As shown in the figure,a charged ball $B$ is suspended from a large charged conducting plate by a silk thread $S$ making an angle $\theta$ with the plate. The surface charge density $\sigma$ of the plate is proportional to ........

The electric field due to an infinitely long thin straight wire with uniform linear charge density of $2.5 \times 10^{-7} \ Cm^{-1}$ at a radial distance of $x$ from the wire is $7.5 \times 10^4 \ NC^{-1}$. Then $x=$ (in $cm$)

The electric field in a region is radially outward and at a point is given by $E = 250 r \, V/m$ (where $r$ is the distance of the point from the origin). Calculate the charge contained in a sphere of radius $20 \, cm$ centered at the origin in Coulombs $(C)$.

$A$ 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:

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 $\frac{23\rho R}{16K\varepsilon_0}$. The value of $K$ is

Two concentric conducting thin spherical shells $A$ and $B$ having radii $r_A$ and $r_B$ $(r_B > r_A)$ are charged to $Q_A$ and $-Q_B$ $(|Q_B| > |Q_A|)$. The electric field along a line passing through the centre is: