${\rho }_0$ is a constant. Assuming the permittivities of the ball and the environment to be equal to unity, find:

$\left(a\right)$ the magnitude of the electric field strength as a function of the distance $r$, both inside and outside the ball;
$\left(b\right)$ the maximum intensity $E_{\max }$ and the corresponding distance $r_{\mathrm{m}}$.