$A$ sphere of radius $R$ has a uniform distribution of electric charge in its volume. At a distance $x$ from its centre,for $x < R$,the electric field is directly proportional to

Find the electric field due to an infinitely long charged cylinder of radius $R$ having a linear charge density $\lambda$ at a distance of $R/2$ from its axis.

$A$ sphere of radius $R$ has a volume charge density $\rho = k r$, where $r$ is the distance from the center of the sphere and $k$ is a constant. The magnitude of the electric field at the surface of the sphere is given by ($\varepsilon_{0} =$ permittivity of free space):

$A$ non-conducting solid sphere has radius $R$ and uniform charge density. $A$ spherical cavity of radius $\frac{R}{4}$ is hollowed out of the sphere. The distance between the center of the sphere and the center of the cavity is $\frac{R}{2}$. If the charge of the sphere is $Q$ after the creation of the cavity and the magnitude of the electric field at the center of the cavity is $E = K \left( \frac{Q}{4 \pi \epsilon_0 R^2} \right)$,determine the approximate value of $K$.

The electric field at a distance $r$ from the centre in the space between two concentric metallic spherical shells of radii $r_1$ and $r_2$ carrying charges $Q_1$ and $Q_2$ respectively is $(r_1 < r < r_2)$.

In the figure,a very large plane sheet of positive charge is shown. $P_{1}$ and $P_{2}$ are two points at distances $l$ and $2l$ from the charge distribution. If $\sigma$ is the surface charge density,then the magnitude of electric fields $E_{1}$ and $E_{2}$ at $P_{1}$ and $P_{2}$ respectively are: