A solid sphere of radius $R$ has a charge $Q$ distributed in its volume with a charge density $\rho =k{r}^a,$ where $k$ and $a$ constants and $r$ is the distance from its centre. If the electric field at $\frac{R}{2}$ is $\frac{1}{8}$ times that at $r=R,$ find the value of $a.$

A spherical portion has been removed from a solid sphere having a charge distributed uniformly in its volume as shown in the Figure (Fig. 3.82). The electric field inside the emptied sphere is

Three infinitely long charge sheets and placed as shown in figure. Fig. 3.83 the electric field at point $P$ is

A charged ball $B$ hangs from a silk thread $S$ white makes an angle $\theta ,$ with a large charged conducting shut $P,$ as shown in figure, Fig. 3.84. The surface charge density $\sigma$ of the sheet is proportional to