Let a total charge $2Q$ be distributed in a sphere of radius $R$,with the charge density given by $\rho(r) = kr$,where $r$ is the distance from the centre. Two charges $A$ and $B$,of $-Q$ each,are placed on diametrically opposite points,at equal distance $a$ from the centre. If $A$ and $B$ do not experience any force,then:

Obtain the expression for the electric field due to:
$(i)$ An infinite plane sheet with uniform charge distribution.
$(ii)$ $A$ thin spherical shell with uniform charge distribution at a point outside it.
$(iii)$ $A$ thin spherical shell with uniform charge distribution at a point inside it.

If $q$ is the charge per unit area on the surface of a conductor,then the electric field intensity at a point on the surface is

$A$ conducting sphere of radius $0.1 \ m$ has a uniform charge density $1.8 \ \mu C/m^2$ on its surface. The electric field in free space at a radial distance of $0.2 \ m$ from the center of the sphere is $(\varepsilon_0 = \text{permittivity of free space})$

$A$ uniformly charged disc of radius $R$ having surface charge density $\sigma$ is placed in the $xy$-plane with its center at the origin. Find the electric field intensity along the $z$-axis at a distance $Z$ from the origin.

$A$ uniform rod $AB$ of mass $m$ and length $l$ is hinged at its midpoint $C$. The left half $(AC)$ of the rod has linear charge density $-\lambda$ and the right half $(CB)$ has $+\lambda$,where $\lambda$ is constant. $A$ large non-conducting sheet of uniform surface charge density $\sigma$ is also present near the rod. Initially,the rod is kept perpendicular to the sheet. The end $A$ of the rod is initially at a distance $d$. Now,the rod is rotated by a small angle $\theta$ in the plane of the paper and released. The time period of small angular oscillations is: