When a dipole is placed inside a cavity in a solid conducting body, then

Two conducting plates $A$ and $B$ each having large surface area $S$ (on one side) are placed parallel to each other. The plate $A$ is given a charge $q$, while the plate $B$ is neutral. Then, the electric field at a point in between the plates is:

The figure shows a charge $q$ placed inside a cavity in an uncharged conductor. Now if an external electric field is switched on

A Gaussian surface $S$ encloses two charges $q_1=q$ and $q_2=-q$. The field at $P$ is, 

where ${\overrightarrow{E}}_1$, ${\overrightarrow{E}}_2$ and $\overrightarrow{E_3}$ are the field contributed by $q_1$, $q_2$ and $q_3$ at $P$ respectively.

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