The electric field strength depends only on the $x$ and $y$ coordinates, according to the law $\overrightarrow{E}=a\frac{(x\hat{\mathrm{i}}+y\hat{\mathrm{j}})}{\left(x^2+y^2\right)}$, where $a$ is a constant, $\hat{\mathrm{i}}$ and $\hat{\mathrm{j}}$ are the unit vectors of $x$ and $y$ axes. Find the flux of the vector $\overrightarrow{\mathit{E}}$, through a sphere of radius $R$, with its centre at the origin of the coordinates.

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 $r_{\mathrm{m}}$.

Determine the electric field strength vector, if the potential of this field depends on $x$ and $y$ coordinates as,
(a) $\mathrm{\phi }=a\left(x^2-y^2\right);$ (b) $\mathrm{\phi }=axy,$

where $a$ is a constant. Draw the approximate shape of these fields using lines of force. (In the $x, y$ plane)