Use Gauss's Law to determine the electric field at$a.$

The "Gaussian Surface" for an infinitely large charged plate is pillbox of surface area A and length $2a$ centred about the origin. The plate has positive surface charge density $\sigma$.

Find the dipole-moment of system

Two molecule each with permanent dipole moment $\overrightarrow{p}$ are aligned as shown. Calculate the force between the molecules if they are separated by a distance that is large compared with the dimension of the dipoles.
[Hint : The relation $(1+x{)}^{-2}\cong 1-2x+3x^2-\dots \dots$ for small $x$ is useful].

Two pith balls each with mass $m$ are suspended from insulating threads. When the pith balls are given equal positive charge $Q$, they hang in equilibrium as shown. We now increase the charge on the left pith ball from $Q$ to $2Q$ while leaving its mass essentially unchanged. Which of the following diagrams best represents the new equilibrium configuration?

For two equal and opposite charges placed at distance $d$, then electric field will be zero at

In which regions, electric field can be zero for given arrangement of an electron and two protons?