An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho$. It has a spherical cavity of radius $R/2$ with its centre on the axis of the cylinder,as shown in the figure. The magnitude of the electric field at the point $P$,which is at a distance $2R$ from the axis of the cylinder,is given by the expression $\frac{23 \rho R}{16 k \varepsilon_0}$. The value of $k$ is

The surface charge density of a thin charged disc of radius $R$ is $\sigma$. The value of the electric field at the centre of the disc is $\frac{\sigma}{2\epsilon_0}$. With respect to the field at the centre,the electric field along the axis at a distance $R$ from the centre of the disc:

In the figure,the inner (shaded) region $A$ represents a sphere of radius $r_A=1$,within which the electrostatic charge density varies with the radial distance $r$ from the center as $\rho_A=k r$,where $k$ is positive. In the spherical shell $B$ of outer radius $r_B$,the electrostatic charge density varies as $\rho_B=\frac{2 k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their $SI$ units. Which of the following statement$(s)$ is(are) correct?

The expression for electric field intensity at a point outside a uniformly charged thin plane sheet is (where $d$ is the distance of the point from the plane sheet):

Let $\rho (r) = \frac{Q}{\pi R^4} r$ be the charge density distribution for a solid sphere of radius $R$ and total charge $Q$. For a point $p$ inside the sphere at distance $r_1$ from the centre of the sphere,the magnitude of the electric field is:

Two parallel large thin metal sheets have equal surface charge densities $(\sigma = 26.4 \times 10^{-12} \, C/m^2)$ of opposite signs. The electric field between these sheets is