The electric field at a distance $R$ from the surface of a uniformly charged nonconducting sphere of radius $R$ is $20 \,V/m$. The electric field at a distance $\frac{R}{2}$ from the centre of the sphere is $.... \,V/m$.

The charge density of a solid sphere of radius $R$ and total charge $Q$ is given by $\rho(r) = \frac{Q}{\pi R^4} r$. The value of the electric field at a point $P$ inside the sphere at a distance $r_1$ from the center is .......

Obtain the expression for the electric field due to a uniformly charged spherical shell at a point outside it.

Consider a thin spherical shell of radius $R$ with its centre at the origin,carrying uniform positive surface charge density. The variation of the magnitude of the electric field $|\vec{E}(r)|$ and the electric potential $V(r)$ with the distance $r$ from the centre,is best represented by which graph?

$A$ 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:

Three infinite plane sheets carrying uniform charge densities $-\sigma, 2 \sigma, 4 \sigma$ are placed parallel to the $XZ$ plane at $Y=a, 3a, 4a$ respectively. The electric field at the point $(0, 2a, 0)$ is